Stability and Derivatives for Fixed Points

#### **Chapter 5**

## On Fixed Point for Derivative of Set-Valued Functions

*Mohamad Muslikh and Adem Kilicman*

#### **Abstract**

In this article, we showed the existence of a fixed point for the derivative of interval-valued functions. The investigation of the existence of such fixed points utilizes the common fixed point concepts for two mappings. Under the condition of compatibility of the hybrid composite mappings in the sense of the Pompei-Hausdorff metric the existence of a fixed point for the derivative is shown. Some examples to support the usability of the result of this study are also given.

**Keywords:** common fixed point theorem, set-valued maps, compatible mappings, differentiable maps, interval-valued functions

#### **1. Introduction**

E. Dyer in [1] conjectured that *f* and *g* must have a common fixed point in 0, 1 ½ � if *fgt* ð Þ¼ ð Þ *gf t* ð Þ ð Þ for each *t*∈ ½ � 0, 1 . In 1967, W.M. Boyce [2] replied in his paper that Dyer's question is negative as well as an answer from Husein [3] and Singh [4]. However, many researchers are curious about conjecture. In 1976, G Jungck [5] shows the existence of the common fixed point for two mappings by the commuting mapping method in general metric spaces. Since then the common fixed point research had quickly grown. In development, some of the researchers not only involved two mappings (single-valued mappings) but also they are more than it is [6]. In fact, some involve the set-valued mapping forms [7, 8].

In progress, the composition mappings are discussed not only between fellow of single-valued mappings or set-valued mappings but also its combination (mixed compositions between of single-valued and set-valued mappings). Since then several authors have studied common fixed point theorems for such mapping in different ways ([9–11] and references therein).

Itoh et al. [12] introduce "commute" term of hybrid composite functions in 1977. By this properties, they have proven common fixed point theorems in topological vector spaces. In 1982, Fisher [13] has introduced common fixed point theorems for commuting mappings in the sense of the other in metric spaces. Then Imdad [14] mentioned the properties *fFx*⊆*Ffx* as "quasi-commute" to distinguish with the latter term. Whereas two commuting mappings *F* and *f* are

weakly commuting, but in general two weakly commuting mappings do not commute as it is shown in Example 1 of [15].

In 1989, Kaneko [16] introduced the concept of "compatible" by using the Hausdorff metric and proved the existence of a common fixed point theorem by the concept. In 1993, Jungck [17] introduced the same things but used the concept of "*δ*-compatible" mappings in metric spaces and proved some common fixed point theorems for *δ*-compatible mappings.

Regarding the fixed point for derivatives has been observed by M. Elekes at all in [18]. In his paper, he shows the compositions of two functions derivatives have fixed points. This result is an affirmative answer to a question of K. Ciesielski, whether the composition of two derivatives on interval closed has a fixed point? The fixed point for a function is usual but for its derivatives is another something. Here have we the quadruplets *X*, *x*, *f*, *f* <sup>0</sup> . How do these problems? By the device of commutativity and compatibility between the function and its derivatives, the author shows that the function derivatives of the real-valued function have a fixed point [19].

Motivated by the results mentioned above, in this article, we introduced the existence theorem of a fixed point for *gh*-derivative of the interval-valued function. To this work, we used hybrid composite mappings involving *gh*-derivative under the compatibility conditions.

#### **2. gh-Differences**

Suppose ð Þ *X*, *d* is a metric spaces. The collection of all non-empty subsets of *X* is denoted by P0ð Þ *X* . Whereas, the notation Bð Þ *X* (resp. CBð Þ *X* , Kð Þ *X* and KCð Þ *X* ) is the collection of all non-empty bounded (resp. closed-bounded, compact and compactconvex) subsets of *X*.

In 1905, In his PhD thesis [20], Pompeiu defined the notions of *e*<sup>0</sup> *cart* between two sets. Hausdorff [21] studies the notion of set distance in the natural setting of metric spaces and with a small modification (the the sum is replaced by the maximum).

Let ð Þ *X*, *d* be a metric spaces and *A*,*B*⊂ *X*. The Hausdorff distance between *A* and *B* is a distance function *H* : P0ð Þ� *X* P0ð Þ! *X* <sup>þ</sup> which is defined as

$$H(A,B) = \sup\{d(A,B), d(B,A)\},\tag{1}$$

where *d A*ð Þ¼ , *B* sup*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>d a*ð Þ , *B* . Certainly value that *d A*ð Þ , *B* 6¼ *d B*ð Þ , *A* . The distance functions *H* to be a metric on the collection of all non-empty closed-bounded subset of *X*, CBð Þ *X* . The metric spaces ð Þ CBð Þ *X* , *H* is called a complete metric spaces if the metric space *X* is a complete.

Suppose Ið Þ¼ *I* ¼ *a*�, *a*<sup>þ</sup> ½ �j*a*�, *a*<sup>þ</sup> ∈ , *a*� <*a*<sup>þ</sup> f g. In [22], R.E. Moore et al. introduced an absolute value of the interval *J* ¼ *x*�, *x*<sup>þ</sup> ½ � asfollows.

$$\|\|f\|\| = \max\{|\mathbf{x}^-|, |\mathbf{x}^+|\}. \tag{2}$$

For a given interval *I* ¼ *a*�, *a*<sup>þ</sup> ½ � define the width, midpoint and radius of *I*, respectively, by

$$\mathfrak{w}(I) = a^+ - a^- \mathfrak{w}(I) = \frac{1}{2}(a^- + a^+) \mathfrak{z} and \quad r(I) = \frac{1}{2}(a^+ - a^-) \ge 0,\tag{3}$$

so that *a*� ¼ *m I*ð Þ� *r I*ð Þ and *a*<sup>þ</sup> ¼ *m I*ð Þþ *r I*ð Þ. Thus the interval notasion *I* ¼ *a*�, *a*<sup>þ</sup> ½ � can be written as the pair *I* ¼ ð Þ *m I*ð Þ;*r I*ð Þ *:*.

The Pompeiu-Hausdorff distance on Ið Þ defined as

$$H(I, f) = \max\left\{|a^- - b^-|, |a^+ - b^+|\right\},\tag{4}$$

where *<sup>I</sup>* <sup>¼</sup> *<sup>a</sup>*�, *<sup>a</sup>*<sup>þ</sup> ½ � and *<sup>J</sup>* <sup>¼</sup> *<sup>b</sup>*�, *<sup>b</sup>*<sup>þ</sup> . The pair ð Þ <sup>I</sup>ð Þ , *<sup>H</sup>* is a complete and separable metric space.

In 1967, M. Hukuhara [23] introduced the difference (*h*-difference) between *U* and *V* defined as *U* � *<sup>h</sup> <sup>V</sup>* <sup>¼</sup> *<sup>W</sup>* if and only if *<sup>U</sup>* <sup>¼</sup> *<sup>V</sup>* <sup>þ</sup> *<sup>W</sup>* for each *<sup>U</sup>*,*V*,*<sup>W</sup>* <sup>∈</sup> KC *<sup>k</sup>* . An important properties of the Hukuhara difference is that *U* � *<sup>h</sup> <sup>U</sup>* <sup>¼</sup> f g <sup>Θ</sup> and ð Þ� *<sup>U</sup>* <sup>þ</sup> *<sup>V</sup> <sup>h</sup> <sup>V</sup>* <sup>¼</sup> *<sup>U</sup>*. The Hukuhara difference is unique, but it does not always exists.

The Hukuhara difference had generalized by Markov in [24]. He defined is following as

$$U^{\underline{gh}} \ V = W \Leftrightarrow (\mathbf{a})U = V + W \quad \text{or} \quad (\mathbf{b})V = U + (-1)W. \tag{5}$$

Furthermore, Hukuhara difference generalized is called the *gh*-difference.

Both the equation *U* ¼ *V* þ *W* and the equation *V* ¼ *U* þ �ð Þ1 *W* can simultaneously holds. It is clear that *h*-difference is part of *gh*-difference. Therefore, the *gh*difference is often said to be a generalization of the *h*-difference. The *gh*-difference of two intervals in Ið Þ always exists.

Proposition 1. Suppose *<sup>I</sup>* <sup>¼</sup> *<sup>a</sup>*�, *<sup>a</sup>*<sup>þ</sup> ½ � and *<sup>J</sup>* <sup>¼</sup> *<sup>b</sup>*�, *<sup>b</sup>*<sup>þ</sup> are intervals in <sup>I</sup>ð Þ . The *gh*difference of two intervals *I* and *J* always exists and

$$I^{\underline{gh}}J = [a^-, a^+]^{\underline{gh}} \left[b^-, b^+\right] = [c^-, c^+] \tag{6}$$

where *<sup>c</sup>*� <sup>¼</sup> min *<sup>a</sup>*� � *<sup>b</sup>*� ð Þ, *<sup>a</sup>*<sup>þ</sup> � *<sup>b</sup>*<sup>þ</sup> and *<sup>c</sup>*<sup>þ</sup> <sup>¼</sup> max *<sup>a</sup>*� � *<sup>b</sup>*� ð Þ, *<sup>a</sup>*<sup>þ</sup> � *<sup>b</sup>*<sup>þ</sup> .

In [25, 26] defined *H I*ð Þ¼k , *J I* � *gh <sup>J</sup>*<sup>k</sup> for each *<sup>I</sup>*,*<sup>J</sup>* <sup>∈</sup>*I*ð Þ . An immediate property of the *gh*-difference for *I*,*J* ∈*I*ð ÞÞ is

$$H(I, I) = \mathbf{0} \Leftrightarrow I \xleftarrow{gh} I = \mathbf{0} \Leftrightarrow I = I \tag{7}$$

It is also well known that ð Þ *I*ð Þ , *H* is complete metric space.

#### **2.1 gh-Derivative of set-valued functions**

The mapping *F* : *X* ! P0ð Þ *Y* is called *set-valued functions* where the maps *F x*ð Þ∈P0ð Þ *Y* for each *x*∈ *X*. The function *f* : *X* ! *Y* is said to be *selection* of *F* if *f x*ð Þ∈*F x*ð Þ for all *x*∈*X*. We say that a point *z*∈*X* is a fixed point of *F* if *z*∈ *F z*ð Þ.

The *gh*-derivative for an interval-valued function, expressed in terms of the difference quotient by *gh*-difference, has been first introduced in 1979 by S. Markov. A very recent and complete description of the algebraic properties of *gh*-derivative can be found in [27].

Definition 1 Let *F* : ½ �! *a*, *b* Ið Þ be an interval-valued function and suppose *t*0,*t*<sup>0</sup> þ *h*∈ð Þ *a*, *b* . The *gh*-derivative *F*<sup>0</sup> *gh*ð Þ *t*<sup>0</sup> ∈Ið Þ defined as

$$F'\_{g^h}(t\_0) = \lim\_{h \to 0} \frac{F(t\_0 + h)^{\frac{gh}{h}} F(t\_0)}{h}. \tag{8}$$

If the limit, lim *<sup>h</sup>*!<sup>0</sup> *F t*ð Þ� <sup>0</sup>þ*<sup>h</sup> ghF t*ð Þ <sup>0</sup> *<sup>h</sup>* exists and satisfies Eq. (6), then *F* is said differentiable in the sense of generalized Hukuhara difference or *gh*-differentiable at a point *t*<sup>0</sup> ∈ð Þ *a*, *b* . The set-valued *F*<sup>0</sup> *gh* is called a generalized Hukuhara derivative.

Theorem 1.1 If interval-valued functions *F* : ½ �! *a*, *b* Ið Þ is a *gh*-differentiable at a point *p*∈ð Þ *a*, *b* then *F* is continuous at *p*.

**Proof:**

$$\begin{aligned} \lim\_{\boldsymbol{x} \to \boldsymbol{p}} F(\boldsymbol{x}) \stackrel{\mathcal{g}h}{=} F(\boldsymbol{p}) &= \lim\_{\boldsymbol{x} \to \boldsymbol{p}} \left[ \frac{F(\boldsymbol{x}) \stackrel{\mathcal{g}h}{=} F(\boldsymbol{p})}{(\boldsymbol{x} - \boldsymbol{p})} (\boldsymbol{x} - \boldsymbol{p}) \right] \\ &= \left[ \lim\_{\boldsymbol{x} \to \boldsymbol{p}} \frac{F(\boldsymbol{x}) \stackrel{\mathcal{g}h}{=} F(\boldsymbol{p})}{(\boldsymbol{x} - \boldsymbol{p})} \right] \left[ \lim\_{\boldsymbol{x} \to \boldsymbol{p}} (\boldsymbol{x} - \boldsymbol{p}) \right] \\ &= F'\_{\mathcal{g}^h}(\boldsymbol{p}) \cdot \mathbf{0} = \mathbf{0}. \end{aligned}$$

So *F* is continuous at the point *p*∈½ � *a*, *b* .

Theorem 1.2 [26] Let *F* : ½ �! *a*, *b* Ið Þ be an interval-valued functions and *F x*ð Þ¼ ½ � *f x*ð Þ, *g x*ð Þ , where *f*,*g* : ½ �! *a*, *b* . *F* is *gh*-differentiable on ð*a*,*b*) if and only if *f* and *g* are differentiable on ð Þ *a*, *b* and

$$F\_{\mathfrak{g}^\hbar}'(\mathfrak{x}) = \left[ \min \left\{ f'(\mathfrak{x}), \ \mathfrak{g}'(\mathfrak{x}) \right\}, \ \max \left\{ f'(\mathfrak{x}), \ \mathfrak{g}'(\mathfrak{x}) \right\} \right],$$

for all *x*∈ð Þ *a*, *b* . This means that

$$F\_{gh}'(\mathfrak{x}) = \begin{cases} \left[f'(\mathfrak{x}), \ g'(\mathfrak{x})\right] & \text{if} \, f'(\mathfrak{x}) < \mathfrak{g}'(\mathfrak{x}),\\ \left[\mathfrak{g}'(\mathfrak{x}), \ f'(\mathfrak{x})\right] & \text{if} \, f'(\mathfrak{x}) < \mathfrak{g}'(\mathfrak{x}) \end{cases}$$

for all *x*∈ð Þ *a*, *b* .

#### **3. Common fixed point**

Definition 2 Suppose ð Þ *X*, *d* is a metric space, *E* ⊂*X*, *F* : *E* ! Bð Þ *X* is a set-valued mapping and *f* : *E* ! *X* is single-valued mapping.


Suppose ð Þ *X*, *d* is a metric space. The mapping *f*,*g* : *X* ! *X* is a single-valued function or function and *F*,*G* : *X* ! Bð Þ *X* is a set-valued function. For each *x*,*y*∈*X*, we used the notation as follows.

$$\mathcal{M}(\mathbf{F}, f) = \max\left\{ d(\mathbf{fx}, \text{ Fx}), d(\mathbf{fy}, \text{ Fy}), d(\mathbf{fx}, \text{ Fy}), d(\mathbf{fy}, \text{ Fx}), d(\mathbf{fx}, \text{ fy}) \right\}. \tag{9}$$

and

$$\mathcal{N}(\mathbf{F}, f) = \max\left\{ d(\mathbf{fx}, f\mathbf{j}), d(\mathbf{fx}, \mathbf{Fx}), d(\mathbf{fy}, \mathbf{Fy}), \frac{1}{2} [d(\mathbf{fx}, \mathbf{Fy}) + d(\mathbf{fy}, \mathbf{Fx})] \right\}. \tag{10}$$

and

$$\mathcal{M}(\mathbf{F}, \mathbf{G}, \mathbf{f}, \mathbf{g}) = \max\left\{ d(\mathbf{fx}, \mathbf{g}y), \delta(\mathbf{fx}, \mathbf{G}y), \delta(\mathbf{g}y, \mathbf{Fx}) \right\}.\tag{11}$$

The following is the existence of common fixed point theorem that result by B. Fisher [13].

Theorem 1.3 Suppose ð Þ *X*, *d* is a complete metric space, *F* : *X* ! Bð Þ *X* is a setvalued mapping and *f* : *X* ! *X* is a single-valued mapping satisfying the inequality

$$\delta(F\mathfrak{x}, F\mathfrak{y}) \le c\mathcal{M}(F, f) \tag{12}$$

for all *x*,*y*∈*X*, where 0 ≤*c*< 1. If.

A.*f* is continuous,

B. *F X*ð Þ ⊆*f X*ð Þ, and

C. *F* and *f* are commute,

then *F* and *f* have a unique common fixed point.

B. Fisher also shown the same with assumes the continuity of *F* in *X* instead of the continuity of *f* [28].

The following theorem is generalization of Theorem 1.3 that has been resulted by M Imdad et al. [14].

Theorem 1.4 Suppose ð Þ *X*, *d* is a complete metric space, *F* : *X* ! Bð Þ *X* is a setvalued mapping and *f* : *X* ! *X* is a single-valued mapping satisfying the inequality

$$
\delta(F\mathfrak{x}, F\mathfrak{y}) \le \psi \mathcal{M}(F, f).
$$

for all *x*,*y*∈*X*, where *ψ* : ½ Þ! 0, ∞ ½ Þ 0, ∞ is a nondecreasing, right continuous and *ψ*ð Þ*t* <*t*, for all *t*>0. If this following is satisfied

A. the function *f* is continuous,

B. the image of *F X*ð Þ is a subset of *f X*ð Þ,

C. the set-valued *F* and single-valued *f* are weakly commute, and

D.∃ *x*<sup>0</sup> ∈ *X* such that supf g *δ*ð Þ *Fxn*, *Fx*<sup>1</sup> : *n* ¼ 0, 1⋯ < þ ∞,

then *F* and *f* have a unique common fixed point on *X*.

Theorem 1.5 Suppose ð Þ *X*, *d* is a complete metric space, *F* : *X* ! Bð Þ *X* is a setvalued mapping and *f* : *X* ! *X* is a single-valued mapping satisfying the inequality

$$
\delta(F\mathfrak{x}, F\mathfrak{y}) \le \mathfrak{y} \, \mathcal{M}(F, f) \tag{13}
$$

for all *x*,*y*∈*X*, where *ψ* : ½ Þ! 0, ∞ ½ Þ 0, ∞ is a non-decreasing, right continuous and *ψ*ð Þ*t* <*t*, for all *t*>0. If this following is satisfied

A. the set-valued mapping *F* or the single-valued mapping *f* are continuous,

B. the image *F X*ð Þ is a subset of the image *f X*ð Þ,

C. the set-valued *F* and the singel valued *f* are slightly commute, and

D.∃ *x*<sup>0</sup> ∈ *X* such that supf g *δ*ð Þ *Fxn*, *Fx*<sup>1</sup> : *n* ¼ 0, 1⋯ < þ ∞,

then *F* and *f* have a unique common fixed point on *X*.

