**3. Some integral inequalities**

#### **3.1 On Minkowski and Hermite-Hadamard fractional inequalities**

In this subsection, we present some fractional integral results related to Minkowski and Hermite-Hadamard integral inequalities. For more details, we refer the reader to [21].

**Theorem 1.9.** Let *α* >0, *p* ≥1 and let *f*, *g* be two positive functions on ½0, ∞½, such that for all *t* >0, *J αf p* ð Þ*t* < ∞, *J <sup>α</sup>gp*ð Þ*<sup>t</sup>* <sup>&</sup>lt; <sup>∞</sup>*:* If 0 <sup>&</sup>lt; *<sup>m</sup>* <sup>≤</sup> *<sup>f</sup>*ð Þ*<sup>τ</sup> <sup>g</sup>*ð Þ*<sup>τ</sup>* <sup>≤</sup> *<sup>M</sup>*, *<sup>τ</sup>* <sup>∈</sup> ½ � 0, *<sup>t</sup>* , then we have

$$[f^a f^p(t)]^{\\\frac{1}{p}} + [f^a g^p(t)]^{\\\frac{1}{p}} \le \frac{1 + M(m+2)}{(m+1)(M+1)} [f^a (f+g)^p(t)]^{\\\frac{1}{p}}.\tag{10}$$

Proof: We use the hypothesis *<sup>f</sup>*ð Þ*<sup>τ</sup> <sup>g</sup>*ð Þ*<sup>τ</sup>* <sup>&</sup>lt; *<sup>M</sup>*, *<sup>τ</sup>* <sup>∈</sup>½ � 0, *<sup>t</sup>* , *<sup>t</sup>* <sup>&</sup>gt;0*:* We can write

$$\begin{aligned} \frac{(\mathcal{M}+1)^p}{\Gamma(\alpha)} \int\_0^t (t-\tau)^{\alpha-1} f^p(\tau) d\tau\\ \leq \frac{\mathcal{M}^p}{\Gamma(\alpha)} \int\_0^t (t-\tau)^{\alpha-1} (f+g)^p(\tau) d\tau. \end{aligned} \tag{11}$$

*J αf <sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

[26] on 0, ½ �*t :*

have

f g 1, 2, … , *n* .

Taking

Proof: It is clear that

*Kp*ð Þ *<sup>τ</sup>*, *<sup>ρ</sup>* <sup>≔</sup> ð Þ *<sup>t</sup>* � *<sup>τ</sup> <sup>α</sup>*�<sup>1</sup>

we observe that

Also, we have

**153**

Γð Þ *α*

following result.

tions on 0, ½ ∞½*:* If *f*

*<sup>p</sup>* þ *J*

*DOI: http://dx.doi.org/10.5772/intechopen.91140*

*p*

*<sup>α</sup>g<sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

*<sup>p</sup>* ≤

*Integral Inequalities and Differential Equations via Fractional Calculus*

<sup>2</sup>�*p*�*q*ð Þ *<sup>f</sup>*ð Þþ <sup>0</sup> *f t*ð Þ *<sup>p</sup>*

≤ *J α t <sup>α</sup>*�1*f <sup>p</sup>* ð Þ ð Þ*<sup>t</sup> <sup>J</sup>*

**3.2 A family of fractional integral inequalities**

functions on ½ � *a*, *b :* Then, the following inequality

h i

ð Þ *<sup>ρ</sup>* � *<sup>a</sup> <sup>δ</sup>* � ð Þ *<sup>τ</sup>* � *<sup>a</sup> <sup>δ</sup>* � � *<sup>f</sup>*

> Y*n i*¼1 *f γi*

Theorem 1.15. Suppose that *fi*

*J <sup>α</sup>* Q*<sup>n</sup> <sup>i</sup>*6¼*<sup>p</sup> <sup>f</sup> γi i f β <sup>p</sup>*ð Þ*t*

*J <sup>α</sup>* Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi <sup>i</sup>* ð Þ*<sup>t</sup>* � � <sup>≥</sup>

1 þ *M m*ð Þ þ 2 ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> ð Þ *<sup>M</sup>* <sup>þ</sup> <sup>1</sup> *<sup>J</sup>*

Remark 1.12. Taking *α* ¼ 1 in this second theorem, we obtain Theorem 2.2 in

Using the notions of concave and *<sup>L</sup>p*�functions, we present to the reader the

Theorem 1.13. Suppose that *α* > 0, *p* >1, *q* >1 and let *f*, *g* be two positive func-

, *<sup>g</sup><sup>q</sup>* are two concave functions on 0, <sup>½</sup> <sup>∞</sup>½, then we have

ð Þ *<sup>g</sup>*ð Þþ <sup>0</sup> *g t*ð Þ *<sup>q</sup> <sup>J</sup>*

*<sup>α</sup>*�1*g<sup>q</sup>* ð Þ ð Þ*<sup>t</sup> :*

*α t*

Lemma 1.14. Let *h* be a concave function on ½ � *a*, *b :* Then for any *x*∈½ � *a*, *b* , we

*h a*ð Þþ *h b*ð Þ<sup>≤</sup> *h b*ð Þþ <sup>þ</sup> *<sup>a</sup>* � *<sup>x</sup> h x*ð Þ<sup>≤</sup> <sup>2</sup>*<sup>h</sup> <sup>a</sup>* <sup>þ</sup> *<sup>b</sup>*

We present to the reader some integral results for a family of functions [22].

*J*

*J*

holds for any *a*< *t*≤ *b*, *α* >0, *δ* >0, *β* ≥*γ<sup>p</sup>* >0, where *p* is a fixed integer in

*β*�*γ<sup>p</sup> <sup>p</sup>* ð Þ� *τ f*

*<sup>δ</sup>* � ð Þ *<sup>τ</sup>* � *<sup>a</sup> <sup>δ</sup>* � � *<sup>f</sup>*

for any fixed *p*∈f g 1, … *n* and for any *β* ≥ *γ<sup>p</sup>* >0, *δ* >0, *τ*, *ρ*∈½ � *a*, *t* ; *a*< *t*≤ *b:*

*<sup>i</sup>* ð Þ*τ* ð Þ *ρ* � *a*

*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup>*

*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup>*

*<sup>δ</sup>*Q*<sup>n</sup> <sup>i</sup>*6¼*<sup>p</sup> <sup>f</sup> γi i f β <sup>p</sup>*ð Þ*t*

h i

*<sup>δ</sup>*Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi <sup>i</sup>* ð Þ*t* h i (19)

> *β*�*γ<sup>p</sup> <sup>p</sup>* ð Þ*ρ*

These results generalize some integral inequalities of [27]. We have

� �

The proof of this theorem is based on the following auxiliary result.

*<sup>α</sup>*ð Þ *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup> <sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

*α t <sup>α</sup>*�<sup>1</sup> ð Þ ð Þ <sup>2</sup>

2

*<sup>i</sup>*¼1, … *<sup>n</sup>* are *<sup>n</sup>* positive, continuous, and decreasing

� �≥0, (20)

*β*�*γ<sup>p</sup> <sup>p</sup>* ð Þ� *τ f*

*Kp*ð Þ *τ*, *ρ* ≥0*:* (22)

*β*�*γ<sup>p</sup> <sup>p</sup>* ð Þ*ρ* � �, (21)

� �*:* (18)

*<sup>p</sup>:* (16)

(17)

Hence, we have

$$J^a f^p(t) \le \frac{M^p}{(M+1)^p} J^a (f+g)^p(t). \tag{12}$$

Thus, it yields that

*J αf <sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup> *<sup>p</sup>* ≤ *M <sup>M</sup>* <sup>þ</sup> <sup>1</sup> *<sup>J</sup> <sup>α</sup>*ð Þ *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup> <sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup> *<sup>p</sup>:* (13)

In the same manner, we have

$$\mathbf{g}\left(\mathbf{1} + \frac{\mathbf{1}}{m}\right)\mathbf{g}(\tau) \le \frac{1}{m} \left(f(\tau) + \mathbf{g}(\tau)\right). \tag{14}$$

And then,

$$\mathbb{E}\left[J^{a}\mathcal{g}^{p}(t)\right]^{\frac{1}{p}} \leq \frac{1}{m+1} \left[J^{a}(f+\mathcal{g})^{p}(t)\right]^{\frac{1}{p}}.\tag{15}$$

Combining (13) and (15), we achieve the proof.

Remark 1.10. Applying the above theorem for *α* ¼ 1, we obtain Theorem 1.2 of [25] on 0, ½ �*t :*

With the same arguments as before, we present the following theorem.

Theorem 1.11. Let *α* >0, *p*≥ 1 and let *f*, *g* be two positive functions on 0, ½ ∞½, such that for all *t* >0, *J αf p* ð Þ*t* < ∞, *J <sup>α</sup>g<sup>p</sup>*ð Þ*<sup>t</sup>* <sup>&</sup>lt; <sup>∞</sup>*:* If 0 <sup>&</sup>lt; *<sup>m</sup>* <sup>≤</sup> *<sup>f</sup>*ð Þ*<sup>τ</sup> <sup>g</sup>*ð Þ*<sup>τ</sup>* <sup>≤</sup> *<sup>M</sup>*, *<sup>τ</sup>* <sup>∈</sup>½ � 0, *<sup>t</sup>* , then we have

*Integral Inequalities and Differential Equations via Fractional Calculus DOI: http://dx.doi.org/10.5772/intechopen.91140*

$$\left[\left[f^a f^p(t)\right]^{\frac{1}{p}} + \left[f^a g^p(t)\right]^{\frac{1}{p}}\right] \le \frac{1 + M(m+2)}{(m+1)(M+1)} [f^a (f+g)^p(t)]^{\frac{1}{p}}.\tag{16}$$

Remark 1.12. Taking *α* ¼ 1 in this second theorem, we obtain Theorem 2.2 in [26] on 0, ½ �*t :*

Using the notions of concave and *<sup>L</sup>p*�functions, we present to the reader the following result.

Theorem 1.13. Suppose that *α* > 0, *p* >1, *q* >1 and let *f*, *g* be two positive functions on 0, ½ ∞½*:* If *f p* , *<sup>g</sup><sup>q</sup>* are two concave functions on 0, <sup>½</sup> <sup>∞</sup>½, then we have

$$\begin{aligned} \mathcal{D}^{-p-q}(f(\mathbf{0}) + f(t))^p (\mathbf{g}(\mathbf{0}) + \mathbf{g}(t))^q (f^a(t^{a-1}))^2 \\ \leq & J^a(t^{a-1}f^p(t))J^a(t^{a-1}\mathbf{g}^q(t)). \end{aligned} \tag{17}$$

The proof of this theorem is based on the following auxiliary result.

Lemma 1.14. Let *h* be a concave function on ½ � *a*, *b :* Then for any *x*∈½ � *a*, *b* , we have

$$h(a) + h(b) \le h(b + a - \varkappa) + h(\varkappa) \le 2h\left(\frac{a + b}{2}\right). \tag{18}$$

#### **3.2 A family of fractional integral inequalities**

We present to the reader some integral results for a family of functions [22]. These results generalize some integral inequalities of [27]. We have

Theorem 1.15. Suppose that *fi* � � *<sup>i</sup>*¼1, … *<sup>n</sup>* are *<sup>n</sup>* positive, continuous, and decreasing functions on ½ � *a*, *b :* Then, the following inequality

$$\frac{J^a\left[\prod\_{i\neq p}^n f\_i^{\gamma\_i} f\_p^{\beta}(t)\right]}{J^a\left[\prod\_{i=1}^n f\_i^{\gamma\_i}(t)\right]} \ge \frac{J^a\left[(t-a)^\delta \prod\_{i\neq p}^n f\_i^{\gamma\_i} f\_p^{\beta}(t)\right]}{J^a\left[(t-a)^\delta \prod\_{i=1}^n f\_i^{\gamma\_i}(t)\right]}\tag{19}$$

holds for any *a*< *t*≤ *b*, *α* >0, *δ* >0, *β* ≥*γ<sup>p</sup>* >0, where *p* is a fixed integer in f g 1, 2, … , *n* .

Proof: It is clear that

$$\left(\left(\rho-\mathfrak{a}\right)^{\delta}-\left(\tau-\mathfrak{a}\right)^{\delta}\right)\left(f\_{p}^{\beta-\gamma\_{p}}\left(\tau\right)-f\_{p}^{\beta-\gamma\_{p}}\left(\rho\right)\right)\geq\mathbf{0},\tag{20}$$

for any fixed *p*∈f g 1, … *n* and for any *β* ≥ *γ<sup>p</sup>* >0, *δ* >0, *τ*, *ρ*∈½ � *a*, *t* ; *a*< *t*≤ *b:* Taking

$$K\_p(\tau,\rho) \coloneqq \frac{(t-\tau)^{a-1}}{\Gamma(a)} \prod\_{i=1}^n f\_i^{\eta\_i}(\tau) \left( (\rho-a)^\delta - (\tau-a)^\delta \right) \left( f\_p^{\beta-\gamma\_p}(\tau) - f\_p^{\beta-\gamma\_p}(\rho) \right), \tag{21}$$

we observe that

$$K\_p(\mathfrak{r}, \rho) \ge \mathbf{0}.\tag{22}$$

Also, we have

**3. Some integral inequalities**

½0, ∞½, such that for all *t* >0, *J*

*J αf <sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

Hence, we have

Thus, it yields that

And then,

[25] on 0, ½ �*t :*

we have

**152**

such that for all *t* >0, *J*

In the same manner, we have

Proof: We use the hypothesis *<sup>f</sup>*ð Þ*<sup>τ</sup>*

the reader to [21].