In the other context, Kaneko and Sessa in [16] introduce the "compatibility" term for the set-valued mapping *F* and the single-valued mapping *f* defined as follows:

Definition 3 Let ð Þ *X*, *d* be a metric spaces. Suppose that *F* : *X* ! CBð Þ *X* is a setvalued mapping and *f* : *X* ! *X* is a single-valued mapping. The mappings *F* and *f* is called **compatible** if the composition *fFx*∈ CBð Þ *X* and the sequence *H Ffxn*, *fFxn* ! 0 whenever f g *xn* is sequence in *X* such that *fxn* ! *t* ∈*B*∈ CBð Þ *X* and *Fxn* ! *B*∈ CBð Þ *X* .

By using such the notion obtained the following theorem and lemma [16].

Theorem 1.6 Suppose ð Þ *X*, *d* is a complete metric space, *F* : *X* ! CBð Þ *X* is a setvalued mapping, and *f* : *X* ! *X* is a single-valued mapping satisfying the inequality

$$H(F\mathbb{X}, F\mathbb{Y}) \le c\mathcal{N}(F, f) \tag{14}$$

for all *x*,*y*∈*X*, where 0 ≤*c*< 1. If this following is satisfied.

A. the set-valued mapping *F* and the single-valued mapping *f* are continuous,

B. the image *F X*ð Þ is a subset of the image *f X*ð Þ, and

C. the set-valued *F* and the single-valued *f* are compatible,

then there exists a point *z*∈ *X* such that *f z*ð Þ∈*F z*ð Þ.

Lemma 1 Let ð Þ *X*, *d* be a metric spaces. Suppose that *F* : *X* ! CBð Þ *X* and *f* : *X* ! *X* are a compatible. If *fw* ∈*Fw* for some *w* ∈*X*, then *Ffw* ¼ *fFw*.

#### **4. Fixed point for derivative**

In this discussion, we shall make frequent use of the following Lemmas.

Lemma 2 [29] Let ð Þ *X*, *d* be a metric spaces. If *C*,*D* ∈ Kð Þ *X* and *c*∈*C*, then there exists the points *d*∈ *D* such that *d c*ð Þ , *d* ≤ *H C*ð Þ , *D* .

Lemma 3 (Lemma 1 [4]) Let *ψ* : ½ Þ! 0, ∞ ½ Þ 0, ∞ be a real function such that nondecreasing, right continuous on 0, ½ Þ ∞ .

$$\lim\_{n \to \infty} \wp''(t) = \mathbf{0}.$$

if and only if for every *t*> 0 and *ψ*ð Þ*t* <*t*.

In this result, we found that Lemma 1 also conversely holds provided its values of mapping are compact sets.

Lemma 4 Let ð Þ *X*, *d* be a metric spaces and the set-valued *F* : *X* ! Kð Þ *X* is a continuous on *X*. If there exists single-valued *f* : *X* ! *X* is continuous on *X* such that *fw* ∈*Fw* for some *w* ∈*X*, then the mappings *F* and *f* are compatible.

**Proof:** Since *Fx* ∈ Kð Þ *X* for each *x*∈ *X* and *f* is continuous, the composition *fFx*∈ Kð Þ *X* for all *x*∈ *X*. Suppose that the sequence f g *xn* on *X* such that the sequence of sets *Fxn* converges to *K* ∈ Kð Þ *X* and the sequence function *fxn* converges to *z*∈*K*. In this case, we choose *z*∈*X* such that *fz*∈*Fz*. Since *F* and *f* are continuous, we obtained

$$\lim\_{n \to \infty} H(F \text{fx}\_n, f \text{Fx}\_n) \le \lim\_{n \to \infty} \left[ H(F \text{fx}\_n, F \text{z}) + H(F \text{z}, \text{ } \{\text{fz}\}) + H(\{\text{fz}\}, f \text{Fx}\_n) \right],$$

$$= H(F \text{z}, F \text{z}) + H(F \text{z}, \{\text{fz}\}) + H(\{\text{fz}\}, f \text{K})$$

$$= \mathbf{0}.$$

The pairs *F* and *f* are proved as compatible by Definition 3.

By using the Lemma 4 we obtain the theorem as follows:

Theorem 1.7 Let ð Þ *X*, *d* be a complete metric space, *F* : *X* ! Kð Þ *X* be a continuous. Suppose there exists *f* : *X* ! *X* is continuous on *X* such that *F X*ð Þ⊆ *f X*ð Þ and for all *x*,*y*∈*X* satisfying the inequality

$$H(F\mathbf{x}, F\mathbf{y}) \le c\mathcal{N}(F, f),\tag{15}$$

where 0 ≤*c*<1. Then *fz*∈ *Fz* for some *z*∈*X* if and only if the pairs *F* and *f* are compatible.

**Proof:** Let *x*<sup>0</sup> ∈*X* be an arbitrary. Since *F X*ð Þ⊆ *f X*ð Þ, we choose the point *x*<sup>1</sup> ∈*X* such that *fx*<sup>1</sup> ∈*Fx*0. If *c* ¼ 0, then

$$d\left(f\mathbb{x}\_1, F\mathbb{x}\_1\right) \le H(F\mathbb{x}\_0, F\mathbb{x}\_1) = \mathbf{0}.$$

Since *Fx*<sup>1</sup> is compact (hence closed), we obtain *fx*<sup>1</sup> ∈*Fx*1.

Now we assume *<sup>c</sup>* 6¼ 0. By Lemma 2 for each *<sup>ε</sup>* <sup>¼</sup> <sup>1</sup>ffi *<sup>c</sup>* <sup>p</sup> >1 there exists a point *y*<sup>1</sup> ∈ *Fx*<sup>1</sup> such that

$$d\left(\mathcal{Y}\_1, F\mathbf{x}\_1\right) \le H(F\mathbf{x}\_1, F\mathbf{x}\_0) < eH(F\mathbf{x}\_1, F\mathbf{x}\_0).$$

Choose *x*<sup>2</sup> ∈*X* such that *y*<sup>1</sup> ¼ *fx*<sup>2</sup> ∈*Fx*<sup>1</sup> and so on. In general, if *xn* ∈*X* there exists *xn*þ<sup>1</sup> <sup>∈</sup> *<sup>X</sup>* such that *yn* <sup>¼</sup> *fxn*þ<sup>1</sup> <sup>∈</sup>*Fxn* and

$$d\left(\mathbf{y}\_n, f\mathbf{x}\_n\right) < \epsilon H(F\mathbf{x}\_n, F\mathbf{x}\_{n-1})$$

for each *n*≥1. By the inequality (10) for each *n* ∈ℕ we have

*d fxn*þ1, *fxn* � � <sup>&</sup>lt; *<sup>ε</sup>H Fx* ð Þ *<sup>n</sup>*, *Fxn*�<sup>1</sup> <sup>≤</sup> *c* ffiffi *<sup>c</sup>* <sup>p</sup> <sup>N</sup> ð Þ¼ *<sup>F</sup>*, *<sup>f</sup>* ffiffi *<sup>c</sup>* <sup>p</sup> <sup>N</sup> ð Þ *<sup>F</sup>*, *<sup>f</sup>* < ffiffi *<sup>c</sup>* <sup>p</sup> max <sup>f</sup>*d fxn*, *fxn*�<sup>1</sup> � �,*d fxn*, *Fxn* � �,*d fxn*�1, *Fxn*�<sup>1</sup> � �, 1 <sup>2</sup> *d fxn*, *Fxn*�<sup>1</sup> � � <sup>þ</sup> *d fxn*�1, *Fxn* � � � � <sup>g</sup>*:* < ffiffi *<sup>c</sup>* <sup>p</sup> max <sup>f</sup>*d fxn*, *fxn*�<sup>1</sup> � �,*d fxn*, *fxn*þ<sup>1</sup> � �,*d fxn*�1, *fxn* � �, 1 2 *d fxn*�1, *fxn*þ<sup>1</sup> � ��g*:* < ffiffi *<sup>c</sup>* <sup>p</sup> max <sup>f</sup>*d fxn*, *fxn*�<sup>1</sup> � �,*d fxn*, *fxn*þ<sup>1</sup> � �,*d fxn*�1, *fxn* � �, 1 <sup>2</sup> *d fxn*�<sup>1</sup>, *fxn* � � <sup>þ</sup> *d fxn*, *fxn*þ<sup>1</sup> � � � � <sup>g</sup>*:* <sup>¼</sup> ffiffi *<sup>c</sup>* <sup>p</sup> max *d fxn*�<sup>1</sup>, *fxn* � �, *d fxn*, *fxn*þ<sup>1</sup> � � � � *:* <sup>¼</sup> ffiffi *<sup>c</sup>* <sup>p</sup> *d fxn*�<sup>1</sup>, *fxn* � �

Since ffiffi *<sup>c</sup>* <sup>p</sup> <sup>&</sup>lt;1, the sequence *fxn* � � is a Cauchy sequence on the complete metric space *X*. Therefore, it converges to a point *z*∈*X*. Likewise f g *Fxn* is a Cauchy sequence on the complete metric space (Kð Þ *X* ,*H*), hence it converges to a set *K* ∈ Kð Þ *X* . As a result

$$d(z, K) \le d\left(z, f\mathfrak{x}\_n\right) + d\left(f\mathfrak{x}\_n, K\right) \le d\left(z, f\mathfrak{x}\_n\right) + H(F\mathfrak{x}\_{n-1}, K).$$

Certainly that *d z*ð Þ¼ , *K* 0 by *d z*, *fxn* � � ! 0 and *H Fx* ð Þ! *<sup>n</sup>*�1, *<sup>K</sup>* 0 as *<sup>n</sup>* ! <sup>∞</sup>. This implies *z*∈*K* since *K* is a compact set. Since *F* and *f* are compatible, we have

$$\begin{split}d(\text{f}\mathbf{z},\text{Fz}) &= \lim\_{n \to \infty} d(\text{fz},\text{Fz}) \leq \lim\_{n \to \infty} \left[ d\left(\text{fz},\text{ffx}\_{n}\right) + d\left(\text{fx}\_{n},\text{Fz}\right) \right] \\ &\leq \lim\_{n \to \infty} \left[ d\left(\text{fz},\text{ffx}\_{n}\right) + H\left(\text{fFx}\_{n},\text{Fz}\right) \right] \\ &\leq \lim\_{n \to \infty} \left[ d\left(\text{fz},\text{ffx}\_{n}\right) + H\left(\text{fFx}\_{n},\text{Ffx}\_{n}\right) + H\left(\text{fFx}\_{n},\text{Fz}\right) \right] \\ &= d(\text{fz},\text{fz}) + H(\text{Fz},\text{Fz}) \\ &= 0. \end{split}$$

So *fz*∈*Fz*. Conversely, it's clear by Lemma.

Remark 1 Theorem 1.7 is a special occurrence of results obtained by H Kaneko and S Sessa [16]. Certainly the provisioning should be satisfied as in Theorem 3.

This result modify of Theorem 1.5 by substituting compatibility with respect to Hausdorff metric on Kð Þ *X* for slight commutativity at once improvement Theorem 1.6 in finding common fixed point for the mapping of the hybrid composite.

Theorem 1.8 Let ð Þ *X*, *d* be a complete metric space, *F* : *X* ! Kð Þ *X* be a set-valued mapping and *f* : *X* ! *X* be a single-valued mapping satisfying the inequality

$$H(F\mathbf{x}, F\mathbf{y}) \le \varphi \mathcal{N}(F, f) \tag{16}$$

for all *x*,*y*∈*X*, where *ψ* : ½ Þ! 0, ∞ ½ Þ 0, ∞ is a nondecreasing, right continuous and *ψ*ð Þ*t* <*t*, for all *t*>0. If this following is satisfied

A. the set-valued mapping *F* and the single-valued mapping *f* are continuous,

B. the image *F X*ð Þ is a subset of the image *f X*ð Þ,

C. the pairs *F* and *f* are compatible, and

D.∃ *x*<sup>0</sup> ∈ *X* such that supf g *H Fx* ð Þ *<sup>n</sup>*, *Fx*<sup>1</sup> : *n* ¼ 0, 1⋯ < þ ∞,

then *F* and *f* have a unique common fixed point on *X*.

**Proof:** This proof is the same as Theorem 5 in [14]

Example 1 Let *<sup>X</sup>* <sup>¼</sup> ½ � 0, 3 with usual metric. Let *F x*ð Þ¼ 0, *<sup>x</sup>*<sup>2</sup> ½ � and *f x*ð Þ¼ <sup>2</sup>*x*<sup>2</sup> � <sup>1</sup> for each *x*∈ ½ � 0, 3 . Its clear that the image *F X*ð Þ¼ *F*ð Þ¼ ½ � 0, 3 ½ � 0, 9 ⊂*f*ð Þ¼ ½ � 0, 3 ½ �¼ �1, 17 *f X*ð Þ and the both *F* and *f* are continuous on 0, 3 ½ �. If the sequence *xn* ! 1, then *Fxn* ! ½ �¼ 0, 1 *<sup>K</sup>* and *fxn* ! <sup>1</sup>∈*K*. We know that *Ffxn* <sup>¼</sup> 0, 2*x*<sup>2</sup> *<sup>n</sup>* � <sup>1</sup> � �<sup>2</sup> h i and *fFxn* ¼ �1, 2*x*<sup>4</sup> *<sup>n</sup>* � <sup>1</sup> � � so that we obtained

$$H(F\mathfrak{X}\_n, f\mathfrak{X}\_n) = |4\mathfrak{x}\_n^4 - 6\mathfrak{x}\_n^2 + 2| \to \mathbf{0})$$

since *xn* ! 1. It is clear supf*H Fx* ð *<sup>n</sup>*,*Fx*<sup>1</sup> : *n* ¼ 0,1,⋯g ¼ 9< þ ∞. Since *F* and *f* are continuous, we have

$$\lim\_{\mathbf{x}\_n \to \mathbf{1}} F \mathbf{f} \mathbf{x}\_n = F(\mathbf{1}) = [0, \mathbf{1}] = K, \quad \text{and} \quad \lim\_{\mathbf{x}\_n \to \mathbf{1}} f \mathbf{x}\_n = \mathbf{1} = f(\mathbf{1}).$$

This means 1 ¼ *f*ð Þ1 ∈ *F*ð Þ¼ 1 *K*.

Remark 2 Simple examples above prove that the condition of the continuity of the both mappings *F* and *f* is important in Theorem 4 other than the other requirements. However, in general the common fixed point theorems for hybrid composite mappings only required one of the mappings *F* or *f* is continuous. In our opinion, such case it can be used if the set *K* is a singleton.

The following main result is a discussion of the existence of a fixed point for the derivative of an interval-valued function.

Theorem 1.9 Suppose that *F* : ½ �! *a*, *b* Ið Þ is a continuously *gh*-differentiable on ð Þ *a*, *b* such that there exists *f* : ½ �! *a*, *b* and *fx* ∈*F*<sup>0</sup> *gh*ð Þ *x* for all *x*∈ ½ � *a*, *b* satisfying the inequality

$$H(F\mathbf{x}, F\mathbf{y}) \le \mathfrak{y} \mathcal{N}(F, f) \tag{17}$$

for all *x*,*y*∈ ½ � *a*, *b* , where *ψ* : ½ Þ! 0, ∞ ½ Þ 0, ∞ is a non-decreasing, right continuous, and *ψ*ð Þ*t* <*t*, for all *t* >0. If this following is satisfied.

A. the image *F a* ð Þ ½ � , *b* is subsets of the image *f a* ð Þ ½ � , *b* ,

B. the pairs *F* and *f* are compatible, and

C. ∃ *x*<sup>0</sup> ∈ ½ � *a*, *b* such that supf g *H Fx* ð Þ *<sup>n</sup>*, *Fx*<sup>1</sup> : *n* ¼ 0, 1⋯ < þ ∞,

then the *gh*-derivative *F*<sup>0</sup> *gh* has a unique fixed point.

**Proof:** From hypothesis (C), suppose *H Fxs* ð Þ , *Fxt* ≤ *H Fxs* ð Þþ , *Fx*<sup>1</sup> *H Fxt* ð Þ , *Fx*<sup>1</sup> ≤ *M* so that

$$\sup\{H(F\mathbb{x}\_t, F\mathbb{x}\_t) : \mathfrak{s}, t = 0, \mathbb{1}, 2\cdots\} = M < +\infty. \tag{18}$$

Suppose that *N* ∈ℕ such that for each *ε*>0

$$
\psi^N L < \varepsilon \tag{19}
$$

by Lemma 3.

Let *x*<sup>0</sup> ∈½ � *a*, *b* be an arbitrary. Since *F a* ð Þ ½ � , *b* ⊆*f a* ð Þ ½ � , *b* , we choose the point *x*<sup>1</sup> ∈½ � *a*, *b* such that *y*<sup>1</sup> ¼ *fx*<sup>1</sup> ∈*Fx*0. In general, if *xn* ∈*X* there exists *xn*þ<sup>1</sup> ∈ *X* such that *yn* ¼ *fxn* ∈*Fxn*�1. By applying inequality (11) to term *H Fx* ð Þ *<sup>m</sup>*, *Fxn* we have for *m*,*n* ≥ *N*:

$$\begin{split} H(\text{Fix}\_{m}, \text{Fix}\_{n}) &\leq \text{y max}\left\{ d(\text{fix}\_{m}, \text{fix}\_{n}), d(\text{fix}\_{n}, \text{Fix}\_{n}), d(\text{fix}\_{m}, \text{Fix}\_{n}), \\ &\quad \frac{1}{2} [d(\text{fix}\_{n}, \text{fix}\_{m}) + d(\text{fix}\_{m}, \text{fix}\_{n})] \right\} \\ &\leq \text{y max}\left\{ H(\text{Fix}\_{m-1}, \text{fix}\_{n-1}), H(\text{Fix}\_{n-1}, \text{fix}\_{n}), H(\text{fix}\_{m-1}, \text{fix}\_{n}) \right\} \\ &\quad \frac{1}{2} [H(\text{fix}\_{n-1}, \text{fix}\_{m}) + H(\text{Fix}\_{m-1}, \text{fix}\_{n})] \right\} \\ &\leq \eta \max\left\{ H(\text{Fix}\_{m-1}, \text{fix}\_{n-1}), H(\text{Fix}\_{n-1}, \text{fix}\_{n}), H(\text{Fix}\_{m-1}, \text{fix}\_{m}) \right\}, \\ &\quad \frac{1}{2} [H(\text{fix}\_{n-1}, \text{fix}\_{m-1}) + H(\text{Fix}\_{m-1}, \text{fix}\_{n})] \} \\ &\quad \frac{1}{2} [H(\text{fix}\_{m-1}, \text{fix}\_{n-1}) + H(\text{Fix}\_{n-1}, \text{fix}\_{n})] \} \\ &= \eta \max\left\{ H(\text{Fix}\_{m-1}, \text{fix}\_{n-1}), H(\text{Fix}\_{n-1}, \text{fix}\_{n}), H(\text{Fix}\_{m-1}, \text{fix}\_{n}) \right\} \end{split} \tag{20}$$

By iterating (14) above as much as *N* times, we deduce for each *m*,*n*> *N* as follows:

$$\begin{split} H(\text{Fx}\_{m}, \text{Fx}\_{n}) &\leq \psi \max\left\{ H(\text{Fx}\_{r}, \text{Fx}\_{i}), H(\text{Fx}\_{r}, \text{Fx}\_{t}), \\ &H(\text{Fx}\_{i}, \text{Fx}\_{k}): m-1 \leq r; t \leq n; n-1 \leq s; k \leq m \right\} \\ &\leq \psi^{2} \max\left\{ H(\text{Fx}\_{r}, \text{Fx}\_{i}), H(\text{Fx}\_{r}, \text{Fx}\_{t}), \\ &H(\text{Fx}\_{i}, \text{Fx}\_{k}): m-2 \\ &\leq r; t \leq n; n-2 \leq s; k \leq m \right\} \leq \dotsb \\ &\leq \psi^{N} \max\left\{ H(\text{Fx}\_{r}, \text{Fx}\_{k}), H(\text{Fx}\_{r}, \text{Fx}\_{t}), \\ &H(\text{Fx}\_{i}, \text{Fx}\_{k}): \begin{array}{c} 4 & m-N \\ &M-N \\ \leq r; t \leq n; n-N \leq s; k \leq m \end{array} \right\} \leq \psi^{N} M < \epsilon, \end{split}$$

by inequality (13).