*Functional Calculus*

then we have

**3.1 On Minkowski and Hermite-Hadamard fractional inequalities**

*αf p*

*<sup>α</sup>g<sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

ð Þ *<sup>M</sup>* <sup>þ</sup> <sup>1</sup> *<sup>p</sup>* Γð Þ *α*

ð*t*

0

*<sup>p</sup>* ≤

*<sup>p</sup>* ≤

*<sup>p</sup>* þ *J*

≤ *M<sup>p</sup>* Γð Þ *α*

*J αf p* ð Þ*t* ≤

*J αf <sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

*J <sup>α</sup>g<sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

*αf p*

Combining (13) and (15), we achieve the proof.

ð Þ*t* < ∞, *J*

1 þ 1 *m* � �

In this subsection, we present some fractional integral results related to Minkowski and Hermite-Hadamard integral inequalities. For more details, we refer

**Theorem 1.9.** Let *α* >0, *p* ≥1 and let *f*, *g* be two positive functions on

ð Þ*t* < ∞, *J*

1 þ *M m*ð Þ þ 2 ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> ð Þ *<sup>M</sup>* <sup>þ</sup> <sup>1</sup> *<sup>J</sup>*

ð Þ *<sup>t</sup>* � *<sup>τ</sup> <sup>α</sup>*�<sup>1</sup>

*f p* ð Þ*τ dτ*

ð Þ *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup> <sup>p</sup>*

*<sup>α</sup>*ð Þ *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup> <sup>p</sup>*

*<sup>α</sup>*ð Þ *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup> <sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

*<sup>α</sup>*ð Þ *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup> <sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

*<sup>α</sup>g<sup>p</sup>*ð Þ*<sup>t</sup>* <sup>&</sup>lt; <sup>∞</sup>*:* If 0 <sup>&</sup>lt; *<sup>m</sup>* <sup>≤</sup> *<sup>f</sup>*ð Þ*<sup>τ</sup>*

*<sup>p</sup>* ≤

ð*t*

0

ð Þ *<sup>t</sup>* � *<sup>τ</sup> <sup>α</sup>*�<sup>1</sup>

*M<sup>p</sup>* ð Þ *<sup>M</sup>* <sup>þ</sup> <sup>1</sup> *<sup>p</sup> <sup>J</sup>*

> *M <sup>M</sup>* <sup>þ</sup> <sup>1</sup> *<sup>J</sup>*

*g*ð Þ*τ* ≤

1 *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> *<sup>J</sup>*

With the same arguments as before, we present the following theorem. Theorem 1.11. Let *α* >0, *p*≥ 1 and let *f*, *g* be two positive functions on 0, ½ ∞½,

1

Remark 1.10. Applying the above theorem for *α* ¼ 1, we obtain Theorem 1.2 of

*<sup>α</sup>gp*ð Þ*<sup>t</sup>* <sup>&</sup>lt; <sup>∞</sup>*:* If 0 <sup>&</sup>lt; *<sup>m</sup>* <sup>≤</sup> *<sup>f</sup>*ð Þ*<sup>τ</sup>*

*<sup>g</sup>*ð Þ*<sup>τ</sup>* <sup>&</sup>lt; *<sup>M</sup>*, *<sup>τ</sup>* <sup>∈</sup>½ � 0, *<sup>t</sup>* , *<sup>t</sup>* <sup>&</sup>gt;0*:* We can write

ð Þ*τ dτ:*

*<sup>α</sup>*ð Þ *<sup>f</sup>* <sup>þ</sup> *<sup>g</sup> <sup>p</sup>* ½ � ð Þ*<sup>t</sup>* <sup>1</sup>

ð Þ*t :* (12)

*<sup>p</sup>:* (13)

*<sup>p</sup>:* (15)

*<sup>g</sup>*ð Þ*<sup>τ</sup>* <sup>≤</sup> *<sup>M</sup>*, *<sup>τ</sup>* <sup>∈</sup>½ � 0, *<sup>t</sup>* , then

*<sup>m</sup>* ð Þ *<sup>f</sup>*ð Þþ *<sup>τ</sup> <sup>g</sup>*ð Þ*<sup>τ</sup> :* (14)

*<sup>g</sup>*ð Þ*<sup>τ</sup>* <sup>≤</sup> *<sup>M</sup>*, *<sup>τ</sup>* <sup>∈</sup> ½ � 0, *<sup>t</sup>* ,

*<sup>p</sup>:* (10)

(11)

$$0 \le \int\_{a}^{t} K\_p(\tau, \rho) d\tau = (\rho - a)^\delta f^a \left[ \prod\_{i \ne p}^n f\_i^{\gamma\_i} f\_p^{\rho}(t) \right] + f\_p^{\beta - \gamma\_p}(\rho) f^a \left[ (t - a)^\delta \prod\_{i = 1}^n f\_i^{\gamma\_i}(t) \right] \tag{23}$$

$$-f^a \left[ (t - a)^\delta \prod\_{i \ne p}^n f\_i^{\gamma\_i} f\_p^{\rho}(t) \right] - (\rho - a)^\delta f^{\beta - \gamma\_p}(\rho) f^a \left[ \prod\_{i = 1}^n f\_i^{\gamma\_i}(t) \right].$$

Remark 1.20. Applying Theorem 1.19 for *α* ¼ 1, *t* ¼ *b*, *n* ¼ 1, we obtain Theorem

In what follows, we present some fractional results on the beta distribution [23].

*B p*ð Þ , *<sup>q</sup> B m*ð Þ , *<sup>n</sup>* , *<sup>α</sup>* <sup>≥</sup>1*:*

*B p*ð Þ , *<sup>q</sup> B m*ð Þ , *<sup>n</sup>* , *<sup>α</sup>*, *<sup>β</sup>* <sup>≥</sup>1*:*

ð Þ *μ* � *t f t*ð Þ*dt*, *α* ≥1*:* (27)

1 A

2

, (28)

Theorem 1.21. Let *X*, *Y*, *U*, and *V* be four random variables, such that *X* � *B p*ð Þ , *q* , *Y* � *B m*ð Þ , *n* , *U* � *B p*ð Þ , *n* , and *V* � *B m*ð Þ , *q* . If ð Þ *p* � *m* ð Þ *q* � *n* ≤ 0, then

For the proof of this result, we can apply a weighted version of the fractional

We propose also the following ð Þ� *α*, *β* version that generalizes the above result.

Remark 1.24. If *α* ¼ *β* ¼ 1, then the above theorem reduces to Theorem 3.1 of [7].