Accordingly the sequence f g *Fxn* is a Cauchy sequence on the complete metric spaces ð Þ Ið Þ , *H* so that converges to an interval *J* ∈Ið Þ . The sequence of singlevalued functions *fxn* � � is also a Cauchy sequence on hence it converges to a point *z*∈ . We have

$$|z - J| \le |z - f\mathbf{x}\_n| + |f\mathbf{x}\_n - J| \le |z - f\mathbf{x}\_n| + H(F\mathbf{x}\_{n-1}, J),\tag{22}$$

as *n* ! ∞, ∣*z* � *J*∣ ¼ 0. This means, *z*∈*J* since *J* ∈Ið Þ . By compatibility of *F* and *f*, we obtain

$$\lim\_{n \to \infty} H(F \mathbf{f} \mathbf{x}\_n, f \mathbf{F} \mathbf{x}\_n) = \mathbf{0}.\tag{23}$$

By using inequality (11), we have

*H Ffxn*þ1, *Fxn* <sup>≤</sup>*<sup>ψ</sup>* max <sup>f</sup>*d f* <sup>2</sup> *xn*þ1, *fxn* ,*d f* <sup>2</sup> *xn*þ1, *Ffxn*þ<sup>1</sup> ,*d fxn*, *Fxn* , 1 <sup>2</sup> *d f* <sup>2</sup> *xn*þ1, *Fxn* <sup>þ</sup> *d fxn*, *Ffxn*þ<sup>1</sup> ≤*ψ* max f*d fFxn*, *fxn* ,*d fFxn*, *Ffxn*þ<sup>1</sup> ,*d fxn*, *Fxn* , 1 <sup>2</sup> *d fFxn*, *Fxn* <sup>þ</sup> *d fxn*, *Ffxn*þ<sup>1</sup> ≤*ψ* max f*d fFxn*, *Ffxn* <sup>þ</sup> *d Ffxn*, *Fxn* <sup>þ</sup> *d Fxn*, *fxn* , *d fFxn*, *Ffxn* <sup>þ</sup> *d Ffxn* ,*fxn*Þ þ *d fxn*, *Ffxn*þ<sup>1</sup> , *d fxn*, *Fxn* , 1 <sup>2</sup> *d fFxn*, *Fxn* <sup>þ</sup> *d fxn*, *Ffxn*þ<sup>1</sup> ≤*ψ* max f*d fFxn*, *Ffxn* <sup>þ</sup> *d Ffxn*, *Fxn* <sup>þ</sup> *d Fxn*, *fxn* , *d fFxn*, *Ffxn* <sup>þ</sup> *d Ffxn*, *fxn* <sup>þ</sup> *d fxn*, *Ffxn*þ<sup>1</sup> <sup>g</sup> ≤*ψ* max f*H fFxn*, *Ffxn* <sup>þ</sup> *H Ffxn*, *Fxn* <sup>þ</sup> *d Fxn*, *fxn* , *H fFxn*, *Ffxn* <sup>þ</sup> *d Ffxn*, *fxn* <sup>þ</sup> *d fxn*, *Ffxn*þ<sup>1</sup> <sup>g</sup> ≤*ψ* max f*H fFxn*, *Ffxn* <sup>þ</sup> *H Ffxn*, *Fxn* <sup>þ</sup> *d Fxn*, *fxn* , *H fFxn*, *Ffxn* <sup>þ</sup> *H Ffxn*, *Fxn*�<sup>1</sup> <sup>þ</sup> *H Fxn*�1, *Ffxn*þ<sup>1</sup> <sup>g</sup>

since *f* 2 *xn*þ<sup>1</sup> ∈*fFxn* and *ψ* are non-decreasing. Since the pairs *F* and *f* are compatible, we obtain

$$H(\text{Fz}, f) \le \psi \max\left\{ 0 + H(\text{Fz}, f) + d(f, \text{ z}), \, 0 + 2H(\text{Fz}, f) \right\}.$$

$$\le \psi \max\left\{ H(\text{Fz}, f), \, 2H(\text{Fz}, f) \right\}$$

$$\le 2\psi H(\text{Fz}, f).$$

Since *ψ*ð Þ*t* < *t* for all *t* >0, we have *H Fz* ð Þ¼ , *J* 0. This means *Fz* ¼ *J*. Since the pairs *F* and *f* are compatible and *F* is continuously differentiable on ½ � *a*, *b* (hence continuous), we have

$$\lim\_{n \to \infty} H(Fx, f) = \lim\_{n \to \infty} H(F\mathfrak{X}\_n, f\mathfrak{X}\_n) = \mathbf{0}.\tag{24}$$

So *Fz* ¼ *fJ*. Since *z*∈ *J*, *f z*ð Þ∈*f J*ð Þ, consequently

$$f(z) \in F(z) = f(f) = f. \tag{25}$$

Since *f* ∈ *F*<sup>0</sup> *gh* and *F*<sup>0</sup> *gh* is continuous, the function *f* is continuous. Of course, the sequence *f* 2 *xn* converges to the point *fz* and the sequence of set *fFxn* converges to a set *fJ*. Since the limit

$$\lim\_{n \to \infty} H(\text{ffc}\_n, ff) \le \lim\_{n \to \infty} \left[ H(\text{ffc}\_n, f\text{Fx}\_n) + H(\text{ffc}\_n, ff) \right] = 0,\tag{26}$$

the sequence of set *Ffxn* � � also converges to a set *fJ*. Since *f* 2 *xn*þ<sup>1</sup> ∈*fFxn* and using inequality (11), we get

$$\left|f^{2}\mathbf{x}\_{n+1} - f\mathbf{x}\_{n+1}\right| \leq H\left(f\mathbf{F}\mathbf{x}\_{n}, F\mathbf{x}\_{n}\right) \leq H\left(f\mathbf{F}\mathbf{x}\_{n}, Ff\mathbf{x}\_{n}\right) + H\left(f\mathbf{f}\mathbf{x}\_{n}, F\mathbf{x}\_{n}\right)$$

$$\leq H\left(f\mathbf{F}\mathbf{x}\_{n}, Ff\mathbf{x}\_{n}\right) + \boldsymbol{\nu} \max\left\{\left|f^{2}\mathbf{x}\_{n} - f\mathbf{x}\_{n}\right|, \left|f^{2}\mathbf{x}\_{n} - Ff\mathbf{x}\_{n}\right|, \ldots\right\}$$

$$\left|f\mathbf{x}\_{n} - F\mathbf{x}\_{n}\right|, \frac{1}{2}\left[\left|f^{2}\mathbf{x}\_{n} - F\mathbf{x}\_{n}\right| + \left|f\mathbf{x}\_{n}, Ff\mathbf{x}\_{n}\right|\right]\}.$$

For *n* ! ∞, it allows from hypothesis the part *B* (compatibility) and the Eq. (19) we obtain

$$\begin{aligned} \left| \left| \left| \hat{z} - z \right| \right| &\leq 0 + \left\| \max \left\{ \left| \left| \hat{z} - z \right|, \left| \left| \hat{z} - f \right| \right|, \left| z - f \right|, \frac{1}{2} \left[ \left| \left| \hat{z} - f \right| + \left| z - Fz \right| \right] \right] \right\} \right\} \\\\ &\leq \psi \max \left\{ \left| \left| \hat{z} - z \right|, 0, \left0, \frac{1}{2} \left[ 0 + 0 \right] \right\} \right. \\\\ &\leq \psi \left| \left| \hat{z} - z \right|. \end{aligned} \right. $$

It implies *z* ¼ *fz*. Meaning the point *z* is a fixed point of *f*. This allows *z* ¼ *fz*∈*F z*ð Þ since the Eq. (19) and hence *z* is also a fixed point of *F*<sup>0</sup> *gh* by *z* ¼ *fz*∈*F*<sup>0</sup> *gh*ð Þ*z* .

Let *u* is another common fixed point of *F* and *f*. By inequality (11), we have that

$$\begin{aligned} \mathcal{H}(\mathcal{F}\mathbf{z}, \mathbf{F}u) &\leq \mathsf{y} \max\left\{ |\mathbf{f}\mathbf{z} - f\mathbf{u}|, |\mathbf{f}\mathbf{z} - \mathbf{F}\mathbf{z}|, |\mathbf{f}\mathbf{u} - \mathbf{F}u|, \frac{1}{2} [|\mathbf{f}\mathbf{z} - \mathbf{F}u| + |\mathbf{f}\mathbf{u} - \mathbf{F}\mathbf{z}|] \right\}. \\\\ &\leq \mathsf{y} \max\left\{ H(\mathbf{Fz}, \mathbf{F}u), \ \mathbf{0}, \ \mathbf{0}, \ \frac{1}{2} [H(\mathbf{Fz}, \mathbf{F}u) + H(\mathbf{F}u, \ \mathbf{F}\mathbf{z})] \right\}. \\\\ &= \mathsf{y} \max\{ H(\mathbf{Fz}, \ \mathbf{F}u) \}. \\\\ &= \mathsf{y} H(\mathbf{Fz}, \ \mathbf{F}u). \end{aligned}$$

It implies that *H Fz* ð Þ¼ , *Fu* 0. Since *d z*ð Þ , *u* ≤ *H Fz* ð Þ¼ , *Fu* 0, we have *z* ¼ *u*. Thus the fixed point *z* is unique. This completes the proof.

Remark 3 To get a common fixed point through the hybrid composite mapping usually contains at least two mappings in its hypothesis. This study shows enough one mapping in its hypothesis. In this case, the mapping given must be differentiable (Theorem 1.9). In addition, the continuity of function is not needed explicitly stated in its hypothesis. Thus this result is more simple than the results reached by past researchers.

Example 2 Let *F x*ð Þ¼ *<sup>x</sup>*½ � ð Þ <sup>2</sup> � *<sup>x</sup>* , *<sup>x</sup>* , be an interval-valued function for all *<sup>x</sup>*∈½ � 0, 2 . It is clear *F* is *gh*-differentiable on 0, 2 ð Þ with derivative

$$F'\_{gh}(\boldsymbol{\pi}) = \begin{cases} [(2\boldsymbol{\pi} - \mathbf{1}), \ \boldsymbol{\pi}] & \text{if } \mathbf{0} \le \boldsymbol{\pi} \le \mathbf{1}, \\\\ [\mathbf{x}, \ (2\boldsymbol{\pi} - \mathbf{1})] & \text{if } \mathbf{1} \le \boldsymbol{\pi} \le \mathbf{2}. \end{cases}$$

In this case, we can take the selector *f x*ð Þ¼ ð Þ 2*x* � 1 ∈*Fgh*ð Þ *x* for all *x*∈½ � 0, 2 . We obtain the image

*On Fixed Point for Derivative of Set-Valued Functions DOI: http://dx.doi.org/10.5772/intechopen.107185*

$$F([\mathbf{0}, \ 2]) = \left[ -\frac{\mathbf{1}}{4}, \mathbf{1} \right] \cup [\mathbf{1}, \ 2] = \left[ -\frac{\mathbf{1}}{4}, \mathbf{2} \right] \subset [-\mathbf{1}, \ 2] = f([\mathbf{0}, \ 2]) \dots$$

This means that the condition in Theorem 4 part (A) is satisfied.

If the sequence *xn* ! 1, then *Fxn* ! ½ �¼ 0, 1 *K* and *fxn* ! 1∈*K*. First, we start with the formula *Ffxn* <sup>¼</sup> ð Þ <sup>2</sup>*xn* � <sup>1</sup> <sup>2</sup> � ð Þ <sup>2</sup>*xn* � <sup>1</sup> , 2ð Þ *xn* � <sup>1</sup> h i and *fFxn* <sup>¼</sup> <sup>2</sup> *<sup>x</sup>*<sup>2</sup> *<sup>n</sup>* � *xn* � � � 1, 2ð Þ *xn* � <sup>1</sup> � �, we obtain

$$H(F\text{fx}\_n, f\text{Fx}\_n) = |2\mathbf{x}\_n^2 - 4\mathbf{x}\_n + 2| \to \mathbf{0})$$

since *xn* ! 1. Thus *F* and *f* are compatible. It is clear that supf g *H Fx* ð Þ *<sup>n</sup>*, *Fx*<sup>1</sup> : *n* ¼ 0, 1, ⋯ ¼ 3< þ ∞. Since *F* is continuously *gh*-differentiable on 0, 2 ð Þ, then implies that *F* and *f* are continuous on 0, 2 ð Þ (see Theorem 1.1). Hence we have

$$\lim\_{\mathbf{x}\_n \to \mathbf{1}} F \mathbf{f} \mathbf{x}\_n = F(\mathbf{1}) = [0, 1] = K, \quad \text{and} \quad \lim\_{\mathbf{x}\_n \to \mathbf{1}} f \mathbf{x}\_n = \mathbf{1} = f(\mathbf{1}).$$

Certainly 1 ¼ *f*ð Þ1 ∈*F*ð Þ¼ 1 *K*. Since *f x*ð Þ∈ *F*<sup>0</sup> *gh*ð Þ *x* for all *x*∈½ � 0, 2 , we obtain 1 ¼ *f*ð Þ1 ∈*F*<sup>0</sup> *gh*ð Þ¼ 1 1. Thus the point *z* ¼ 1 is a unique fixed point of *F*<sup>0</sup> *gh*.

Furthermore, if *f* ∈*F*, then we have the following.

Corollary 1 Suppose that *F* : ½ �! *a*, *b* Ið Þ is a continuously *gh*-differentiable on ð Þ *a*, *b* such that there exists *f* : ½ �! *a*, *b* and *fx* ∈*F x*ð Þ for all *x*∈½ � *a*, *b* . If the function *f* and the derivative *F*<sup>0</sup> *gh* satisfies the inequality

$$H\left(F'\_{gh}\mathcal{X}, F'\_{gh}\mathcal{Y}\right) \le \mathcal{Y}\mathcal{N}\left(F'\_{gh}, f\right).$$

for all *x*,*y*∈½ � *a*, *b* , where *ψ* : ½ Þ! 0, ∞ ½ Þ 0, ∞ is a nondecreasing, right continuous, and *ψ*ð Þ*t* <*t*, for all *t* >0 and satisfies the condition.

A. the image *F*<sup>0</sup> ð Þ ½ � *a*, *b* is subsets of the image *f a* ð Þ ½ � , *b* ,

B. the pairs *F*<sup>0</sup> and *f* are compatible, and

C. ∃ *x*<sup>0</sup> ∈ ½ � *a*, *b* such that supf g *H Fx* ð Þ *<sup>n</sup>*, *Fx*<sup>1</sup> : *n* ¼ 0, 1⋯ < þ ∞,

then *F*<sup>0</sup> *gh* has a unique fixed point on *X*.

Example 3 Let *X* ¼ �½ � 2, 2 with usual metric. Let *F* : ½ �! �2, 2 Ið Þ with the formula

$$F(\mathbf{x}) = \begin{cases} \left[ \left( \mathbf{x} + \sin \left( \mathbf{x} + \frac{1}{2} \right) \right), \ \mathbf{x} \right] & \text{if } -2 \le \mathbf{x} \le -\frac{1}{2}, \\\\ \left[ \mathbf{x}, \ \left( \mathbf{x} + \sin \left( \mathbf{x} + \frac{1}{2} \right) \right) \right] & \text{if } -\frac{1}{2} \le \mathbf{x} \le -2. \end{cases}$$

It is clear that *F* is *gh*-differentiable on ð Þ �2, 2 by Theorem 1.2 with derivative

$$F'\_{g^h}(\mathbf{x}) = \begin{cases} \begin{bmatrix} \mathbf{1}, \ \left(\mathbf{1} + \cos\left(\mathbf{x} + \frac{\mathbf{1}}{2}\right)\right) \end{bmatrix} & \text{if } -2 \le x \le 1, \\\\ \begin{bmatrix} \left(\mathbf{1} + \cos\left(\mathbf{x} + \frac{\mathbf{1}}{2}\right)\right), \ \mathbf{1} \end{bmatrix} & \text{if } \mathbf{1} \le x \le 2. \end{cases}$$

If we choose *f x*ð Þ¼ *x*∈*F x*ð Þ for all *x*∈½ � �2, 2 , then we obtain

$$F\_{gh}'X = F\_{gh}'([-2, \ 2]) = [1, \ 2] \cup [0.198, 1] = [0.198, 2] \subset [-2, \ 2] = f([-2, \ 2]) = fX.$$

This means the condition in Corollary 1 part (A) is satisfied. If the sequence *xn* ! 1, then *F*<sup>0</sup> *gh*ð Þ! *xn* f g1 ¼ *K* and *fxn* ! 1∈*K*. First, we start with the formula *F*0 *ghfxn* ¼ *F*<sup>0</sup> *gh*ð Þ¼ *xn* <sup>1</sup> <sup>þ</sup> cos *xn* <sup>þ</sup> <sup>1</sup> 2 � � � � , 1 � � <sup>∪</sup> 1, 1 <sup>þ</sup> cos *xn* <sup>þ</sup> <sup>1</sup> 2 � � � � � and *f F*0 *gh*ð Þ¼ *xn F*<sup>0</sup> *gh*ð Þ¼ *xn* <sup>1</sup> <sup>þ</sup> cos *xn* <sup>þ</sup> <sup>1</sup> 2 � � � � , 1 � � <sup>∪</sup> 1, 1 <sup>þ</sup> cos *xn* <sup>þ</sup> <sup>1</sup> 2 � � � � � , we obtain

$$H(F'\_{gh}f\mathfrak{x}\_n, fF'\_{gh}\mathfrak{x}\_n) = \mathbf{0}.$$

Thus *F* and *f* are compatible. Since *F* is continuously *gh*-differentiable on ð Þ �2, 2 , this implies that *F* and *f* are continuous on ð Þ �2, 2 (see Theorem 1.1). Hence we have

$$\lim\_{x\_n \to 1} F'\_{gh} f \mathbf{x}\_n = F'\_{gh}(\mathbf{1}) = \{\mathbf{1}\}, \quad \text{and} \quad \lim\_{x\_n \to 1} f \mathbf{x}\_n = \mathbf{1} = f(\mathbf{1}).$$

Consequently, 1 ¼ *f*ð Þ1 ∈*F*<sup>0</sup> *gh*ð Þ¼ 1 f g1 .