In the following theorem, the fractional covariance of *X* and *g X*ð Þ is expressed with the derivative of *g X*ð Þ. It can be considered as a generalization of a covariance

Theorem 1.25. Let *X* be a random variable having a *p:d:f* defined on ½ � *a*, *b* ;

*a*

We can prove this result by the application of the covariance definition in the

The following theorem establishes a lower bound for *Varα*ð Þ *g X*ð Þ of any function

Theorem 1.26. Let *X* be a random variable having a *p:d:f* defined on ½ � *a*, *b* , such

*a*

ð Þ *<sup>b</sup>* � *<sup>t</sup> <sup>α</sup>*�<sup>1</sup>

ð Þ *μ* � *t f t*ð Þ*dt*

ð Þ *<sup>b</sup>* � *<sup>t</sup> <sup>α</sup>*�<sup>1</sup>

Theorem 1.23. Let *X*, *Y*, *U*, and *V* be four random variables, such that *X* � *B p*ð Þ , *q* , *Y* � *B m*ð Þ , *n* , *U* � *B p*ð Þ , *n* , and *V* � *B m*ð Þ , *q* . If ð Þ *p* � *m* ð Þ *q* � *n* ≤ 0, then

*<sup>E</sup><sup>α</sup> <sup>U</sup><sup>r</sup>* ð Þ*E<sup>β</sup> <sup>V</sup><sup>r</sup>* ð Þþ *<sup>E</sup><sup>β</sup> <sup>U</sup><sup>r</sup>* ð Þ*E<sup>α</sup> <sup>V</sup><sup>r</sup>* ð Þ <sup>≥</sup> *B p*ð Þ , *<sup>n</sup> B m*ð Þ , *<sup>q</sup>*

identity established by the authors of [28]. So, we prove the result:

Γð Þ *α*

1 *VarX*,*<sup>α</sup>*

ð Þ ½ � *a*, *b* .

ð *b*

*a g*0 ð Þ *<sup>x</sup> dx* <sup>ð</sup> *x*

1 Γð Þ *α*

0 @ ð *b*

*a g*0 ð Þ *<sup>x</sup> dx* <sup>ð</sup> *x*

*<sup>E</sup><sup>α</sup> <sup>U</sup><sup>r</sup>* ð Þ*E<sup>α</sup> <sup>V</sup><sup>r</sup>* ð Þ <sup>≥</sup> *B p*ð Þ , *<sup>n</sup> B m*ð Þ , *<sup>q</sup>*

Remark 1.22. The above theorem generalizes Theorem 3.1 of [7].

4 of [27].

We have

**3.3 Some estimations on random variables**

*DOI: http://dx.doi.org/10.5772/intechopen.91140*

So let us prove the following *α*�version.

Chebyshev inequality as is mentioned in [1].

*3.3.2 Identities and lower bounds*

*μ* ¼ *E X*ð Þ. Then, we have

case where *ω*ð Þ¼ *x* 1*:*

ð Þ ½ � *a*, *b* . We have

that *μ* ¼ *E X*ð Þ. Then, we have

*Varα*ð Þ *g X*ð Þ ≥

for any *g* ∈*C*<sup>1</sup>

*g* ∈*C*<sup>1</sup>

**155**

*Covα*ð Þ¼ *<sup>X</sup>*, *g X*ð Þ <sup>1</sup>

*3.3.1 Bounds for fractional moments of beta distribution*

*Integral Inequalities and Differential Equations via Fractional Calculus*

*<sup>E</sup><sup>α</sup> <sup>X</sup><sup>r</sup>* ð Þ*E<sup>α</sup> <sup>Y</sup><sup>r</sup>* ð Þ

*<sup>E</sup><sup>α</sup> <sup>X</sup><sup>r</sup>* ð Þ*E<sup>β</sup> <sup>Y</sup><sup>r</sup>* ð Þþ *<sup>E</sup><sup>β</sup> <sup>X</sup><sup>r</sup>* ð Þ*E<sup>α</sup> <sup>Y</sup><sup>r</sup>* ð Þ

Hence, we get

$$J^a \left[ (t-a)^\delta \prod\_{i=1}^n f\_i^{\gamma\_i}(t) \right] J^a \left[ \prod\_{i \neq p}^n f\_i^{\gamma\_i} f\_p^\beta(t) \right] \ge J^a \left[ \prod\_{i=1}^n f\_i^{\gamma\_i}(t) \right] J^a \left[ (t-a)^\delta \prod\_{i \neq p}^n f\_i^{\gamma\_i} f\_p^\beta(t) \right]. \tag{24}$$

The proof is thus achieved.

Remark 1.16. Applying Theorem 1.15 for *α* ¼ 1, *t* ¼ *b*, *n* ¼ 1, we obtain Theorem 3 in [27].

Using other sufficient conditions, we prove the following generalization.

Theorem 1.17. Suppose that *fi* � � *<sup>i</sup>*¼1, … *<sup>n</sup>* are positive, continuous, and decreasing functions on ½ � *a*, *b :* Then for any fixed *p* in 1, 2, f g … , *n* and for any *a*< *t*≤ *b*, *α* >0,*ω* >0, *δ* >0, *β* ≥*γ<sup>p</sup>* > 0, we have

$$\frac{J^a\left[\prod\_{i\neq p}^n f\_i^{r\_i} f\_p^{\theta}(t)\right] I^a\left[(t-a)^\delta \prod\_{i=1}^n f\_i^{r\_i}(t)\right] + J^w\left[\prod\_{i\neq p}^n f\_i^{r\_i} f\_p^{\theta}(t)\right] I^a\left[(t-a)^\delta \prod\_{i=1}^n f\_i^{r\_i}(t)\right]}{J^a\left[(t-a)^\delta \prod\_{i\neq p}^n f\_i^{r\_i} f\_p^{\theta}(t)\right] I^a\left[\prod\_{i=1}^n f\_i^{r\_i}(t)\right] + J^w\left[(t-a)^\delta \prod\_{i\neq p}^n f\_i^{r\_i} f\_p^{\theta}(t)\right] J^a\left[\prod\_{i=1}^n f\_i^{r\_i}(t)\right]} \tag{25}$$

Proof: Multiplying both sides of (23) by ð Þ *<sup>t</sup>*�*<sup>ρ</sup> <sup>ω</sup>*�<sup>1</sup> Γð Þ *ω* Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi <sup>i</sup>* ð Þ*ρ* ,*ω* >0, then integrating the resulting inequality with respect to *ρ* over ð Þ *a*, *t* , *a*< *t*≤ *b* and using Fubini's theorem, we obtain the desired inequality.

Remark 1.18.


Introducing a positive increasing function *g* to the family *fi* � � *<sup>i</sup>*¼1, … *<sup>n</sup>*, we establish the following theorem.