#### **5. Conclusions**

The existence of a fixed point for the derivative of set-valued mappings can be obtained by using the method of the compatibility of the hybrid composite mappings in the sense of the Pompei-Hausdorff metric.

#### **Conflict of interest**

The authors declare no conflict of interest.

*On Fixed Point for Derivative of Set-Valued Functions DOI: http://dx.doi.org/10.5772/intechopen.107185*

### **Author details**

Mohamad Muslikh<sup>1</sup> \*† and Adem Kilicman2†

1 Universitas Brawijaya, Malang, Indonesia

2 Universiti Putra malaysia, Serdang, Malaysia

\*Address all correspondence to: mslk@ub.ac.id

† These authors contributed equally.

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Baxter G. Mint: On fixed points of the composite of commuting functions. 1963. Available from: http://www.ams. org/journal-terms-of-use. [Accessed: 2018-12-31]

[2] Boyce WM. Mint: Commuting functions with no common fixed point. 1967. Available from: http://www.ams. org/journal-terms-of-use. [Accessed: 2017-10-3]

[3] Husein SA, Sehgal VM. Mint: On common fixed point for a family of mappings. Bulletin of the Australian Mathematical Society. 1975;**13**:261-267. [Accessed: 2018-12-31]

[4] Singh SL, Meade BA. Mint: On common fixed point theorems. Bulletin of the Australian Mathematical Society. 1977;**16**:49-53. [Accessed: 2018-12-31]

[5] Jungck G. Mint: Commuting mappings and fixed points. The American Mathematical Monthly. 1976; **83**(4):261-263. Available from: http:// www.jstor.org/stable/2318216. [Accessed: 2017-10-31]

[6] Mehta JG, Joshi ML. Mint: On common fixed point theorem in complete metric spaces. General Mathematics Notes. 2012;**2**(1):55-63

[7] Fisher B. Mint: Result on common fixed points on complete metric spaces. Glasgow Mathematical Journal. 1980;**21**: 65-67

[8] Khan MS. Mint: Common fixed point theorems for multivalued mappings. Pacific Journal of Mathematics. 1981; **95**(2):337-347

[9] Abdou AAN. Mint: Common fixed point result for multi-valued mappings with some examples. Journal of

Nonlinear Sciences and Applications. 2016;9:787-798. Available from: www. tjnsa.com [Accessed: 2017-07-31]

[10] Fisher B. Mint: Common fixed point theorems for mappings and set-valued mappings. Rostocker Mathematisches Kolloquium. 1981;**18**:69-77

[11] Singh SL, Kamal R, Sen MDL, Chugh R. Mint: New a type of coincidence and common fixed point theorem with applications. Abstract and Apllied Analysis. 2014;**2014**:1-11. [Accessed: 2017-08-31]

[12] Itoh S, Takahashi W. Mint: Singlevalued mappings, set-valued mappings and fixed point theorrems. Journal of Mathematical Analysis and Applications. 1977;**59**:514-521

[13] Fisher B. Mint: Fixed point of mappings and set-valued mappings. The Journal of the University of Kuwait, Science. 1982;**9**:175-180

[14] Imdad M, Khan MS, Sessa S. On some weak conditions of commutativity in common fixed point theorems. International Journal of Mathematics and Mathematical Sciences. 1988;**11**(2): 289-296

[15] Sessa S, Khan MS, Imdad M. Mint: Common fixed point theorem with a weak commutativity condition. Glasnik Mathematicki. 1986;**21**(41):225-235

[16] Kaneko H, Sessa S. Mint: Fixed point theorems for multivalued and single mappings. International Journal of Mathematics and Mathematical Sciences. 1989;**12**(2):257-262

[17] Jungck G, Rhoades BE. Mint: Some fixed point theorems for compatible maps. International Journal of

*On Fixed Point for Derivative of Set-Valued Functions DOI: http://dx.doi.org/10.5772/intechopen.107185*

Mathematics and Mathematical Sciences. 1993;**16**(3):417-428

[18] Elekes M, Keleti T, Prokaj V. Mint: The fixed point of the composition of derivatives. Real Analysis Exchange. 2001-2002;**27**(1):131-140

[19] Muslikh M, Kilicman A. On common fixed point of a function and its derivative. Advances Fixed Point Theory. 2017;**7**(3):359-371

[20] Pompeiu D. Sur a Continuité des fonctions devariables complexes, [Theses]. Paris: Gauthier-Villars; 1905

[21] Hausdorff F. Grundzughe der Mengenlehre. Leipzig: Viet; 1914

[22] Moore RE, Kearfott RB, Cloud MJ. Introduction to Interval Analysis. Philadelphia, USA: The society for Industrial and applied Mathematics; 2009

[23] Hukuhara M. Intégration des applications mesurables dont la valeur est uncompact convex. Fako de l'Funkcialaj Ekvacioj Japana Matematika Societo. 1967;**10**:205-229

[24] Markov S. Calculus for Interval Function of a real variables. Computing. 1969;**22**:325-337

[25] Stefanini L. A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets and System. 2010;**161**(11):1564-1584

[26] Stefanini L, Bede B. Generalized Hukuhara differentiability of interval valued functions and interval differential equations. Nonlinear Analysis. 2009;**71**: 1311-1328

[27] Chalcao-Chano Y, Machui-Huaman GG, Silva GN, Jimenez-Gamero MD. Algebra of generalized Hukuhara

differentiable interval-valued functions: Review and new properties. Fuzzy Sets and System. 2019;**375**:53-69

[28] Fisher B. Common fixed point theorems for mappings and set-valued mappings. The Journal of the University of Kuwait, Science. 1984;**11**:15-21

[29] Nadler SB Jr. Multivalued contraction mappings. Pacific Journal of Mathematics. 1969;**30**(2):475-488

#### **Chapter 6**

## Stability Estimates for Fractional Hardy-Schrödinger Operators

*Konstantinos Tzirakis*

#### **Abstract**

In this chapter, we derive optimal Hardy-Sobolev type improvements of fractional Hardy inequalities, formally written as L*su*≥ *w x*ð Þ j j *<sup>x</sup> <sup>θ</sup> <sup>u</sup>*<sup>2</sup> <sup>∗</sup> �1, for the fractional Schrödinger operator <sup>L</sup>*su* ¼ �ð Þ <sup>Δ</sup> *<sup>s</sup> <sup>u</sup>* � *kn*,*<sup>s</sup> <sup>u</sup>* j j *<sup>x</sup>* <sup>2</sup>*<sup>s</sup>* associated with *<sup>s</sup>*-th powers of the Laplacian for *<sup>s</sup>*∈ð Þ 0, 1 , on bounded domains in *<sup>n</sup>:* Here, *kn*,*<sup>s</sup>* denotes the optimal constant in the fractional Hardy inequality, and 2 <sup>∗</sup> <sup>¼</sup> <sup>2</sup>ð Þ *<sup>n</sup>*�*<sup>θ</sup> <sup>n</sup>*�2*<sup>s</sup>* , for 0 <sup>≤</sup>*<sup>θ</sup>* <sup>≤</sup>2*<sup>s</sup>* <sup>&</sup>lt;*n:* The optimality refers to the singularity of the logarithmic correction *w* that has to be involved so that an improvement of this type is possible. It is interesting to note that Hardy inequalities related to two distinct fractional Laplacians on bounded domains admit the same optimal remainder terms of Hardy-Sobolev type. For deriving our results, we also discuss refined trace Hardy inequalities in the upper half space which are rather of independent interest.

**Keywords:** fractional Laplacian, hardy-Sobolev inequalities, Schrödinger operator

#### **1. Introduction**

Fractional Laplacian operators have attracted considerable attention in various areas of pure and applied mathematics, see for instance [1] and the review articles [2–4]. Such non-local operators appear naturally in several branches of the applied sciences to model phenomena where long-range interactions take place, in fluid dynamics, quantum mechanics, biological populations, materials science, finance, image processing, and game theory, to name a few, for example, [5–16]. They have a prominent interest from a mathematical point of view, arising in analysis and partial differential equations (pdes), geometry, probability, and financial mathematics, see for instance [17–22].

For 0 <sup>&</sup>lt;*<sup>s</sup>* <sup>&</sup>lt;1, the fractional Laplacian ð Þ �<sup>Δ</sup> *<sup>s</sup>* of a function *<sup>f</sup>* in the Schwartz space of rapidly decaying *C*<sup>∞</sup> functions on *<sup>n</sup>*, is defined as a pseudodifferential operator (e.g., [1, 23, 24])

$$(-\Delta)^{\sharp}f = \mathcal{F}^{-1}\left(|\xi|^{2s}(\mathcal{F}f)\right), \qquad \forall \xi \in \mathbb{R}^n,\tag{1}$$

where, F*f* denotes the Fourier transform of *f* defined by

$$\mathcal{F}f(\xi) = \frac{1}{(2\pi)^{n/2}} \int\_{\mathbb{R}^n} e^{-i\xi \cdot x} f(x) \, d\infty.$$

It can be shown that the operator ð Þ �<sup>Δ</sup> *<sup>s</sup>* can be equivalently defined as the singular integral operator (see for instance [1], Proposition 3.3])

$$\begin{split} (-\Delta)^{\mathfrak{t}} f(\mathbf{x}) &= c(n, \mathfrak{s}) \text{P.V.} \int\_{\mathbb{R}^{n}} \frac{f(\mathbf{x}) - f(\mathbf{y})}{|\mathbf{x} - \mathbf{y}|^{n + 2s}} \, d\mathbf{y} \\ &= c(n, \mathfrak{s}) \lim\_{\epsilon \to 0^{+}} \int\_{\{|\mathbf{x} - \mathbf{y}| > \epsilon\}} \frac{f(\mathbf{x}) - f(\mathbf{y})}{|\mathbf{x} - \mathbf{y}|^{n + 2s}} \, d\mathbf{y}, \qquad \forall \mathbf{x} \in \mathbb{R}^{n}, \end{split} \tag{2}$$

where

$$c(n,s) = \frac{s\mathfrak{A}^s}{\mathfrak{a}^{n/2}} \frac{\Gamma\left(\frac{n+2s}{2}\right)}{\Gamma(1-s)}\tag{3}$$

and <sup>Γ</sup> stands for the usual Gamma function defined by <sup>Γ</sup>ðÞ¼ *<sup>s</sup>* <sup>Ð</sup> <sup>∞</sup> 0 *t <sup>s</sup>*�<sup>1</sup>*e*�*<sup>t</sup> dt:* Notice that, if *s* <1*=*2, then the integrand exhibits an integrable singularity, thus the principal value (P*:*V*:*) may be dropped. Moreover, by a change of variable, we can avoid the principal value and transform the singular integral in (2) as

$$(-\Delta)^{\circ}f(\varkappa) = \frac{1}{2}c(n,\varkappa)\int\_{\mathbb{R}^n} \frac{\mathcal{D}f(\varkappa) - f(\varkappa+\jmath) - f(\varkappa-\jmath)}{|\jmath|^{n+2\varkappa}}d\jmath.$$

We caution the reader to take into account the conventional value imposed for the constant *c n*ð Þ , *s* when comparing different definitions for fractional Laplacian. Here, we fix the value (3) so that the singular integral representation (2) accords with the characterization (1) as a Fourier multiplier operator, and notice that lim *<sup>s</sup>*!1� ð Þ �<sup>Δ</sup> *<sup>s</sup> f* ¼ �Δ*<sup>f</sup>* and lim *<sup>s</sup>*!0<sup>þ</sup> ð Þ �<sup>Δ</sup> *<sup>s</sup> f* ¼ *f:* Note that the definition (1) allows for a wider range of the fractional Laplace's exponents *s*, while the expression (2) is defined for *s*<1*:* We point out that the characterization via Fourier transform is reduced to the standard Laplacian as *s* ! 1, which, however cannot be defined by the pointwise expression (2). Let us also remark that from the definition in the Schwartz space it is possible to extend ð Þ �<sup>Δ</sup> *<sup>s</sup>* by duality in a large class of tempered distributions; see, for example [25]. For a further discussion on the fractional Laplacian and the associated fractional Sobolev spaces we refer the readers to ([1], §§2–3]).

In the literature, other characterizations for ð Þ �<sup>Δ</sup> *<sup>s</sup>* are also used, that turn out to be equivalent to the definitions (1), (2). A further discussion on the different definitions of the fractional Laplacian on *<sup>n</sup>* and a proof of their equivalence can be found in [26]. Each of these equivalent characterizations allows for different approaches for the related problems, and in our context, we exploit a characterization realizing the nonlocal operator via an appropriate extended local problem (see Section 3), where local pdes techniques can be applied.

Regarding the corresponding quadratic form for ð Þ �<sup>Δ</sup> *<sup>s</sup>* ,

$$\rho((-\Delta)^s f, f) \coloneqq \int\_{\mathbb{R}^n} f \, (-\Delta)^s f \, \, d\mathfrak{x} = \int\_{\mathbb{R}^n} |\xi|^{2s} \, (\mathcal{F}f)^2(\xi) \, \, d\xi$$

we have (see Aronszajn-Smith [27], page 402)

*Stability Estimates for Fractional Hardy-Schrödinger Operators DOI: http://dx.doi.org/10.5772/intechopen.109606*

$$\int\_{\mathbb{R}^n} |\xi|^{2s} (\mathcal{F}\mathcal{f})^2(\xi) \, d\xi = \frac{c(n,s)}{2} \int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{|f(\varkappa) - f(\jmath)|^2}{|\varkappa - \jmath|^{n+2s}} \, d\varkappa \, d\jmath. \tag{4}$$

We consider the homogeneous fractional Sobolev space *<sup>H</sup>*\_ *<sup>s</sup> <sup>n</sup>* ð Þ, defined as the completion of *C*<sup>∞</sup> <sup>0</sup> *<sup>n</sup>* ð Þ with respect to

$$||f||\_{\dot{H}'(\mathbb{R}^n)} := \int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{|f(\boldsymbol{x}) - f(\boldsymbol{y})|^2}{|\boldsymbol{x} - \boldsymbol{y}|^{n + 2s}} \, d\boldsymbol{x} \, d\boldsymbol{y}. \tag{5}$$

The sharp fractional Sobolev inequality, associated to ð Þ �<sup>Δ</sup> *<sup>s</sup>* , states that

$$\mathcal{S}\_{n,\mathfrak{s}}\left(\int\_{\mathbb{R}^{n}}|f|^{2^{\*}\_{\mathfrak{s}}}\left(\infty\right)d\mathfrak{x}\right)^{2/2^{\*}\_{\mathfrak{s}}}\leq\int\_{\mathbb{R}^{n}}\int\_{\mathbb{R}^{n}}\frac{|f(\mathfrak{x})-f^{\flat}(\mathfrak{y})|^{2}}{|\mathfrak{x}-\mathfrak{y}|^{n+2\mathfrak{s}}}d\mathfrak{x}\ dy,\quad\forall\mathfrak{f}\in\dot{H}^{i}(\mathbb{R}^{n}),\tag{6}$$

where 2 <sup>∗</sup> *<sup>s</sup>* <sup>¼</sup> <sup>2</sup>*<sup>n</sup> <sup>n</sup>*�2*<sup>s</sup>* , and the best constant

$$\mathcal{S}\_{n,s} = \frac{2^{2s} \pi^{\prime} \Gamma\left(\frac{n+2s}{2}\right)}{\Gamma\left(\frac{n-2s}{2}\right)} \left(\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma(n)}\right)^{2s/n}$$

is achieved in *<sup>H</sup>*\_ *<sup>s</sup> <sup>n</sup>* ð Þ, exactly by the multiples, dilates, and translates of the function 1 þ j j *x* <sup>2</sup> � �ð Þ <sup>2</sup>*s*�*<sup>n</sup> <sup>=</sup>*<sup>2</sup> ; see [28, 29]. Sobolev inequality (6) yields the continuous embedding *<sup>H</sup>*\_ *<sup>s</sup> <sup>n</sup>* ð Þ↪*L*<sup>2</sup> <sup>∗</sup> *<sup>s</sup> <sup>n</sup>* ð Þ, which is sharp within the framework of Lebesgue spaces, in the sense that the embedding fails for any other Lebesgue subspace. In terms of Lorentz spaces, this embedding reads as *<sup>H</sup>*\_ <sup>1</sup> *<sup>n</sup>* ð Þ↪*L*<sup>2</sup> <sup>∗</sup> *<sup>s</sup>* ,2 <sup>∗</sup> *<sup>s</sup> <sup>n</sup>* ð Þ, which admits an extension within the whole Lorentz space scale *L*<sup>2</sup> <sup>∗</sup> *<sup>s</sup>* ,*<sup>p</sup> <sup>n</sup>* ð Þ, *<sup>p</sup>*<sup>≥</sup> <sup>2</sup>*:* As a matter of fact, the embeddings for *p*> 2, follow from the continuous inclusions *L*<sup>2</sup> <sup>∗</sup> *<sup>s</sup>* ,2 *<sup>n</sup>* ð Þ↪*L*<sup>2</sup> <sup>∗</sup> *<sup>s</sup>* ,*<sup>p</sup> <sup>n</sup>* ð Þ, and the continuous embedding

$$\dot{H}'(\mathbb{R}^n) \hookrightarrow L^{2, \text{"}2}(\mathbb{R}^n),\tag{7}$$

which, in turn, follows from the fractional Hardy inequality

$$k\_{n, \boldsymbol{\nu}} \int\_{\mathbb{R}^n} \frac{\left| \boldsymbol{f}(\boldsymbol{x}) \right|^2}{\left| \boldsymbol{\infty} \right|^{2s}} \, d\boldsymbol{x} \le \int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{\left| \boldsymbol{f}(\boldsymbol{x}) - \boldsymbol{f}\_{-}(\boldsymbol{y}) \right|^2}{\left| \boldsymbol{x} - \boldsymbol{y} \right|^{n + 2s}} \, d\boldsymbol{x} \, \, d\boldsymbol{y}. \tag{8}$$

Indeed, one can derive (7) from (8), by the fact that under radially decreasing rearrangement the *<sup>H</sup>*\_ *<sup>s</sup> <sup>n</sup>* ð Þ norm does not increase [30] and the left hand side of (8) does not decrease, while the Lorentz quasinorm ∣∣ � jj*L*<sup>2</sup> <sup>∗</sup> *<sup>s</sup>* ,2 is invariant and proportional to the left hand side of (8).

In this sense, Hardy's inequality (8) is stronger than Sobolev's inequality (6). The value

$$k\_{n,s} = \frac{2\pi^{n/2}\Gamma(\mathbf{1}-s)\,\Gamma^2\left(\frac{n+2s}{4}\right)}{s\,\Gamma^2\left(\frac{n-2s}{4}\right)\,\Gamma\left(\frac{n+2s}{2}\right)}$$

is the best possible constant in (8). It is well known that the best constant *kn*,*<sup>s</sup>* in (8) is not attained in *<sup>H</sup>*\_ *<sup>s</sup> <sup>n</sup>* ð Þ, yet no *Lp* improvement is possible in *<sup>H</sup>*\_ *<sup>s</sup> <sup>n</sup>* ð Þ, as demonstrated by testing with suitable perturbations of the solution j j *<sup>x</sup>* <sup>2</sup>*s*�*<sup>n</sup>* <sup>2</sup> , of the corresponding Euler–Lagrange equation.

An application of Hölder's inequality together with (6) and (8), yield the following Hardy-Sobolev inequality:

$$\Lambda\_{\mathfrak{n},\theta,\mathfrak{t}} \int\_{\mathbb{R}^{n}} \frac{|f|^{2\_{\ast}(\theta)}}{|\mathfrak{x}|^{\theta}} \, d\mathfrak{x} \le \int\_{\mathbb{R}^{n}} \int\_{\mathbb{R}^{n}} \frac{|f(\mathfrak{x}) - f\_{\mathfrak{n}}(\mathfrak{y})|^{2}}{|\mathfrak{x} - \mathfrak{y}|^{n + 2s}} \, d\mathfrak{x} \, \, d\mathfrak{y}, \quad f \in C\_{0}^{\infty}(\mathbb{R}^{n}), \tag{9}$$

where 2 <sup>∗</sup> ð Þ¼ *<sup>θ</sup>* <sup>2</sup>ð Þ *<sup>n</sup>*�*<sup>θ</sup> <sup>n</sup>*�2*<sup>s</sup>* , 0≤*<sup>θ</sup>* <sup>&</sup>lt;2*s:* The best constant in (9), contrary to the borderline case (8) i.e. *<sup>θ</sup>* <sup>¼</sup> <sup>2</sup>*s*, is achieved in *<sup>H</sup>*\_ *<sup>s</sup> <sup>n</sup>* ð Þ; cf. [31].

In view of (3)–(4), inequality (8) is equivalent to

$$h\_{n, \mathfrak{s}} \int\_{\mathbb{R}^n} \frac{f^2(\mathfrak{x})}{|\mathfrak{x}|^{2s}} \, d\mathfrak{x} \le \int\_{\mathbb{R}^n} |\mathfrak{s}|^{2\mathfrak{s}} (\mathcal{F}f)^2(\mathfrak{x}) \, d\mathfrak{x}, \quad \forall f \in \dot{H}^{\mathfrak{f}}(\mathbb{R}^n), \tag{10}$$

with the sharp constant

$$h\_{n, \mathfrak{s}} = 4^{\mathfrak{s}} \Gamma^2 \left( \frac{n + 2\mathfrak{s}}{4} \right) / \Gamma^2 \left( \frac{n - 2\mathfrak{s}}{4} \right). \tag{11}$$

The dual form of (10), formulated in terms of Riesz integral operator, is a special case of Stein-Weiss inequalities [32], and the best constant *hn*,*<sup>s</sup>* is identified by Herbst [33]; see also Beckner [34], Yafaev [35].

By Hardy-Littlewood and Pólya-Szegö type rearrangement inequalities, it suffices to prove (10) for radial decreasing *f*; see Almgren and Lieb [30] where it is shown that (4) does not increase if *f* is replaced by its equimeasurable symmetric decreasing rearrangement. Then, we will show that the inequality is equivalent to a convolution inequality on the multiplicative group <sup>þ</sup> equipped with the Haar measure <sup>1</sup> *<sup>r</sup> dr*.

In particular, (10) is equivalent to the following doubly weighted Hardy-Littlewood-Sobolev inequality of Stein-Weiss [32].

$$\int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{f(\mathbf{x})}{\left| \mathbf{x} \right|^\delta} \frac{1}{\left| \mathbf{x} - \mathbf{y} \right|^{n-2s}} \frac{f(\mathbf{y})}{\left| \mathbf{y} \right|^\delta} \, d\mathbf{x} d\mathbf{y} \le C\_{n,\delta} \int\_{\mathbb{R}^n} \left| f(\mathbf{x}) \right|^2 \, d\mathbf{x},\tag{12}$$

with sharp constant

$$\mathcal{C}\_{n,s} = \frac{\pi^{n/2} \Gamma^2\left(\frac{n-2s}{4}\right) \Gamma(s)}{\Gamma^2\left(\frac{n+2s}{4}\right) \Gamma\left(\frac{n-2s}{2}\right)} \dots$$

Since we can assume that *f* is radial, we set *f x*ð Þ¼ fð Þ*r* , and *x* ¼ *rx*<sup>0</sup> , *y* ¼ *ρy*<sup>0</sup> where ∣*x*0 ∣ ¼ ∣*y*<sup>0</sup> ∣ ¼ 1*:* Regarding the convolution integral of the left side in (12), we employ polar coordinates to get

*Stability Estimates for Fractional Hardy-Schrödinger Operators DOI: http://dx.doi.org/10.5772/intechopen.109606*

ð *n* ð *n f x*ð Þ <sup>1</sup> j j *x s* 1 j j *<sup>x</sup>* � *<sup>y</sup> <sup>n</sup>*�2*<sup>s</sup>* 1 j j *<sup>y</sup> <sup>s</sup> f y*ð Þ *dxdy* <sup>¼</sup> ∞ð 0 ∞ð 0 ð ∣*x*0 ∣¼1 ð ∣*y*0 ∣¼1 fð Þ*r rn*�<sup>1</sup> *rs* 1 *rx*<sup>0</sup> � *<sup>ρ</sup>y*<sup>0</sup> j j*n*�2*<sup>s</sup> ρn*�<sup>1</sup> *<sup>ρ</sup><sup>s</sup>* <sup>f</sup>ð Þ*<sup>ρ</sup> <sup>d</sup><sup>σ</sup> <sup>x</sup>*<sup>0</sup> ð Þ*d<sup>σ</sup> <sup>y</sup>*<sup>0</sup> ð Þ*drd<sup>ρ</sup>* <sup>¼</sup> ∞ð 0 ∞ð 0 ð ∣*x*0 ∣¼1 fð Þ*r r <sup>n</sup>=*<sup>2</sup> h i 1 *r* 2*s*�*n* 2 *K r*ð Þ , *ρ* 1 *ρ* 2*s*�*n* 2 <sup>f</sup>ð Þ*<sup>ρ</sup> <sup>ρ</sup>n=*<sup>2</sup> h i *<sup>d</sup><sup>σ</sup> <sup>x</sup>*<sup>0</sup> ð Þ *dr r dρ ρ* (13)

where *dσ* denotes ð Þ *n* � 1 -dimensional Lebesgue integration over the unit sphere *n*�<sup>1</sup> <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>n</sup>* : <sup>j</sup>*x*<sup>0</sup> f g j¼ <sup>1</sup> , and we set

$$K(r,\rho) \coloneqq \int\_{|\mathbf{y}'|=1} \frac{1}{|r\mathbf{x}'-\rho\mathbf{y}'|^{n-2s}} \cdot d\sigma(\mathbf{y}').\tag{14}$$

Notice that *K r*ð Þ , *<sup>ρ</sup>* in (14) is independent of *<sup>x</sup>*<sup>0</sup> <sup>∈</sup>*<sup>n</sup>*�<sup>1</sup> *:* To show this independence, we may assume *r* ¼ 1, *ρ* ¼ *τ*, or more generally, to use the variable *τ* ¼ *ρ=r* and then it suffices to show that

$$K(\tau) := \int\_{|\mathbf{y}'| = 1} \frac{1}{|\mathbf{x}' - \mathbf{y}\mathbf{y}'|^{n-2s}} \, d\sigma(\mathbf{y}')$$

is independent of *x*<sup>0</sup> ∈*<sup>n</sup>*�<sup>1</sup> *:* Indeed, take an arbitrary *z*<sup>0</sup> ∈*<sup>n</sup>*�<sup>1</sup> *:* Then there exists a rotation *<sup>R</sup>* such that *<sup>z</sup>*<sup>0</sup> <sup>¼</sup> *Rx*<sup>0</sup> and we denote by *<sup>R</sup><sup>T</sup>* its transpose. Performing the change of variables *<sup>w</sup>*<sup>0</sup> <sup>¼</sup> *<sup>R</sup>Ty*<sup>0</sup> , we get

$$\int\_{|y'|=1} \frac{1}{|x'-\tau y'|^{n-2s}} \, d\sigma(y') = \int\_{|w'|=1} \frac{1}{|x'-\tau w'|^{n-2s}} \, d\sigma(w') = K(\tau),$$

since <sup>∣</sup>*detR*<sup>∣</sup> <sup>¼</sup> 1 and <sup>∣</sup>*R*v1 � *<sup>R</sup>*v2<sup>∣</sup> <sup>¼</sup> <sup>∣</sup>v1 � v2∣, for every v1, v2 <sup>∈</sup> *<sup>n</sup>:* Since *K r*ð Þ , *<sup>ρ</sup>* is independent of *x*<sup>0</sup> ∈ *<sup>n</sup>*�<sup>1</sup> we have

$$\int\_{\mathbb{S}^{n-1}} K(r,\rho)d\sigma(\mathbf{x}') \;=\; K(r,\rho)\Big|\_{\mathbb{S}^{n-1}}\mathbf{1}d\sigma(\mathbf{x}')\;=\; K(r,\rho)\frac{2\pi^{n/2}}{\Gamma\left(\frac{n}{2}\right)}.\tag{15}$$

Moreover, in (14), we can choose *x*<sup>0</sup> to be the first direction unit vector in *<sup>n</sup>* , that is ^*e*<sup>1</sup> ¼ ð Þ *x*1, *x*2, ⋯, *xn* with *x*<sup>1</sup> ¼ 1, *x*<sup>2</sup> ¼ *x*<sup>3</sup> ¼ ⋯ ¼ *xn* ¼ 0, hence

$$K(r,\rho) = \int\_{|\mathbf{y}'|=1} \frac{1}{\left(r^2 - 2r\rho\mathbf{y}\_1 + \rho^2\right)^{\frac{n-2}{2}}} \,d\sigma(\mathbf{y}')$$

thus

$$\frac{1}{r^{\frac{2-n}{2}}} \ K(r,\rho) \quad \frac{1}{\rho^{\frac{2-n}{2}}} = \int\_{|\mathbf{y}'|=1} \frac{1}{\left(\frac{r}{\rho} - 2\mathbf{y}\_1 + \frac{\varrho}{r}\right)^{\frac{n-2s}{2}}} \,d\sigma(\mathbf{y}')$$

and substituting (15) into (13), we get

$$\int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} f(\mathbf{x}) \, \frac{1}{\left| \mathbf{x} \right|^\varepsilon} \, \frac{1}{\left| \mathbf{x} - \mathbf{y} \right|^{n-2s}} \, \frac{1}{\left| \mathbf{y} \right|^\varepsilon} \, f(\mathbf{y}) \, \, d\mathbf{x} \, d\mathbf{y} = \frac{2\pi^{n/2}}{\Gamma\left(\frac{n}{2}\right)} \int\_0^\infty \int\_0^\infty h(r) \, \, \Psi\left(\frac{r}{\rho}\right) \, h(\rho) \, \, \frac{dr}{r} \frac{d\rho}{\rho} \, \, dr \,\,\tag{16}$$

where

$$h(r) \coloneqq \mathbf{f}(r)r^{n/2} \quad \text{and} \quad \boldsymbol{\nu}(\boldsymbol{\tau}) = \boldsymbol{\nu}\left(\frac{\mathbf{1}}{\boldsymbol{\tau}}\right) = \int\_{|\boldsymbol{\nu}'|=1} \frac{\mathbf{1}}{\left(\boldsymbol{\tau} - \mathbf{2}\boldsymbol{\nu}\_1 + \frac{1}{\tau}\right)^{\frac{n-2}{2}}} \, d\boldsymbol{\sigma}(\boldsymbol{\nu}').$$

As for the right side of the fractional integral inequality (12), we use again polar coordinates to get

ð *n* j j *f x*ð Þ <sup>2</sup> *dx* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> n* 2 Γ *<sup>n</sup>* 2 � � <sup>ð</sup><sup>∞</sup> 0 j j *h r*ð Þ <sup>2</sup> *dr <sup>r</sup> :* (17)

Finally, substituting (16), (17) in (12), we conclude that the fractional Hardy inequality (10) is written equivalently as the convolution inequality

$$\int\_{0}^{\infty} \int\_{0}^{\infty} h(r) \left. \psi \left( \frac{r}{\rho} \right) \right| h(\rho) \left. \frac{dr}{r} \frac{d\rho}{\rho} \leq \mathcal{C}\_{n, \varepsilon} \int\_{0}^{\infty} \left| h(r) \right|^{2} \frac{dr}{r} . \tag{18}$$

Inequality (18) is a convolution inequality on the multiplicative group <sup>þ</sup> equipped with the Haar measure <sup>1</sup> *<sup>r</sup> dr*, and using the sharp Young's inequality for convolution on certain noncompact Lie groups, we recover the sharpness of the constant and the non-existence of extremals for the fractional Hardy inequality (10).

#### **2. Fractional hardy-Sobolev inequalities on bounded domains**

In the sequel, we will discuss Hardy type inequalities for fractional powers of Laplacian associated with bounded domains, and, more precisely, defined for functions satisfying homogeneous Dirichlet boundary or exterior conditions. So hereafter let us fix a bounded domain Ω ⊂ *<sup>n</sup>*, with *n* >2*s:*

In opposition to the case of the whole of *<sup>n</sup>*, distinct definitions of such non-local operators have been introduced as mathematical models in various applications. In particular, we consider two of the most commonly used operators of this type, which are the so-called spectral Laplacian (see e.g. [36–38] and references therein) and the Dirichlet (also referred to as *restricted* or *regional* or integral, see e.g. [39, 40], and references therein). Both operators are deeply associated with the theory of stochastic processes. They can be characterized as generators of a 2ð Þ*s* -stable Lévy process with jumps resulting from two consecutive modifications of Wiener process, the subordination and the stopping (killing the process when leaves the domain), which reflect the homogeneous Dirichlet-type boundary (or exterior) conditions. Depending on which of these modifications is first applied, we take two different stochastic processes and their corresponding infinitesimal generators.

**The Dirichlet fractional Laplacian** Next, we will discuss improved versions of fractional Hardy inequalities, involving sharp Sobolev-Hardy type correction terms. We begin with the Dirichlet fractional Laplacian which we again denote by ð Þ �<sup>Δ</sup> *<sup>s</sup> :* We merely extend any function *f* ∈*C*<sup>∞</sup> <sup>0</sup> ð Þ <sup>Ω</sup> in the entire *<sup>n</sup>* by defining *f x*ð Þ¼ 0, for any *<sup>x</sup>* <sup>∉</sup> <sup>Ω</sup>, and then we define ð Þ �<sup>Δ</sup> *<sup>s</sup> f* as the standard fractional Laplacian on the whole space, acting on the extension of *f* to *<sup>n</sup>:* More precisely, we define

$$(-\Delta)^{\mathfrak{s}}f = \mathcal{F}^{-1}\left(\left|\xi\right|^{2s}(\mathcal{F}f)\right), \qquad \forall \xi \in \mathbb{R}^n.$$

The Dirichlet fractional Laplacian can be equivalently characterized as the singular integral operator (2) for the *c n*ð Þ , *s* given in (3).

Passing from *<sup>n</sup>* to a bounded domain Ω, containing the origin, inequality (8) is still valid with the same best possible constant

$$k\_{n, \boldsymbol{\varepsilon}} \int\_{\Omega} \frac{f^2(\boldsymbol{x})}{|\boldsymbol{x}|^{2s}} \, d\boldsymbol{x} \leq \int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{|f(\boldsymbol{x}) - f(\boldsymbol{y})|^2}{|\boldsymbol{x} - \boldsymbol{y}|^{n + 2s}} \, d\boldsymbol{x} \, d\boldsymbol{y}, \qquad \forall \boldsymbol{\xi} \in H\_0^\prime(\Omega), \tag{19}$$

where *H<sup>s</sup>* <sup>0</sup>ð Þ Ω is the homogeneous fractional Sobolev space, defined as the completion of the functions in *C*<sup>∞</sup> <sup>0</sup> ð Þ Ω , extended by zero outside Ω, with respect to the norm (5). Clearly the constant *kn*,*<sup>s</sup>* can not be achieved in *H<sup>s</sup>* <sup>0</sup>ð Þ Ω , and various improved versions of (19) have been established by many authors, which amount to adding *L<sup>p</sup>* norms of *u* or its fractional gradients in the left hand side.

In particular, Frank, Lieb and Seiringer have shown among others in [40], that for any 1≤*q*< 2 <sup>∗</sup> *<sup>s</sup>* <sup>≔</sup> <sup>2</sup>*n=*ð Þ *<sup>n</sup>* � <sup>2</sup>*<sup>s</sup>* and any bounded domain <sup>Ω</sup> <sup>⊂</sup> *<sup>n</sup>* there exists a positive constant *c* ¼ *c n*ð Þ , *s*, *q*, j j Ω such that

$$k\_{\varepsilon,\eta} \int\_{\Omega} \frac{f^2(\mathbf{x})}{|\mathbf{x}|^{2s}} \, d\mathbf{x} + c \left( \int\_{\Omega} |f(\mathbf{x})|^q \, d\mathbf{x} \right)^{2/q} \le \int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{|f(\mathbf{x}) - f(\mathbf{y})|^2}{|\mathbf{x} - \mathbf{y}|^{n + 2s}} \, d\mathbf{x} \, \, d\mathbf{y}, \quad f \in C\_0^\infty(\Omega). \tag{20}$$

Using the Dirichlet to Neumann mapping for the representation of the fractional Laplacian [39] (see Section 3 for details), a partial extension of (20) has been obtained in [41], replacing the remainder term with the *p*�norm of a fractional gradient, *p*<2.