Theorem 1.19. Let *fi* � � *<sup>i</sup>*¼1, … *<sup>n</sup>* and *<sup>g</sup>* be positive continuous functions on ½ � *<sup>a</sup>*, *<sup>b</sup>* , such that *g* is increasing and *fi* � � *<sup>i</sup>*¼1, … *<sup>n</sup>* are decreasing on ½ � *<sup>a</sup>*, *<sup>b</sup> :* Then, the following inequality

$$\frac{J^{a}\left[\prod\_{i\neq p}^{n}f\_{i}^{\gamma\_{i}}f\_{p}^{\beta}(t)\right]J^{a}\left[\mathbf{g}^{\delta}(t)\prod\_{i=1}^{n}f\_{i}^{\gamma\_{i}}(t)\right]}{J^{a}\left[\mathbf{g}^{\delta}(t)\prod\_{i\neq p}^{n}f\_{i}^{\gamma\_{i}}f\_{p}^{\beta}(t)\right]J^{a}\left[\prod\_{i=1}^{n}f\_{i}^{\gamma\_{i}}(t)\right]}\geq 1\tag{26}$$

holds for any *a*< *t*≤ *b*, *α* >0, *δ* >0, *β* ≥*γ<sup>p</sup>* > 0, where *p* is a fixed integer in f g 1, 2, … , *n :*

*Integral Inequalities and Differential Equations via Fractional Calculus DOI: http://dx.doi.org/10.5772/intechopen.91140*

Remark 1.20. Applying Theorem 1.19 for *α* ¼ 1, *t* ¼ *b*, *n* ¼ 1, we obtain Theorem 4 of [27].

#### **3.3 Some estimations on random variables**

0≤ ð*t*

*J*

3 in [27].

h i

Remark 1.18.

of [27].

inequality

f g 1, 2, … , *n :*

**154**

establish the following theorem. Theorem 1.19. Let *fi*

such that *g* is increasing and *fi*

� �

*J <sup>α</sup>* Q*<sup>n</sup> <sup>i</sup>*6¼*<sup>p</sup> <sup>f</sup> γi i f β <sup>p</sup>*ð Þ*t*

*J <sup>α</sup> <sup>g</sup><sup>δ</sup>*ð Þ*<sup>t</sup>*

*<sup>δ</sup>*Q*<sup>n</sup> <sup>i</sup>*6¼*<sup>p</sup> <sup>f</sup> γi i f β <sup>p</sup>*ð Þ*t*

h i

*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup>*

*J <sup>α</sup>* Q*<sup>n</sup> <sup>i</sup>*6¼*<sup>p</sup> <sup>f</sup> γi i f β <sup>p</sup>*ð Þ*t*

*J*

*a*

*Functional Calculus*

Hence, we get

*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup> <sup>δ</sup>* <sup>Y</sup>*<sup>n</sup>*

*i*¼1 *f γi <sup>i</sup>* ð Þ*t*

The proof is thus achieved.

Theorem 1.17. Suppose that *fi*

*J*

*a*< *t*≤ *b*, *α* >0,*ω* >0, *δ* >0, *β* ≥*γ<sup>p</sup>* > 0, we have

*J <sup>ω</sup>* Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi <sup>i</sup>* ð Þ*<sup>t</sup>* � � <sup>þ</sup> *<sup>J</sup>*

Proof: Multiplying both sides of (23) by ð Þ *<sup>t</sup>*�*<sup>ρ</sup> <sup>ω</sup>*�<sup>1</sup>

*<sup>δ</sup>*Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi <sup>i</sup>* ð Þ*t*

h i

*<sup>ω</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup>*

theorem, we obtain the desired inequality.

" #

*Kp*ð Þ *τ*, *ρ dτ* ¼ ð Þ *ρ* � *a*

�*J*

2 4

> *J <sup>α</sup>* <sup>Y</sup>*<sup>n</sup>*

2 4 *δ J <sup>α</sup>* <sup>Y</sup>*<sup>n</sup> i*6¼*p f γi i f β <sup>p</sup>*ð Þ*t*

*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup> <sup>δ</sup>* <sup>Y</sup>*<sup>n</sup>*

*i*6¼*p f γi i f β <sup>p</sup>*ð Þ*t*

*i*6¼*p f γi i f β <sup>p</sup>*ð Þ*t*

> 3 5≥*J*

2 4 3 5 þ *f β*�*γ<sup>p</sup> <sup>p</sup>* ð Þ*ρ J*

3

*<sup>α</sup>* <sup>Y</sup>*<sup>n</sup> i*¼1 *f γi <sup>i</sup>* ð Þ*t* " #

Remark 1.16. Applying Theorem 1.15 for *α* ¼ 1, *t* ¼ *b*, *n* ¼ 1, we obtain Theorem

Using other sufficient conditions, we prove the following generalization.

þ *J <sup>ω</sup>* Q*<sup>n</sup> <sup>i</sup>*6¼*<sup>p</sup> <sup>f</sup> γi i f β <sup>p</sup>*ð Þ*t*

ing the resulting inequality with respect to *ρ* over ð Þ *a*, *t* , *a*< *t*≤ *b* and using Fubini's

ii. Applying Theorem 1.17 for *α* ¼ *ω* ¼ 1, *t* ¼ *b*, *n* ¼ 1, we obtain Theorem 3

i. Applying Theorem 1.17 for *α* ¼ *ω*, we obtain Theorem 1.15.

Introducing a positive increasing function *g* to the family *fi*

� �

h i

Q*<sup>n</sup> <sup>i</sup>*6¼*<sup>p</sup> <sup>f</sup> γi i f β <sup>p</sup>*ð Þ*t*

h i

*J <sup>α</sup> <sup>g</sup><sup>δ</sup>*ð Þ*<sup>t</sup>*

holds for any *a*< *t*≤ *b*, *α* >0, *δ* >0, *β* ≥*γ<sup>p</sup>* > 0, where *p* is a fixed integer in

h i

*<sup>δ</sup>*Q*<sup>n</sup> <sup>i</sup>*6¼*<sup>p</sup> <sup>f</sup> γi i f β <sup>p</sup>*ð Þ*t*

Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi*

*<sup>i</sup>*¼1, … *<sup>n</sup>* and *<sup>g</sup>* be positive continuous functions on ½ � *<sup>a</sup>*, *<sup>b</sup>* ,

Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi <sup>i</sup>* ð Þ*<sup>t</sup>* � �

> *J <sup>α</sup>* Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi <sup>i</sup>* ð Þ*<sup>t</sup>* � �

*<sup>i</sup>*¼1, … *<sup>n</sup>* are decreasing on ½ � *<sup>a</sup>*, *<sup>b</sup> :* Then, the following

h i

*<sup>ω</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup>*

Γð Þ *ω*

� �

functions on ½ � *a*, *b :* Then for any fixed *p* in 1, 2, f g … , *n* and for any

<sup>5</sup> � ð Þ *<sup>ρ</sup>* � *<sup>a</sup> <sup>δ</sup>*

*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup>*

*f <sup>β</sup>*�*γ<sup>p</sup>* ð Þ*<sup>ρ</sup> <sup>J</sup>*

*J*

2 4

*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup> <sup>δ</sup>* <sup>Y</sup>*<sup>n</sup>*

*<sup>i</sup>*¼1, … *<sup>n</sup>* are positive, continuous, and decreasing

*J*

*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>a</sup>*

� �

*J <sup>α</sup>* Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi <sup>i</sup>* ð Þ*<sup>t</sup>* � �

*<sup>δ</sup>*Q*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>f</sup> γi <sup>i</sup>* ð Þ*t*

h i

*<sup>i</sup>* ð Þ*ρ* ,*ω* >0, then integrat-

*<sup>i</sup>*¼1, … *<sup>n</sup>*, we

≥1 (26)

*<sup>δ</sup>* <sup>Y</sup>*<sup>n</sup> i*¼1 *f γi <sup>i</sup>* ð Þ*t*

*<sup>α</sup>* <sup>Y</sup>*<sup>n</sup> i*¼1 *f γi <sup>i</sup>* ð Þ*t* " #

> *i*6¼*p f γi i f β <sup>p</sup>*ð Þ*t*

*:*

3 5*:*

(24)

≥1*:*

(25)

(23)

" #

#### *3.3.1 Bounds for fractional moments of beta distribution*

In what follows, we present some fractional results on the beta distribution [23]. So let us prove the following *α*�version.

Theorem 1.21. Let *X*, *Y*, *U*, and *V* be four random variables, such that *X* � *B p*ð Þ , *q* , *Y* � *B m*ð Þ , *n* , *U* � *B p*ð Þ , *n* , and *V* � *B m*ð Þ , *q* . If ð Þ *p* � *m* ð Þ *q* � *n* ≤ 0, then

$$\frac{E\_a(X^r)E\_a(Y^r)}{E\_a(U^r)E\_a(V^r)} \ge \frac{B(p,n)B(m,q)}{B(p,q)B(m,n)}, a \ge 1.$$

For the proof of this result, we can apply a weighted version of the fractional Chebyshev inequality as is mentioned in [1].

Remark 1.22. The above theorem generalizes Theorem 3.1 of [7].

We propose also the following ð Þ� *α*, *β* version that generalizes the above result. We have

Theorem 1.23. Let *X*, *Y*, *U*, and *V* be four random variables, such that *X* � *B p*ð Þ , *q* , *Y* � *B m*ð Þ , *n* , *U* � *B p*ð Þ , *n* , and *V* � *B m*ð Þ , *q* . If ð Þ *p* � *m* ð Þ *q* � *n* ≤ 0, then

$$\frac{E\_a(X^r)E\_\beta(Y^r) + E\_\beta(X^r)E\_a(Y^r)}{E\_a(U^r)E\_\beta(V^r) + E\_\beta(U^r)E\_a(V^r)} \ge \frac{B(p,n)B(m,q)}{B(p,q)B(m,n)}, a, \beta \ge 1.$$

Remark 1.24. If *α* ¼ *β* ¼ 1, then the above theorem reduces to Theorem 3.1 of [7].

#### *3.3.2 Identities and lower bounds*

In the following theorem, the fractional covariance of *X* and *g X*ð Þ is expressed with the derivative of *g X*ð Þ. It can be considered as a generalization of a covariance identity established by the authors of [28]. So, we prove the result:

Theorem 1.25. Let *X* be a random variable having a *p:d:f* defined on ½ � *a*, *b* ; *μ* ¼ *E X*ð Þ. Then, we have

$$\text{Cov}\_a(\mathbf{X}, \mathbf{g}(\mathbf{X})) = \frac{1}{\Gamma(a)} \int\_a^b \mathbf{g}'(\mathbf{x}) d\mathbf{x} \int\_a^\mathbf{x} (b-t)^{a-1} (\mu - t) f(t) dt, a \ge 1. \tag{27}$$

We can prove this result by the application of the covariance definition in the case where *ω*ð Þ¼ *x* 1*:*

The following theorem establishes a lower bound for *Varα*ð Þ *g X*ð Þ of any function *g* ∈*C*<sup>1</sup> ð Þ ½ � *a*, *b* . We have

Theorem 1.26. Let *X* be a random variable having a *p:d:f* defined on ½ � *a*, *b* , such that *μ* ¼ *E X*ð Þ. Then, we have

$$Var\_a(g(X)) \ge \frac{1}{Var\_{X,a}} \left( \frac{1}{\Gamma(a)} \int\_a^b g'(x) dx \int\_a^x (b-t)^{a-1} (\mu - t) f(t) dt \right)^2,\tag{28}$$

for any *g* ∈*C*<sup>1</sup> ð Þ ½ � *a*, *b* .

To prove this result, we use fractional Cauchy-Schwarz inequality established in [29].

Remark 1.27. Let us consider <sup>Ω</sup> <sup>∈</sup>*C a* ð Þ ½ � , *<sup>b</sup>* that satisfies <sup>Ð</sup> *<sup>x</sup> <sup>a</sup>* ð Þ *<sup>b</sup>* � *<sup>t</sup> <sup>α</sup>*�<sup>1</sup> ð Þ *μ* � *t f t*ð Þ*dt* <sup>¼</sup> ð Þ *<sup>b</sup>* � *<sup>x</sup> <sup>α</sup>*�<sup>1</sup> *<sup>σ</sup>*2Ωð Þ *<sup>x</sup> f x*ð Þ. Then, we present the following result.

Theorem 1.28. Let *X* be a random variable having a *p:d:f:* defined on ½ � *a*, *b* , such *x*

$$\text{that } \mu = E(X), \sigma^2 = Var(X) \text{ and } \mathfrak{Q} \in C([a, b]); \int\_a (b - t)^{a - 1} (\mu - t) f(t) dt = 0$$

ð Þ *<sup>b</sup>* � *<sup>x</sup> <sup>α</sup>*�<sup>1</sup> *<sup>σ</sup>*2Ωð Þ *<sup>x</sup> f x*ð Þ. Then, we have

$$Var\_a(\mathbf{g}(X)) \ge \frac{\sigma^4(X)}{Var\_a(X)} E\_a^2(\mathbf{g}'(X)\boldsymbol{\Omega}(X)).\tag{29}$$

where *g* ∈*C*<sup>1</sup>

Proof: We have

<sup>Γ</sup>ð Þ *<sup>α</sup>* <sup>¼</sup> <sup>Ð</sup>

*Covα*ð Þ¼ *h X*ð Þ, *g X*ð Þ <sup>1</sup>

*x a*

*DOI: http://dx.doi.org/10.5772/intechopen.91140*

*Z x*ð Þ*f x*ð Þð Þ *<sup>b</sup>*�*<sup>x</sup> <sup>α</sup>*�<sup>1</sup>

and

ð Þ ½ � *a*, *b* , with *E*∣*Z X*ð Þ*g*<sup>0</sup>

Γð Þ *α*

*<sup>E</sup><sup>α</sup> <sup>g</sup>*<sup>0</sup> ð Þ¼ ð Þ *<sup>X</sup> Z X*ð Þ <sup>1</sup>

Γð Þ *α*

þ 1 Γð Þ *α*

ð*μ a*

> ð*b μ*

**4. A class of differential equations of fractional order**

*<sup>t</sup>* <sup>∈</sup>½ � 0, 1 , *<sup>λ</sup>*<sup>∈</sup> <sup>∗</sup>

0< *α*≤ 1, 0 ≤ *δ*< *α*,

The definition of *Z X*ð Þ implies that

*<sup>E</sup><sup>α</sup> <sup>g</sup>*<sup>0</sup> ð Þ¼ ð Þ *<sup>X</sup> Z X*ð Þ <sup>1</sup>

us consider the following problem:

associated with the conditions

*β* ∈ , such that *βsin*ð Þ *λη* 6¼ *sin* ð Þ*λ :*

We recall the following result [20]:

**4.1 Integral representation**

where *<sup>c</sup>*

**157**

*c*

Hence, we obtain

of [10].