An improvement involving a 2-norm of a fractional gradient, has been obtained in [42], using the following representation of the remainder term ([40], Proposition 4.1),

$$\begin{split} &k\_{n,\varepsilon} \int\_{\mathbb{R}^{n}} \frac{f^{2}(\mathbf{x})}{|\boldsymbol{\varkappa}|^{2\varepsilon}} \, d\boldsymbol{\varkappa} - \int\_{\mathbb{R}^{n}} \int\_{\mathbb{R}^{n}} \frac{|f(\mathbf{x}) - f(\mathbf{y})|^{2}}{|\boldsymbol{\varkappa} - \boldsymbol{\jmath}|^{n+2\varepsilon}} \, d\boldsymbol{\varkappa} \, d\boldsymbol{\jmath} \\ &= c(n, \varepsilon) \int\_{\mathbb{R}^{n}} \int\_{\mathbb{R}^{n}} \frac{|\boldsymbol{\nu}(\mathbf{x}) - \boldsymbol{\nu}(\mathbf{y})|^{2}}{|\boldsymbol{\varkappa} - \boldsymbol{\jmath}|^{n+2\varepsilon}} \frac{1}{|\boldsymbol{\varkappa}|^{\frac{n-2\varepsilon}{2}}} \frac{1}{|\boldsymbol{\jmath}|^{\frac{n-2\varepsilon}{2}}} \, d\boldsymbol{\varkappa} \, d\boldsymbol{\jmath} \end{split} \tag{21}$$

with the ground state substitution

$$\nu(\mathbf{x}) = f(\mathbf{x}) |\mathbf{x}|^{\frac{n-2}{2}}.\tag{22}$$

We point out that the exponent *q* in (20) is strictly smaller than the critical fractional Sobolev exponent 2 <sup>∗</sup> *<sup>s</sup>* and the inequality fails for *<sup>q</sup>* <sup>¼</sup> <sup>2</sup> <sup>∗</sup> *<sup>s</sup> :* In [43] we have shown that introducing a logarithmic relaxation we can have a critical Sobolev

improvement of (19). More precisely, it has been shown the existence of a positive constant *C*, depending only on *n* and *s*, such that for *f* ∈ *H<sup>s</sup>* <sup>0</sup>ð Þ Ω ,

$$k\_{n, \varepsilon} \int\_{\Omega} \frac{\left| f(\mathbf{x}) \right|^2}{\left| \mathbf{x} \right|^{2\varepsilon}} \, d\mathbf{x} + C \left( \int\_{\Omega} X^{\frac{2(n-\varepsilon)}{n-2\varepsilon}} \left( \frac{|\mathbf{x}|}{D} \right) |f(\mathbf{x})|^{\frac{2\varepsilon}{n-2\varepsilon}} \, d\mathbf{x} \right)^{\frac{n-2\varepsilon}{n}} \leq \int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{\left| f(\mathbf{x}) - f\_{\mathbf{x}} \left( \mathbf{y} \right) \right|^2}{\left| \mathbf{x} - \mathbf{y} \right|^{n+2\varepsilon}} \, d\mathbf{x} \, d\mathbf{y}, \tag{23}$$

where *D* ¼ sup*x*<sup>∈</sup> <sup>Ω</sup>∣*x*∣ and

*X r*ð Þ¼ ð Þ <sup>1</sup> � ln *<sup>r</sup>* �<sup>1</sup> , 0< *r*≤ 1*:*

Moreover, the weight *X* 2ð Þ *n*�*s <sup>n</sup>*�2*<sup>s</sup>* cannot be replaced by a smaller power of *X:* We emphasize that inequality (23) involves the critical exponent but contrary to the subcritical case, that is (20), it has a logarithmic correction. However inequality (23) is sharp in the sense that inequality fails for smaller powers of the logarithmic correction *X:* This result may be seen as the fractional version of (see [44, 45])

$$\frac{\left(n-2\right)^{2}}{4}\int\_{\Omega} \frac{\left|f(\mathbf{x})\right|^{2}}{\left|\mathbf{x}\right|^{2}} \,d\mathbf{x} + c\_{\pi} \left(\int\_{\Omega} \left|f(\mathbf{x})\right|^{\frac{2n}{n-2}} \mathbf{X}^{\frac{2(n-1)}{n-2}}(|\mathbf{x}|/D) \,d\mathbf{x}\right)^{\frac{n-2}{n}} \leq \int\_{\Omega} \left|\nabla f\right|^{2} \,d\mathbf{x},\tag{24}$$

in the sense that (23) reduces to (24) when *s* ! 1�.

Moreover, in [43] we have shown, for some constant *C*>0,

$$k\_{n, \mathfrak{s}} \int\_{\Omega} \frac{|f(\mathbf{x})|^2}{|\mathbf{x}|^{2s}} \, d\mathbf{x} + C \int\_{\Omega} X^2 \, \left(\frac{|\mathbf{x}|}{D}\right) |f(\mathbf{x})|^2 \, d\mathbf{x} \le \int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{|f(\mathbf{x}) - f(\mathbf{y})|^2}{|\mathbf{x} - \mathbf{y}|^{n + 2s}} \, d\mathbf{x} \, \, d\mathbf{y}, \tag{25}$$

where the weight *X*<sup>2</sup> cannot be replaced by a smaller power of *X:*

Let us notice that contrary to the Hardy-Sobolev inequalities obtained in [46], where the Hardy potential entails the distance to the boundary, the Hardy-Sobolev inequalities involving the distance from the origin, miss the critical-Sobolev exponent by a logarithmic correction which cannot be removed. Let us also emphasize that our results cover the full range *s*∈ð Þ 0, 1 , in contrast to the case involving the distance from the boundary, where Hardy inequalities associated with the spectral and Dirichlet fractional Laplacians fail within the range 0 <*s* <1*=*2*:*

In view of (23) and (25), we can apply Hölder inequality to get the following Hardy-Sobolev improvement of (19).

**Theorem 1.** *Let s*∈ð Þ 0, 1 , 0<sup>≤</sup> *<sup>θ</sup>* <sup>≤</sup>2*s*, <sup>Ω</sup> *be a bounded domain in <sup>n</sup> with n*>2*s: Then there exists a positive constant C* ¼ *C n*ð Þ , *s*, *θ such that*

$$h\_{n, \boldsymbol{\epsilon}} \left[ \frac{\left| \boldsymbol{f}(\boldsymbol{x}) \right|^{2}}{\left| \boldsymbol{\alpha} \right|^{2s}} \, d\boldsymbol{x} + \mathcal{C} \left( \int\_{\Omega} \frac{\mathbf{X}^{p(\boldsymbol{\theta})}}{\left| \boldsymbol{\alpha} \right|^{\theta}} \, \left| \boldsymbol{f} \right|^{2\_{\boldsymbol{\epsilon}}(\boldsymbol{\theta})} d\boldsymbol{x} \right)^{\frac{2}{2\_{\boldsymbol{\epsilon}}(\boldsymbol{\theta})}} \leq ((-\Delta)^{\boldsymbol{\epsilon}} \boldsymbol{f}, \boldsymbol{f}), \quad$$

for any *f* ∈*C*<sup>∞</sup> <sup>0</sup> ð Þ Ω , or equivalently,

$$k\_{n,\*} \int\_{\Omega} \frac{\left| f(\mathbf{x}) \right|^2}{\left| \mathbf{x} \right|^2} \, d\mathbf{x} + \mathcal{C} \left( \int\_{\Omega} \frac{X^{p(\theta)}}{\left| \mathbf{x} \right|^{\theta}} \left| f \right|^{2\_\* \cdot (\theta)} \, d\mathbf{x} \right)^{\frac{2}{2\_\* (\theta)}} \le \int\_{\mathbb{R}^n} \int\_{\mathbb{R}^n} \frac{\left| f(\mathbf{x}) - f(\mathbf{y}) \right|^2}{\left| \mathbf{x} - \mathbf{y} \right|^{n + 2s}} \, d\mathbf{x} \, \, d\mathbf{y}, \tag{26}$$

*Stability Estimates for Fractional Hardy-Schrödinger Operators DOI: http://dx.doi.org/10.5772/intechopen.109606*

where 2 <sup>∗</sup> ð Þ¼ *<sup>θ</sup>* <sup>2</sup>ð Þ *<sup>n</sup>*�*<sup>θ</sup> <sup>n</sup>*�2*<sup>s</sup>* , *<sup>p</sup>*ð Þ¼ *<sup>θ</sup>* <sup>2</sup>*n*�*θ*�2*<sup>s</sup> <sup>n</sup>*�2*<sup>s</sup>* and *<sup>X</sup>* <sup>¼</sup> *<sup>X</sup>*ð Þ <sup>j</sup>*x*j*=<sup>D</sup>* with *<sup>D</sup>* <sup>¼</sup> sup*x*<sup>∈</sup> <sup>Ω</sup>∣*x*∣*:* The logarithmic weight cannot be replaced by a smaller power of *X:*

The optimality of the exponent *<sup>p</sup>* <sup>≔</sup> *<sup>p</sup>*ð Þ¼ *<sup>θ</sup> 2 n*ð Þ� �*<sup>s</sup> <sup>θ</sup> <sup>n</sup>*�*2s* of the logarithmic weight, for the range *θ* ∈½ Þ 0, 2*s* can be deduced by the optimality of the exponent of the weight *<sup>X</sup>2*, for the case *<sup>θ</sup>* <sup>¼</sup> <sup>2</sup>*s*, jointly with Hölder inequality; cf. ([43], Remark), [47].

In view of (21), under the substitution (22) inequality (26) yields sharp limiting cases of certain fractional Caffarelli-Kohn-Nirenberg inequalities established in [48, 49].

**The spectral fractional Laplacian** We proceed with another reasonable approach in defining a nonlocal operator related to fractional powers of the Laplacian on the bounded domain Ω*:* We consider an orthonormal basis of *L*<sup>2</sup> ð Þ Ω , consisting of eigenfunctions of �Δ with homogeneous Dirichlet boundary conditions, say *ϕ*1, … , *ϕk*, … , with corresponding eigenvalues

$$0 < \lambda\_1 < \lambda\_2 \le \lambda\_3 \le \cdots \qquad \text{with} \qquad \lambda\_k \to \infty.$$

More precisely,

$$\begin{cases} -\Delta \phi\_k = \lambda\_k \phi\_k, & \text{in } \Omega, \\ \phi\_k = 0, & \text{on } \partial \Omega. \end{cases}$$

Then we have.

$$f = \sum\_{k=1}^{\infty} c\_k \phi\_k \qquad \text{where} \qquad c\_k = \int\_{\Omega} f \phi\_k \, d\mathbf{x}.$$

For any 0<*s* <1, the spectral fractional Laplacian, denoted hereafter by *As*, is defined, similarly to the spectral decomposition of the standard Laplacian, by

$$A\_{\sharp}f = \sum\_{k=1}^{\infty} \lambda\_k^{\sharp} c\_k \,\phi\_k, \quad \forall f \in C\_0^{\infty}(\Omega).$$

Notice that the operator *As* can be extended by approximation for functions in the Hilbert space

$$H = \left\{ f = \sum\_{k=1}^{\infty} c\_k \phi\_k \in L^2(\mathfrak{Q}) : ||f||\_H = \left(\sum\_{k=1}^{\infty} \lambda\_k^\varepsilon c\_k^2\right)^{1/2} < \infty \right\}.$$

The quadratic form corresponding to *As* is given by

$$(A\_{\emptyset}f, f) \coloneqq \int\_{\Omega} f A f \, \, d\mathfrak{x} = \sum\_{k=1}^{\infty} \lambda\_k^s c\_k^2 \dots$$

Let us point out that, contrary to the case of the whole space *<sup>n</sup>*, the fractional operators *As* and ð Þ �<sup>Δ</sup> *<sup>s</sup>* , as they defined above on bounded domains, differ in several aspects. For example, the natural functional domains of their definition are different, as the definition for the Dirichlet Laplacian ð Þ �<sup>Δ</sup> *<sup>s</sup>* requires the prescribed zero values of the functions on the whole of the exterior of the domain Ω, while the definition of the spectral Laplacian requires only zero values on boundary (local boundary conditions). They have essential differences even if we consider them as operators on a restricted class of functions, where they are both defined, e.g. in *C*<sup>∞</sup> <sup>0</sup> ð Þ <sup>Ω</sup> <sup>⊂</sup>*C*<sup>∞</sup> *<sup>c</sup> <sup>n</sup>* ð Þ*:* For example, the spectral Laplacian depends on the domain Ω through its eigenvalue and eigenfunctions. A further discussion on the differences between the operators *As* and ð Þ �<sup>Δ</sup> *<sup>s</sup>* can be found in [50].

The Hardy inequality corresponding to the spectral Laplacian *As*, involving the distance to the origin, reads

$$h\_{n, \boldsymbol{\varsigma}} \left[ \frac{f^2(\boldsymbol{x})}{|\boldsymbol{x}|^{2s}} \mid d\boldsymbol{x} \le (A\_{\boldsymbol{\zeta}}f, f), \qquad \forall f \in C\_0^\infty(\Omega), \tag{27}$$

with the constant *hn*,*<sup>s</sup>* given by (11), and this constant is the best possible in the case of 0∈ Ω*:* Observe that the Hardy inequalities (10), (27) associated with two distinct non-local operators share the same optimal constant. This is not the case when the distance is taken from the boundary, where the optimal constants for the corresponding Hardy inequalities are different, as it was shown among others in [46].

Similarly to Theorem 1, one can show that (27) may be improved by adding a critical Sobolev norm with the same sharp logarithmic corrective weight appearing in (26).

#### **3. Extension problems related to the fractional Laplacians**

In the following, we denote a point in *<sup>n</sup>*þ<sup>1</sup> as ð Þ *<sup>x</sup>*, *<sup>y</sup>* with *<sup>x</sup>*<sup>∈</sup> *<sup>n</sup>*, and *<sup>y</sup>*<sup>∈</sup> , and let us set *∂<sup>n</sup>*þ<sup>1</sup> <sup>þ</sup> <sup>¼</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>n</sup>*þ<sup>1</sup> : *<sup>x</sup>*<sup>∈</sup> *<sup>n</sup>*, *<sup>y</sup>* <sup>¼</sup> <sup>0</sup> � �*:* A fundamental property of the fractional Laplacian ð Þ �<sup>Δ</sup> *<sup>s</sup>* is its non-local character, which can be expressed as an operator that maps Dirichlet boundary conditions to a Neumann-type condition via an extension problem posed on the upper half space

$$\mathbb{R}\_+^{n+1} = \left\{ (\varkappa, \jmath) \in \mathbb{R}^{n+1} : \varkappa \in \mathbb{R}^n, \jmath > 0 \right\}.$$

The realization of the fractional Laplacian by a Dirichlet-to-Neumann map is known to Probabilists since the work [51] for any *s*, while for *s* ¼ 1 we refer to [52]. It is also widely used in the study of PDEs since the work of Caffarelli and Silvestre [39]. The authors in [39] introduced the extended problem

$$\begin{cases} \operatorname{div}(y^{1-2s}\nabla u(\mathbf{x},y)) = \mathbf{0}, & \mathbf{x} \in \mathbb{R}^n, \ y > \mathbf{0}, \\ u(\mathbf{x}, \mathbf{0}) = f(\mathbf{x}), & \mathbf{x} \in \mathbb{R}^n \end{cases} \tag{28}$$

and then showed that

$$(-\Delta)^{\sharp}f(\mathfrak{x}) = \mathcal{C}\_{\mathfrak{s}} \lim\_{\mathcal{Y} \to \mathbf{0}^{+}} \mathcal{Y}^{1-2\mathfrak{s}} u\_{\mathfrak{y}}(\mathfrak{x}, \mathfrak{y}),$$

where *Cs* > 0 is a constant depending only on *s:* The dimensional independence of *Cs* has been shown in ([39], Section 3.2) and its concrete expression can be found for instance in [38, 53],

*Stability Estimates for Fractional Hardy-Schrödinger Operators DOI: http://dx.doi.org/10.5772/intechopen.109606*

$$\mathbf{C}\_{s} = -\frac{2^{2s-1}\Gamma(s)}{\Gamma(1-s)}.\tag{29}$$

The partial differential equation in (28) is a linear degenerate elliptic equation with weight *<sup>w</sup>* <sup>¼</sup> *<sup>y</sup>*1�2*<sup>s</sup> :* Since *s*∈ð Þ 0, 1 , the weight *w* belongs to the class of the so-called Muckenhoupt *A*2-weights [54], comprising the nonnegative functions *w* defined in *<sup>n</sup>*þ<sup>1</sup> such that, for some constant *C*>0 independent of balls *B*⊂ *<sup>n</sup>*þ<sup>1</sup> ,

$$\left( |B|^{-1} \int\_B w(\mathfrak{x}, \mathfrak{y}) \, d\mathfrak{x} d\mathfrak{y} \right) \left( |B|^{-1} \int\_B w^{-1}(\mathfrak{x}, \mathfrak{y}) \, d\mathfrak{x} d\mathfrak{y} \right) < \mathbb{C}.$$

Fabes et al. [55, 56] studied systematically differential equations of divergence form with *<sup>A</sup>*2-weights, therefore we can obtain quantitative properties on ð Þ �<sup>Δ</sup> *<sup>s</sup> f* from the corresponding properties of solutions of the extension problem (28).

Regarding the operators *As*,ð Þ �<sup>Δ</sup> *<sup>s</sup>* , which are defined on bounded domains, several authors, motivated by the work in [39], have considered equivalent definitions by means of an extra auxiliary variable. Next we recall the associated extension problems for these two operators.

We start with the Dirichlet Laplacian ð Þ �<sup>Δ</sup> *<sup>s</sup>* in <sup>Ω</sup>, as defined in the introduction, which is plainly the fractional Laplacian ð Þ �<sup>Δ</sup> *<sup>s</sup>* in the whole space, of the functions supported in <sup>Ω</sup>*:* Then following [39], the fractional Laplacian ð Þ �<sup>Δ</sup> *<sup>s</sup>* is connected with the extended problem (cf. (28))

$$\begin{cases} \operatorname{div}(\boldsymbol{y}^{1-2\epsilon}\nabla\boldsymbol{u}(\boldsymbol{x},\boldsymbol{y})) = \mathbf{0}, & \text{in} \quad \mathbb{R}^{n} \times (\mathbf{0}, \infty), \\\boldsymbol{u}(\boldsymbol{x},\mathbf{0}) = \boldsymbol{f}(\boldsymbol{x}), & \boldsymbol{x} \in \mathbb{R}^{n}. \end{cases} \tag{30}$$

In particular, the so-called 2*s*�harmonic extension *u* is related to the fractional Laplacian of the original function *f* through the pointwise formula

$$(-\Delta)^{s}f(\mathbf{x}) = \mathbf{C}\_{\mathfrak{s}} \lim\_{\mathcal{V} \to \mathbf{0}^{\prime}} \mathcal{Y}^{1-2s} u\_{\mathcal{V}}(\mathbf{x}, \mathbf{y}), \qquad \forall \mathbf{x} \in \mathbb{R}^{n}, \tag{31}$$

where the constant *Cs* is given in (29).