ð *b*

ð Þ *<sup>b</sup>* � *<sup>x</sup> <sup>α</sup>*�<sup>1</sup>

Γð Þ *α*

ð *b*

ð Þ *<sup>b</sup>* � *<sup>x</sup> <sup>α</sup>*�<sup>1</sup>

*g*0

*E<sup>α</sup> Z X*ð Þ*g*<sup>0</sup> ð Þ¼ ð Þ *X Covα*ð Þ *g X*ð Þ, *h X*ð Þ *:* (38)

*Dδ u t*ð Þ � �,

*u*ð Þ¼ 0 0, *u*00ð Þ¼ 0 0, *u*ð Þ¼ 1 *βu*ð Þ*η* , *η*∈ð Þ 0, 1 , (40)

*D<sup>α</sup>* denotes the Caputo fractional derivative of fractional order *α*, *D*<sup>2</sup> is

*a*

ð Þ *<sup>g</sup>*ð Þ� *<sup>μ</sup> g t*ð Þ ð Þ *<sup>b</sup>* � *<sup>t</sup> <sup>α</sup>*�<sup>1</sup>

Remark 1.31. Taking *α* ¼ 1, in the above theorem, we obtain Theorem 2.2

Inspired by the work in [4, 20], in what follows we will be concerned with a more general class of Langevin equations of fractional order. The considered class will contain a nonlinearity that depends on a fractional derivative of order *δ:* So, let

*<sup>D</sup><sup>α</sup> <sup>D</sup>*<sup>2</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup> � �*u t*ðÞ¼ *f t*, *u t*ð Þ, *<sup>c</sup>*

þ

the two-order classical derivative, *f* : ½ �� 0, 1 � ! is a given function, and

ð Þ *g t*ðÞ� *<sup>g</sup>*ð Þ *<sup>μ</sup>* ð Þ *<sup>b</sup>* � *<sup>t</sup> <sup>α</sup>*�<sup>1</sup>

*a*

ð Þ *EhX* ð Þ� ð Þ *h t*ð Þ ð Þ *<sup>b</sup>*�*<sup>t</sup> <sup>α</sup>*�<sup>1</sup>

*Integral Inequalities and Differential Equations via Fractional Calculus*

<sup>Γ</sup>ð Þ *<sup>α</sup> f t*ð Þ*dt:*

ð Þ *X* ∣< ∞, *h x*ð Þ is a given function and

ð Þ *h x*ð Þ� *h*ð Þ *μ* ð Þ *g x*ð Þ� *g*ð Þ *μ f x*ð Þ*dx* (35)

ð Þ *x Z x*ð Þ*f x*ð Þ*dx:* (36)

ð Þ *h*ð Þ� *μ h t*ð Þ *f t*ð Þ*dt* (37)

(39)

ð Þ *h t*ð Þ� *h*ð Þ *μ f t*ð Þ*dt:*

Proof: We have

$$\text{Cov}\_{a}^{2}(\mathbf{X}, \mathbf{g}(\mathbf{X})) = \left[ \frac{\mathbf{1}}{\Gamma(a)} \Big|\_{a}^{b} \Big( \mathbf{g}'(\mathbf{x}) d\mathbf{x} (b - \mathbf{x})^{a - 1} \sigma^{2} \Omega(\mathbf{x}) f(\mathbf{x}) d\mathbf{x} \Big)^{2} \right] \,. \tag{30}$$

On the other hand, we can see that

$$\left[\frac{1}{\Gamma(a)}\Big|\limits\_{a}^{b}(\varkappa)d\varkappa(b-\varkappa)^{a-1}\sigma^{2}\Omega(\varkappa)f(\varkappa)d\varkappa\right]^{2}=\sigma^{4}\mathcal{E}\_{a}^{2}(\mathcal{g}'(X)\Omega(X))\tag{31}$$

Thanks to the fractional version of Cauchy Schwarz inequality [29], and using the fact that

$$\text{Cov}\_a^2(X, \mathcal{g}(X)) \le \text{Var}\_a(X)\text{Var}\_a(\mathcal{g}(X)),\tag{32}$$

we obtain

$$
\sigma^4 E\_a^2(\mathcal{g}'(X)\Omega(X)) \le \operatorname{Var}\_a(X)\operatorname{Var}\_a(\mathcal{g}(X)).\tag{33}
$$

This ends the proof.

Remark 1.29. Thanks to (30) and (31), we obtain the following fractional covariance identity

$$
\sigma^2 E\_a(\mathfrak{g}'(X)\mathfrak{Q}(X)) = \text{Cov}\_a(X, \mathfrak{g}(X)).
$$

It generalizes the good standard identity obtained in [28] that corresponds to *α* ¼ 1 and it is given by

$$
\sigma^2 E(\mathfrak{g}'(X)\Omega(X)) = Cov(X, \mathfrak{g}(X)) .
$$

We end this section by proving the following fractional integral identity between covariance and expectation in the fractional case.

Theorem 1.30. Let *X* be a continuous random variable with a *p:d:f:* having a support an interval ½ � *a*, *b* , *E X*ð Þ¼ *μ*. Then, for any *α* ≥1, the following general covariance identity holds

$$\text{Cov}\_a(h(X), \mathbf{g}(X)) = E\_a(\mathbf{g}'(X)Z(X)),\tag{34}$$

*Integral Inequalities and Differential Equations via Fractional Calculus DOI: http://dx.doi.org/10.5772/intechopen.91140*

where *g* ∈*C*<sup>1</sup> ð Þ ½ � *a*, *b* , with *E*∣*Z X*ð Þ*g*<sup>0</sup> ð Þ *X* ∣< ∞, *h x*ð Þ is a given function and *Z x*ð Þ*f x*ð Þð Þ *<sup>b</sup>*�*<sup>x</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup>* <sup>¼</sup> <sup>Ð</sup> *x a* ð Þ *EhX* ð Þ� ð Þ *h t*ð Þ ð Þ *<sup>b</sup>*�*<sup>t</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup> f t*ð Þ*dt:*