A Dirichlet-to-Neumann mapping characterization, similar to (30)–(31), is also available for the spectral fractional Laplacian on Ω (see [36–38]), where the proper extended local problem is posed on the cylinder Ω � ð Þ 0, ∞ in place of the upper-half space. More precisely, for a function *f* ∈*C*<sup>∞</sup> <sup>0</sup> ð Þ Ω , we consider the problem

$$\begin{cases} \operatorname{div}(y^{1-2\epsilon}\nabla u(\mathbf{x},y)) = \mathbf{0}, & \text{in} \quad \Omega \times (\mathbf{0}, \infty), \\ u = \mathbf{0}, & \text{on} \quad \partial\Omega \times [\mathbf{0}, \infty), \\ u(\mathbf{x}, \mathbf{0}) = f(\mathbf{x}), & \mathbf{x} \in \Omega, \end{cases} \tag{32}$$

with Ð <sup>∞</sup> 0 Ð Ω*y*<sup>1</sup>�2*<sup>s</sup>* j j ∇*u* <sup>2</sup> *dxdy* < ∞*:* Then the extension function *u* is related to the spectral Laplacian of the original function *f* through the pointwise formula

$$(A\_{\mathfrak{J}}f)(\mathfrak{x}) = \mathbb{C}\_{\mathfrak{s}} \lim\_{\mathcal{y} \to \mathfrak{0}} y^{1-2\mathfrak{s}} u\_{\mathfrak{I}}(\mathfrak{x}, \mathfrak{y}), \qquad \forall \mathfrak{x} \in \mathfrak{Q}, \tag{33}$$

where the constant *Cs* is given by (29).

#### **4. Weighted trace hardy inequality**

An alternative proof of (8) and its improvement (26) may be given following local variational techniques exploiting the characterization of [39]. In particular, using the representation of ð Þ �<sup>Δ</sup> *<sup>s</sup>* in terms of a Dirichlet to Neumann map, we consider the proper extended local problem with test functions in *C*<sup>∞</sup> <sup>0</sup> *<sup>n</sup>*þ<sup>1</sup> � �*:* Then we can get (8) by applying, for the solution *u* ¼ *u x*ð Þ , *y* of the extended problem, the following trace Hardy inequality (cf. [57], Proposition 1)

$$H\_{n, \boldsymbol{\epsilon}} \int\_{\mathbb{R}^{n}} \frac{u^{2}(\boldsymbol{x}, \mathbf{0})}{|\boldsymbol{x}|^{2s}} \, d\boldsymbol{x} \le \int\_{0}^{\infty} \int\_{\mathbb{R}^{n}} \mathcal{Y}^{1-2s} |\nabla \boldsymbol{u}|^{2} \, d\boldsymbol{x} \, d\boldsymbol{y}, \qquad \forall \boldsymbol{u} \in C\_{0}^{\infty}(\mathbb{R}^{n+1}), \tag{34}$$

where the constant

$$H\_{n,s} = \frac{2s\Gamma^2\left(\frac{n+2s}{4}\right)\Gamma(1-s)}{\Gamma(1+s)\Gamma^2\left(\frac{n-2s}{4}\right)}\tag{35}$$

is the best possible. This argumentation has been applied by Filippas, Moschini and Tertikas [46, 58] to obtain fractional Hardy and Hardy-Sobolev inequalities involving the distance to the boundary.

In the case of bounded domains, we have

$$H\_{n,\varepsilon} \int\_{\Omega} \frac{u^2(\varkappa, \mathbf{0})}{|\varkappa|^{2\varepsilon}} \, d\varkappa \le \int\_0^\infty \int\_{\mathbb{R}^n} y^{1-2s} \left| \nabla u \right|^2 \, d\varkappa \, \, dy \tag{36}$$

for any *u*∈*C*<sup>∞</sup> <sup>0</sup> *<sup>n</sup>*þ<sup>1</sup> � � with *u x*ð Þ¼ , 0 0, *<sup>x</sup>* �∈ Ω*:* By a scaling argument it is clear that (34), (36) share the same optimal constant. Then the key estimate in deriving (26) turn out to be the sharpened versions of (34). A proof of (34) is given by the author [57], after identifying the energetic solution *ψ* ¼ *ψ*ð Þ *x*, *y* of the Euler Lagrange equations (see [57], Proposition 1)

$$\begin{cases} \operatorname{div}(\mathcal{y}^{1-2\varepsilon}\nabla\mathcal{y}) = \mathbf{0}, & \text{in } \mathbb{R}^{n+1}\_+, \\\lim\_{\mathbf{y}\to 0} \mathcal{y}^{1-2\varepsilon} \frac{\partial \nu(\mathbf{x}, \mathbf{y})}{\partial \mathbf{y}} = -H\_{n, \varepsilon} \frac{\mathcal{y}^{\varepsilon}}{|\mathbf{x}|^{2\varepsilon}}, & \text{on } \partial \mathbb{R}^{n+1}\_+ \backslash \{\mathbf{0}\}. \end{cases} \tag{37}$$

In the following, we set

*<sup>β</sup>* <sup>≔</sup> <sup>2</sup>*<sup>s</sup>* � *<sup>n</sup>* <sup>2</sup> *:*

Noticing the invariant properties of problem (37), we search for solutions of the form

$$\left|\psi(z) = \left|x\right|^{\beta}B(t), \quad x \in \mathbb{R}^{n}, \quad y \ge 0, \quad z = (x, y) \ne (0, 0) \tag{38}$$

where

$$t(\mathfrak{x}, \mathfrak{y}) := \frac{\mathfrak{y}}{|\mathfrak{x}|}.$$

Then, by direct manipulations and a normalization, we can see that problem (37) has a solution of the form (38) for the solution *B* : ½ Þ! 0, ∞ of the boundary conditions problem

$$\begin{cases} t(\mathbf{1}+t^2)\mathcal{B}''(t) + \left[ (\mathbf{3}-\mathbf{2})t^2 + (\mathbf{1}-\mathbf{2}) \right] \mathcal{B}'(t) + \frac{\beta(\mathbf{2}+n-4)}{2}t\mathcal{B}(t) = \mathbf{0}, \quad t>0, \quad (\mathbf{a}) \\ t(\mathbf{1}+t^2)\mathcal{B}''(t) = \mathbf{0}, \quad (\mathbf{a}) \end{cases}$$

$$\begin{cases} \mathcal{B}(\mathbf{0}) = \mathbf{1}, \\ \lim\_{t \to \infty} t^{-\beta} \mathcal{B}(t) \in \mathbb{R}. \end{cases} \tag{\mathbf{b}}$$

$$\lim\_{t \to \infty} t^{-\beta} B(t) \in \mathbb{R}.\tag{c}$$

$$\left(\mathbf{39}\right)$$

Let us remark that the boundary value (39b) comes from a normalization, and it plays no essential role in our subsequent analysis, contrary to condition (39c) which yields a solution of (39) with the less possible singularity. Note also that the ground state *ψ* ¼ *ψ*ð Þ *x*, *y* is well defined for *x* ¼ 0 with *y*>0, by virtue of (39b). Furthermore, it is useful to notice that (39a) is transformed into divergence form, after multiplying by *t* �2*s* ,

$$\left(t^{1-2s}(1+t^2)B'(t)\right)' + \frac{\beta(2s+n-4)}{2}t^{1-2s}B(t) = 0, \qquad t>0. \tag{40}$$

Clearly, in the special instance *n* ¼ 3 with *s* ¼ 1*=*2, problem (39) can be solved directly and more precisely, *B t*ðÞ¼ <sup>1</sup> � <sup>2</sup> *<sup>π</sup>* arctanð Þ*t :* For the general case, we perform the change of variable *z* ¼ �*t* <sup>2</sup> and then problem (39) is reduced to the boundary conditions problem for the hypergeometric equation, for the function *ω*ð Þ¼ *z B t*ð Þ,

$$\int z(1-z)\frac{d^2\alpha}{dz^2} + \left[1-\varsigma - (2-\varsigma)z\right]\frac{d\alpha}{dz} + \frac{\beta(4-n-2\varsigma)}{8}\alpha(z) = 0, \quad -\infty < z < 0, \quad (\mathbf{a})$$

$$\begin{cases} \boldsymbol{\alpha}(\mathbf{0}) = \mathbf{1}, \\ \lim\_{\mathbf{z} \to -\infty} (-\mathbf{z})^{-\beta/2} \boldsymbol{\alpha}(\mathbf{z}) \in \mathbb{R}. \end{cases} \tag{\mathbf{b}}$$

$$\lim\_{z \to -\infty} (-z)^{-\beta/2} w(z) \in \mathbb{R}.\tag{c}$$

(41)

For convenience of the reader, next we just record the properties of *B* that we shall need, and give their proof in Section 5. See also ([57], Lemma 1) and ([59], (42)–(48)). In the following, we use the notation *g* � *h* for real functions *g*, *h* to denote that *c*1*g* ≤*h*≤*c*<sup>2</sup> *g* on their domain, for some constants *c*1,*c*<sup>2</sup> >0*:*

It can be shown (see Section 5) that problem (39) has a positive decreasing solution *B* and

$$B \sim \left(\mathbf{1} + t^2\right)^{\beta/2} \qquad \text{and} \qquad B' \sim -t^{2\nu - 1} \left(\mathbf{1} + t^2\right)^{\frac{1}{2}}, \qquad \forall t > 0,\tag{42}$$

with

$$t\mathcal{B}' - \beta B(t) = O\left(t^{\beta - 2}\right), \quad a\mathfrak{s} \quad t \to \infty. \tag{43}$$

Moreover, we have

$$\lim\_{t \to 0^{+}} t^{1-2s} B'(t) = -H\_{n,s},\tag{44}$$

with the constant *Hn*,*<sup>s</sup>* given in (35). Moreover, in view of (38), we can see that

$$
\nabla \psi \cdot z = \frac{2s - n}{2} \psi(z), \quad \forall z \in \mathbb{R}\_+^{n+1} \backslash \{0\}. \tag{45}
$$

Using (42)–(44), (45), we obtain the following uniform asymptotic behavior of the ground state *ψ*; cf. ([57], Lemma 2).

**Lemma.** There holds

$$
\omega \cdot \omega \sim \left( |\varkappa|^2 + \jmath^2 \right)^{\frac{2r-n}{4}}, \quad in \quad \mathbb{R}^{n+1}\_+. \tag{46}
$$

Moreover, for *s*∈ ½ Þ 1*=*2, 1 , there holds

$$|\nabla \boldsymbol{\nu}| \sim \left( |\boldsymbol{\omega}|^2 + \boldsymbol{\mathcal{Y}}^2 \right)^{\frac{2-n-2}{4}}, \qquad \boldsymbol{in} \quad \mathbb{R}\_+^{n+1}.$$

If *s*∈ ð Þ 0, 1*=*2 , then there holds

$$|\nabla \boldsymbol{\nu}| \sim \left( |\boldsymbol{\varepsilon}|^2 + \boldsymbol{\mathcal{y}}^2 \right)^{-\frac{n+2s}{4}} \boldsymbol{\mathcal{y}}^{2s-1}, \qquad \boldsymbol{in} \quad \mathbb{R}\_+^{n+1}.$$

#### **5. Ground state**

In this section we prove the properties of the function *B* of the ground state *ψ* given in (38).

The differential eq. (41a) is a special instance of the general class of hypergeometric equations and the relevant theory of the subsequent discussion, can be found in ([60], §15), ([61], Chap. II) and ([62], §§2.1.2–2.1.5). In the following, we also refer to ([57], §3) and the Appendix of [59].

We will denote by *F*ð Þ a, b; c; *z* the hypergeometric function which is defined in the open unit disk through the series ([60], 15.1.1)

$$F(\mathbf{a}, \mathbf{b}; \mathbf{c}; z) = \sum\_{k=0}^{\infty} \frac{(\mathbf{a})\_k (\mathbf{b})\_k}{(\mathbf{c})\_k} \frac{z^k}{k!} \tag{47}$$

and then by analytic continuation into n½ Þ 1, ∞ *:* In (45) we set að Þ*<sup>k</sup>* ¼ a að Þ þ 1 ⋯ð Þ a þ *k* � 1 and að Þ<sup>0</sup> ¼ 1*:* It is clear that

$$F(\mathbf{a}, \mathbf{b}; \mathbf{c}; z) = F(\mathbf{b}, \mathbf{a}; \mathbf{c}; z).$$

We consider the hypergeometric differential equation

$$z(\mathbf{1} - \mathbf{z})\boldsymbol{\alpha}^{\prime}(\mathbf{z}) + \left[\mathbf{c} - (\mathbf{a} + \mathbf{b} + \mathbf{1})\mathbf{z}\right]\boldsymbol{\alpha}^{\prime}(\mathbf{z}) - \mathbf{a}\mathbf{b}\boldsymbol{\alpha}(\mathbf{z}) = \mathbf{0} \tag{48}$$

for complex functions *ω* ¼ *ω*ð Þ*z* with *z*∈ , and real parameters a, b, c satisfying the conditions

$$\mathbf{c} - \mathbf{a} - \mathbf{b} \ge \mathbf{0}, \qquad \mathbf{b} > \mathbf{0}, \qquad \mathbf{c} > \mathbf{0}.\tag{49}$$

By formulae ([60], 15.5.3, 15.5.4), we have the following expression for the (general) solution of (48), defined in n½ Þ 1, ∞ ,

$$\rho\_{\mathbf{b}}(\mathbf{z}) = \mathbb{C}\_1 F(\mathbf{a}, \mathbf{b}; \mathbf{c}; \mathbf{z}) + \mathbb{C}\_2 z^{1-\mathbf{c}} F(\mathbf{a} - \mathbf{c} + \mathbf{1}, \mathbf{b} - \mathbf{c} + \mathbf{1}; \mathbf{2} - \mathbf{c}; \mathbf{z}) \tag{50}$$

with any *C*1,*C*<sup>2</sup> ∈ *:* Let us next derive an explicit formula for the analytic continuation of the series (47) into the domain f g *z*∈ :j*z*j> 1, *z* �∈ð Þ 1, ∞ *:* To this end, we consider ∣*z*∣>1 with *z* �∈ð Þ 1, ∞ and we discriminate among four cases, depending on *n*, *s*, as follows.

We begin with the case that all of the three numbers a, c � b, and a � b are different from any non-positive integer *m* ¼ 0, � 1, � 2, … *:* Then by expression ([60], 15.3.7) we get

$$\begin{split} F(\mathbf{a}, \mathbf{b}; \mathbf{c}; z) &= \frac{\Gamma(\mathbf{c}) \Gamma(\mathbf{b} - \mathbf{a})}{\Gamma(\mathbf{b}) \Gamma(\mathbf{c} - \mathbf{a})} (-z)^{-\mathbf{a}} F\left(\mathbf{a}, \mathbf{a} - \mathbf{c} + \mathbf{1}; \mathbf{a} - \mathbf{b} + \mathbf{1}; \frac{\mathbf{1}}{z}\right) \\ &+ \frac{\Gamma(\mathbf{c}) \Gamma(\mathbf{a} - \mathbf{b})}{\Gamma(\mathbf{a}) \Gamma(\mathbf{c} - \mathbf{b})} (-z)^{-\mathbf{b}} F\left(\mathbf{b}, \mathbf{b} - \mathbf{c} + \mathbf{1}; \mathbf{b} - \mathbf{a} + \mathbf{1}; \frac{\mathbf{1}}{z}\right). \end{split} \tag{51}$$

As for the case of a ¼ b 6¼ �*m*, ∀*m* ¼ 0, � 1, � 2, … , and c � a 6¼ *l*, for any *l* ¼ 1, 2, … , we have, by ([60], 15.3.13),

$$F(\mathbf{a}, \mathbf{a}; \mathbf{c}; \mathbf{z}) = \frac{\Gamma(\mathbf{c})(-\mathbf{z})^{-\mathbf{a}}}{\Gamma(\mathbf{a})\Gamma(\mathbf{c} - \mathbf{a})} \sum\_{k=0}^{\bullet} \frac{(\mathbf{a})\_k (\mathbf{1} - \mathbf{c} + \mathbf{a})\_k}{(k!)^2} \mathbf{z}^{-k} [\ln(-\mathbf{z}) + 2\Psi(k + \mathbf{1}) - \Psi(\mathbf{a} + k) - \Psi(\mathbf{c} - \mathbf{a} - k)],\tag{52}$$

where we set <sup>Ψ</sup>ð Þ¼� *<sup>z</sup> <sup>γ</sup>* � <sup>P</sup><sup>∞</sup> *k*¼0 1 *<sup>z</sup>*þ*<sup>k</sup>* � <sup>1</sup> *k*þ1 � � with the so-called Euler's constant *γ* ≈0*:*5772156649*:*

Let us next proceed with the case where b � a ¼ *m*, *m* ¼ 1, 2, … , and a 6¼ �*k*, for any *k* ¼ 0,1,2, … *:* Firstly, if c � a 6¼ *l*, for any *l* ¼ 1, 2, … , then the formula ([60], 15.3.14) yields

$$F(\mathbf{a}, \mathbf{a} + m; \mathbf{c}; x) = \frac{\Gamma(\mathbf{c})(-x)^{-\mathbf{a} - m}}{\Gamma(\mathbf{a} + m)\Gamma(\mathbf{c} - \mathbf{a})} \sum\_{k=0}^{\bullet \text{e}} \frac{(\mathbf{a})\_{k+m}(\mathbf{1} - \mathbf{c} + \mathbf{a})\_{k+m}}{(k+m)!k!} x^{-k} [\ln(-x) + \Psi(\mathbf{1} + m + k) + \Psi(\mathbf{1} + k)], \tag{51}$$

$$-\Psi(\mathbf{a} + m + k) - \Psi(\mathbf{c} - \mathbf{a} - m - k)] + (-x)^{-1} \frac{\Gamma(\mathbf{c})}{\Gamma(\mathbf{a} + m)} \sum\_{k=0}^{m-1} \frac{\Gamma(m - k)(\mathbf{a})\_k}{k!\Gamma(\mathbf{c} - \mathbf{a} - k)} x^{-k}. \tag{53}$$

Otherwise, if c � a ¼ *l*, for some *l* ¼ 1, 2, … , such that *l* > *m*, then we get from formula ([61], (19) in §2.1.4),

$$F(\mathbf{a}, \mathbf{a} + m; \mathbf{a} + l; x) = \frac{\Gamma(\mathbf{a} + l)}{\Gamma(\mathbf{a} + m)} (-x)^{-k} \left[ (-\mathbf{1})^l (-x)^{-m} \sum\_{k = l - m}^{\mathbf{a}} \frac{(\mathbf{a})\_{k + m} (k + m - l)!}{(k + m)! k!} x^{-k} \right] \tag{54}$$

$$+ \sum\_{k = 0}^{m - 1} \frac{(m - k - 1)! (\mathbf{a})\_k}{(l - k - 1)! k!} x^{-k} + \frac{(-x)^{-m}}{(l - 1)!} \sum\_{k = 0}^{l - m - 1} \frac{(\mathbf{a})\_{k + m} (1 - l)\_{k + m}}{(k + m)! k!} x^{-k} \times$$

$$\times \left[ \ln(-x) + \Psi(\mathbf{1} + m + k) + \Psi(\mathbf{1} + k) - \Psi(\mathbf{a} + m + k) - \Psi(l - m - k) \right].$$

We conclude with the case that some of the parameters a or c � b equals a nonpositive integer. In this case, *F*ð Þ a, b; c; *z* is an elementary function of *z:* In particular, if a ¼ �*m* for some *m* ¼ 0,1,2, … then, ([60], 15.4.1), the hypergeometric series in (47) is the polynomial

$$F(-m, \mathbf{b}; \mathbf{c}; z) = \sum\_{k=0}^{m} \frac{(-m)\_k (\mathbf{b})\_k}{(\mathbf{c})\_k} \frac{z^k}{k!}. \tag{55}$$

Otherwise, if c � b ¼ �*l*, for some *l* ¼ 0,1,2, … , then from formula ([60], 15.3.3), *F*ð Þ a, b; c; *z* is given by

$$F(\mathbf{a}, \mathbf{b}; \mathbf{c}; z) = (\mathbf{1} - z)^{-\mathbf{a} - l} F(\mathbf{c} - \mathbf{a}, -l; \mathbf{c}; z) \tag{56}$$

and notice by (55) that the hypergeometric function of the right side is a polynomial of degree *l:*

In the following, we will also use the differentiation formula ([60], 15.2.1), that is

$$\frac{d}{dz}F(\mathbf{a}, \mathbf{b}; \mathbf{c}; z) = \frac{\mathbf{a}\mathbf{b}}{\mathbf{c}}\ \ F(\mathbf{a} + \mathbf{1}, \mathbf{b} + \mathbf{1}; \mathbf{c} + \mathbf{1}; z). \tag{57}$$

Let us now proceed to prove that *B* is positive and monotone, and also derive the asymptotics (42)–(44). To simplify the presentation, we set

$$\begin{aligned} a\_1 &= \frac{4-n-2s}{4}, a\_2 = a\_1 - c\_1 + 1 = \frac{4-n+2s}{4}, c\_1 = 1-s, \\ b\_1 &= -\frac{\beta}{2} = \frac{n-2s}{4}, b\_2 = b\_1 - c\_1 + 1 = \frac{n+2s}{4}, c\_2 = 2-c\_1 = 1+s. \end{aligned}$$

For these values, and recalling the assumption *n*> 2*s* with 0 <*s*<1, it is easily seen that the parameters f g *a*1, *b*1,*c*<sup>1</sup> and f g *a*2, *b*2,*c*<sup>2</sup> , satisfy the assumptions (49), so we can apply the aforementioned formulas. The first main step is to get an explicit expression of *B t*ðÞ¼ *ω*ð Þ*z :* In view of (50) the general solution of (41a) is given by

$$u(z) = \mathbb{C}\_1 F(a\_1, b\_1; c\_1; z) + \mathbb{C}\_2 (-z)^{1-c\_1} F(a\_2, b\_2; c\_2; z), \qquad z \le 0,\tag{58}$$

for certain constants *C*1,*C*2*:* We apply (41b) to (58), and take into account that *F a*ð Þ¼ 1, *b*1;*c*1; 0 *F a*ð Þ¼ 2, *b*2;*c*2; 0 1, to get that *C*<sup>1</sup> ¼ 1*:*

The constant *C*<sup>2</sup> will be determined by the condition at ∞, and to this aim we will get an expression for *ω*ð Þ*z* for *z*< � 1*:* By considering separately the cases for *n*, *s*, corresponding to the formulas (51)–(56), which give the explicit expression for the hypergeometric functions in (58), we get, in all instances, that

$$\mathcal{C}\_2 = -\frac{\Gamma(c\_1)\Gamma(b\_2)\Gamma(c\_2 - a\_2)}{\Gamma(c\_2)\Gamma(b\_1)\Gamma(c\_1 - a\_1)},\tag{59}$$

and the asymptotics

$$
\rho a(z) = O\left( (-z)^{-b\_1} \right), \quad a\mathfrak{s} \quad z \to -\infty. \tag{60}
$$

In order to determine the limit

$$H\_{n, \iota} := -\lim\_{t \to 0^+} t^{1-2\iota} B'(t) = 2 \lim\_{z \to 0^-} (-z)^{1-\iota} w'(z)$$

we differentiate (58) and using (57) we obtain

$$\begin{aligned} \alpha'(z) &= \frac{a\_1 b\_1}{c\_1} F(a\_1 + \mathbf{1}, b\_1 + \mathbf{1}; c\_1 + \mathbf{1}; z) - \mathbf{C}\_2 \varsigma(-z)^{\iota - 1} F(a\_2, b\_2; c\_2; z) \\ &+ \mathbf{C}\_2 \frac{a\_2 b\_2}{c\_2} \left(-z\right)^{\iota} F(a\_2 + \mathbf{1}, b\_2 + \mathbf{1}; c\_2 + \mathbf{1}; z) \end{aligned}$$

and then let *z* ! 0� to get

$$H\_{n, \mathfrak{s}} = 2 \lim\_{\mathbf{z} \to \mathbf{0}^-} (-\mathbf{z})^{1-\mathfrak{s}} a'(\mathbf{z}) = -2 \mathbf{s} \mathbf{C}\_2$$

and taking into account (59) we obtain (44).

Let us next show that *B* is decreasing and positive. We first assume that 4 � *n* � 2*s*< 0*:* In this case, the positivity of *B* follows from the fact that if there exist *t*<sup>0</sup> > 0 such that *B t*ð Þ¼ <sup>0</sup> 0, then since lim*<sup>t</sup>*!<sup>∞</sup> *B t*ðÞ¼ 0, there exists *tm* <sup>&</sup>gt; *<sup>t</sup>*<sup>0</sup> where *<sup>B</sup>* attains local non-negative maximum or local non-positive minimum which disagree with the differential eq. (39a). Therefore *B* is positive and the same argument shows that *B* is decreasing.

For the case that 4 � *n* � 2*s* ≥0, we perform the transformation *g t*ðÞ¼ 1 þ *t* <sup>2</sup> ð Þ*<sup>b</sup>*<sup>1</sup> *B t*ð Þ which reduces (39) to the problem

$$\begin{cases} t(\mathbf{1} + t^2)^2 g''(t) + [1 - 2s + (3 - n)t^2](\mathbf{1} + t^2)g'(t) - \beta^2 t g(t) = \mathbf{0}, & t > \mathbf{0}, \quad (\mathbf{a})\\ g(\mathbf{0}) = \mathbf{1}, & (\mathbf{b}) \\ \lim\_{t \to \mathbf{0}} g(t) \in \mathbb{R}. \end{cases} \quad \text{(b)} \quad \text{(c)}$$

One can verify condition (61c) directly from the explicit formula of *B t*ðÞ¼ *ω*ð Þ*z :* Then, by a standard minimum principle argumentation for the boundary conditions problem (61), we can verify that *g* is not negative, and as a consequence *B* is nonnegative. Then the fact that *B* is monotone and positive follows from (40) together with the negativity of the derivative of *B* near the origin.

To show the asymptotics for *B* in (42), we use conditions (39b)-(39c) taking into account that *B* is positive, and to show the asymptotics of *B*<sup>0</sup> in (42), we differentiate the expression (58) exploiting (57).

To conclude, it is straightforward to show (43) by substituting the concrete expression for *B t*ðÞ¼ *ω* �*t* <sup>2</sup> ð Þ through the corresponding formulas (depending on the parameters *n*, *s*) and the *B*<sup>0</sup> *:*

*Fixed Point Theory and Chaos*

### **Author details**

Konstantinos Tzirakis Institute of Applied and Computational Mathematics, FORTH, Heraklion, Greece

\*Address all correspondence to: kostas.tzirakis@gmail.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bulletin des Sciences Mathematiques. 2012;**136**(5):521-573

[2] Daoud M, Laamri EH. Fractional Laplacians: A short survey. Discrete & Continuous Dynamical Systems-S. 2022; **15**(1):95-116

[3] Duo S, Wang H, Zhang Y. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete & Continuous Dynamical Systems-B. 2019;**24**(1): 231-256

[4] Lischke A, Pang G, Gulian M, Song F, Glusa C, Zheng X, et al. What is the fractional Laplacian? A comparative review with new results. Journal of Computational Physics. 2020;**404**:109009

[5] Vázquez JL. The mathematical theories of diffusion: Nonlinear and fractional diffusion. In: Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Cham, Switzerland: Springer International Publishing AG; 2017. pp. 205-278

[6] Dalibard A-L, Gérard-Varet D. On shape optimization problems involving the fractional Laplacian. ESAIM. 2013; **19**:976-1013

[7] Laskin N. Fractional quantum mechanics and Lévy path integrals. Physics Letters A. 2000;**268**(4–6): 298-305

[8] Laskin N. Fractional Schrödinger equation. Physical Review E. 2002;**66**: 056108

[9] Laskin N. Fractional Quantum Mechanics. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd.; 2018 [10] Massaccesi A, Valdinoci E. Is a nonlocal diffusion strategy convenient for biological populations in competition? Journal of Mathematical Biology. 2017;**74**:113-147

[11] Bates PW. On some nonlocal evolution equations arising in materials science. In: Nonlinear Dynamics and Evolution Equations. Vol. 48, Amer. Math. Soc. Providence, RI: Fields Inst. Commun; 2006. pp. 13-52

[12] Cont R, Tankov P. Financial Modelling with Jump Processes. Boca Raton, FL: Chapman & Hall/CRC Financial Mathematics Series; 2004

[13] Schoutens W. Lévy Processes in Finance: Pricing Financial Derivatives. New York: Wiley; 2003

[14] Levendorski SZ. Pricing of the American put under Lévy processes. International Journal of Theory & Applied Finance. 2004;**7**(3):303-335

[15] Gilboa G, Osher S. Nonlocal operators with applications to image processing. Multiscale Modeling and Simulation. 2008;**7**:1005-1028

[16] Caffarelli L. Non-local diffusions, drifts and games. In: Nonlinear partial differential equations, Abel Symp. Vol. 7. Heidelberg: Springer; 2012. pp. 37-52

[17] Ros-Oton X. Nonlocal elliptic equations in bounded domains: A survey. Publicacions Matemàtiques. 2016;**60**:3-26

[18] Danielli D, Salsa S. Obstacle problems involving the fractional Laplacian. In: Recent Developments in Nonlocal Theory. Poland: De Gruyter Open Poland; 2018. pp. 81-164

[19] González M. Recent Progress on the fractional Laplacian in conformal geometry. In: Palatucci G, Kuusi T, editors. Recent Developments in Nonlocal Theory. Warsaw, Poland: De Gruyter Open Poland; 2017. pp. 236-273

[20] Applebaum D. Lévy processes and stochastic calculus. In: Cambridge Studies in Advanced Mathematics. Second ed. Vol. 116. Cambridge, UK: Cambridge University Press; 2009

[21] Bertoin J. Lévy Processes. In: Cambridge Tracts in Mathematics. Vol. 121. Cambridge: Cambridge University Press; 1996

[22] Bogdan K, Burdzy K, Chen Z-Q. Censored stable processes. Probability Theory and Related Fields. 2003;**127**: 89-152

[23] Stein EM. Singular integrals and differentiability properties of functions. In: Princeton Mathematical Series. Vol. 30. Princeton: Princeton University Press; 1970

[24] Landkof NS. Foundations of modern potential theory, translated from the Russian by. In: Doohovskoy AP, editor. Die Grundlehren der mathematischen Wissenschaften. Vol. Band 180. New York-Heidelberg: Springer-Verlag; 1972

[25] Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Communications on Pure and Applied Mathematics. 2007; **60**(1):67-112

[26] Kwaśnicki M. Ten equivalent definitions of the fractional Laplace operator. Fractional Calculas and Applied Analysis. 2017;**20**(1):7-51

[27] Aronszajn N, Smith KT. Theory of Bessel potentials I. Annals of the Fourier Institute. 1961;**11**:385-475

[28] Cotsiolis A, Travoularis NK. Best constants for Sobolev iequalities for higher order fractional derivatives. Journal of Mathematical Analysis and Applications. 2004;**295**:225-236

[29] Lieb EH. Sharp constants in the hardy-Littlewood-Sobolev and related inequalities. Annals of Mathematics. 1983;**118**:349-374

[30] Almgren FJ, Lieb EH. Symmetric decreasing rearrangement is sometimes continuous. Journal of the American Mathematical Society. 1989;**2**(4):683-773

[31] Yang J. Fractional hardy-Sobolev inequality in *<sup>N</sup>*. Nonlinear Analysis. 2015;**119**:179-185

[32] Stein EM, Weiss G. Fractional integrals on *n*-dimensional Euclidean space. Journal of Mathematics and Mechanics On JSTOR. 1958;**7**:503-514

[33] Herbst IW. Spectral theory of the operator *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> ð Þ<sup>1</sup>*=*<sup>2</sup> � *Ze*<sup>2</sup> *=r*. Communications in Mathematical Physics. 1977;**53**(3):255-294

[34] Beckner W. Pitt's inequality and the uncertainty principle. Proceedings of the American Mathematical Society. 1995; **123**(1):1897-1905

[35] Yafaev D. Sharp constants in the hardy-Rellich inequalities. Journal of Functional Analysis. 1999;**168**(1): 121-144

[36] Cabre X, Tan J. Positive solutions of nonlinear problems involving the square root of the Laplacian. Advances in Mathematics. 2010;**224**(5):2052-2093

[37] Capella A, Davila J, Dupaigne L, Sire Y. Regularity of radial extremal solutions for some non-local semilinear equations. Communications in Partial

*Stability Estimates for Fractional Hardy-Schrödinger Operators DOI: http://dx.doi.org/10.5772/intechopen.109606*

Differential Equations. 2011;**36**(8): 1353-1384

[38] Stinga PR, Torrea JL. Extension problem and Harnack's inequality for some fractional operators. Communications in Partial Differential Equations. 2010;**35**(11):2092-2122

[39] Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Communications in Partial Differential Equations. 2007;**32**: 1245-1260

[40] Frank RL, Lieb EH, Seiringer R. Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. Journal of the American Mathematical Society. 2008;**21**(4):925-950

[41] Fall MM. Semilinear elliptic equations for the fractional Laplacian with hardy potential. Nonlinear Analysis. 2020;**193**:111311

[42] Abdellaoui B, Peral I, Primo A. A remark on the fractional hardy inequality with a remainder term. Proceedings of the Academy of Sciences Series I. 2014;**352**:299-303

[43] Tzirakis K. Sharp trace hardy-Sobolev inequalities and fractional hardy-Sobolev inequalities. Journal of Functional Analysis. 2016;**270**:413-439

[44] Adimurthi S, Filippas A. Tertikas, on the best constant of hardy Sobolev inequalities. Nonlinear Analysis. 2009; **70**:2826-2833

[45] Filippas S, Tertikas A. Optimizing improved hardy inequalities. Journal of Functional Analysis. 2002;**192**(1): 186-233

[46] Filippas S, Moschini L, Tertikas A. Sharp trace hardy-Sobolev-Mazya inequalities and the fractional Laplacian. Archive for Rational Mechanics and Analysis. 2013;**208**:109-161

[47] Psaradakis G, Spector D. A Leray-Trudinger inequality. Journal of Functional Analysis. 2015;**269**(1): 215-228

[48] Abdellaoui B, Bentifour R. Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications. Journal of Functional Analysis. 2017;**272**:3998-4029

[49] Nguyen H-M, Squassina M. Fractional Caffarelli-Kohn-Nirenberg inequalities. Journal of Functional Analysis. 2018;**274**:2661-2672

[50] Servadei R, Valdinoci E. On the spectrum of two different fractional operators. Proceedings of the Royal Society of Edinburgh. 2014;**144**:831-855

[51] Molchanov SA, Ostrovskii E. Symmetric stable processes as traces of degenerate diffusion processes. Theory of Probability and its Applications. 1969; **14**:128-131

[52] Spitzer F. Some theorems concerning 2-dimensional Brownian motion. Transactions of the American Mathematical Society. 1958;**87**:187-197

[53] Cabré X, Sire Y. Non-linear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates. Annals of the Institut Henri Poincaré C, Nonlinear Analysis. 2014;**31**:23-53

[54] Muckenhoupt B. Weighted norm inequalities for the hardy maximal function. Transactions of the American Mathematical Society. 1972;**165**:207-226

[55] Fabes EB, Kenig CE, Serapioni RP. The local regularity of solutions of degenerate elliptic equations.

Communications in Partial Differential Equations. 1982;**7**(1):77-116

[56] Fabes E, Jerison D, Kenig C. The wiener test for degenerate elliptic equations. Annals of the Fourier Institute. 1982;**32**(3):151-182

[57] Tzirakis K. Improving interpolated hardy and trace hardy inequalities on bounded domains. Nonlinear Analysis. 2015;**127**:17-34

[58] Filippas S, Moschini L, Tertikas A. Trace hardy-Sobolev-Maz'ya inequalities for the half fractional Laplacian. Communications on Pure and Applied Analysis. 2015;**14**(2):373-382

[59] Tzirakis K. Series expansion of weighted Finsler-Kato-hardy inequalities. Nonlinear Analysis. 2022; **222**:113016

[60] Abramowitz M, Stegun IA. Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables. New York: Dover Publicationss, Inc.; 1992

[61] Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG. Higher Higher Transcendental Functions. Vol. 1. New York: McGraw-Hill Book Company; 1953

[62] Polyanin AD, Zaitsev VF. Handbook for Exact Solutions for Ordinary Differential Equations. New York: Chapman & Hall/CRC; 2003

*Edited by Guillermo Huerta-Cuellar*

The field of mathematics has produced many beautiful results, and among them, fixed points hold great importance as they can be used to describe and explain various mathematical curiosities, even those of the most complex technical applications. This book explores some of the latest and most fascinating results on the study of fixed points from a scientific perspective and presents new approaches to studying them along with possible applications.

Published in London, UK © 2023 IntechOpen © skrotov / iStock

Fixed Point Theory and Chaos

Fixed Point Theory

and Chaos

*Edited by Guillermo Huerta-Cuellar*