Proof: We have

$$\text{Cov}\_a(h(X), \mathbf{g}(X)) = \frac{1}{\Gamma(a)} \Big|\_{a}^{b} (b - \mathbf{x})^{a-1} (h(\mathbf{x}) - h(\mu)) (\mathbf{g}(\mathbf{x}) - \mathbf{g}(\mu)) f(\mathbf{x}) d\mathbf{x} \tag{35}$$

and

To prove this result, we use fractional Cauchy-Schwarz inequality established

*<sup>σ</sup>*2Ωð Þ *<sup>x</sup> f x*ð Þ. Then, we present the following result. Theorem 1.28. Let *X* be a random variable having a *p:d:f:* defined on ½ � *a*, *b* , such

*Varα*ð Þ *<sup>X</sup> <sup>E</sup>*<sup>2</sup>

ð Þ *<sup>x</sup> dx b*ð Þ � *<sup>x</sup> <sup>α</sup>*�<sup>1</sup>

3 5

2

Ωð Þ *x f x*ð Þ*dx*

Remark 1.29. Thanks to (30) and (31), we obtain the following fractional

*E<sup>α</sup> g*<sup>0</sup> ð Þ¼ ð Þ *X* Ωð Þ *X Covα*ð Þ *X*, *g X*ð Þ *:*

It generalizes the good standard identity obtained in [28] that corresponds to

*E g*<sup>0</sup> ð Þ¼ ð Þ *X* Ωð Þ *X Cov X*ð Þ , *g X*ð Þ *:*

We end this section by proving the following fractional integral identity

Theorem 1.30. Let *X* be a continuous random variable with a *p:d:f:* having a support an interval ½ � *a*, *b* , *E X*ð Þ¼ *μ*. Then, for any *α* ≥1, the following general

Thanks to the fractional version of Cauchy Schwarz inequality [29], and using

Ð *x a*

ð Þ *<sup>b</sup>* � *<sup>t</sup> <sup>α</sup>*�<sup>1</sup>

*σ*2

<sup>¼</sup> *<sup>σ</sup>*<sup>4</sup>*E*<sup>2</sup>

*<sup>α</sup>*ð Þ *X*, *g X*ð Þ ≤ *Varα*ð Þ *X Varα*ð Þ *g X*ð Þ , (32)

*<sup>α</sup> g*<sup>0</sup> ð Þ ð Þ *X* Ωð Þ *X* ≤ *Varα*ð Þ *X Varα*ð Þ *g X*ð Þ *:* (33)

*Covα*ð Þ¼ *h X*ð Þ, *g X*ð Þ *E<sup>α</sup> g*<sup>0</sup> ð Þ ð Þ *X Z X*ð Þ , (34)

Ωð Þ *x f x*ð Þ*dx*

*<sup>a</sup>* ð Þ *<sup>b</sup>* � *<sup>t</sup> <sup>α</sup>*�<sup>1</sup>

ð Þ *μ* � *t f t*ð Þ*dt* ¼

*<sup>α</sup> g*<sup>0</sup> ð Þ ð Þ *X* Ωð Þ *X :* (29)

3 5

2

*<sup>α</sup> g*<sup>0</sup> ð Þ ð Þ *X* Ωð Þ *X* (31)

*:* (30)

ð Þ *μ* � *t*

Remark 1.27. Let us consider <sup>Ω</sup> <sup>∈</sup>*C a* ð Þ ½ � , *<sup>b</sup>* that satisfies <sup>Ð</sup> *<sup>x</sup>*

*Varα*ð Þ *g X*ð Þ <sup>≥</sup> *<sup>σ</sup>*4ð Þ *<sup>X</sup>*

ð *b*

*a g*0

*σ*2

Γð Þ *α*

2 4

ð Þ *<sup>x</sup> dx b*ð Þ � *<sup>x</sup> <sup>α</sup>*�<sup>1</sup>

*Cov*<sup>2</sup>

*σ*<sup>4</sup>*E*<sup>2</sup>

*σ*2

*σ*2

between covariance and expectation in the fractional case.

that *<sup>μ</sup>* <sup>¼</sup> *E X*ð Þ, *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *Var X*ð Þ and <sup>Ω</sup> <sup>∈</sup>*C a* ð Þ ½ � , *<sup>b</sup>* ;

*<sup>α</sup>*ð Þ¼ *<sup>X</sup>*, *g X*ð Þ <sup>1</sup>

On the other hand, we can see that

*<sup>σ</sup>*2Ωð Þ *<sup>x</sup> f x*ð Þ. Then, we have

in [29].

ð Þ *<sup>b</sup>* � *<sup>x</sup> <sup>α</sup>*�<sup>1</sup>

*f t*ð Þ*dt* <sup>¼</sup> ð Þ *<sup>b</sup>* � *<sup>x</sup> <sup>α</sup>*�<sup>1</sup>

*Functional Calculus*

Proof: We have

*Cov*<sup>2</sup>

1 Γð Þ *α*

2 4

the fact that

we obtain

covariance identity

*α* ¼ 1 and it is given by

covariance identity holds

**156**

ð *b*

*a g*0

This ends the proof.

$$E\_a(\mathcal{g}'(X)Z(X)) = \frac{1}{\Gamma(a)} \int\_a^b (b-x)^{a-1} \mathcal{g}'(x) Z(x) f(x) d\mathfrak{x}.\tag{36}$$

The definition of *Z X*ð Þ implies that

$$E\_a(\mathbf{g'}(X)Z(X)) = \frac{1}{\Gamma(a)} \int\_a^\mu (\mathbf{g}(\mu) - \mathbf{g}(t))(b - t)^{a - 1}(h(\mu) - h(t))\mathbf{f}(t)dt \tag{37}$$

$$+ \frac{1}{\Gamma(a)} \int\_\mu^b (\mathbf{g}(t) - \mathbf{g}(\mu))(b - t)^{a - 1}(h(t) - h(\mu))\mathbf{f}(t)dt.$$

Hence, we obtain

$$E\_a(Z(X)\mathbf{g}'(X)) = \text{Cov}\_a(\mathbf{g}(X), h(X)). \tag{38}$$

Remark 1.31. Taking *α* ¼ 1, in the above theorem, we obtain Theorem 2.2 of [10].

### **4. A class of differential equations of fractional order**

Inspired by the work in [4, 20], in what follows we will be concerned with a more general class of Langevin equations of fractional order. The considered class will contain a nonlinearity that depends on a fractional derivative of order *δ:* So, let us consider the following problem:

$$\begin{cases} \,^cD^a(D^2 + \lambda^2)u(t) = f\left(t, u(t), \,^cD^\delta u(t)\right), \\ t \in [0, 1], \quad \lambda \in \mathbb{R}\_+^\* \\ 0 < a \le 1, \quad 0 \le \delta < a, \end{cases} \tag{39}$$

associated with the conditions

$$u(\mathbf{0}) = \mathbf{0}, \quad u''(\mathbf{0}) = \mathbf{0}, \quad u(\mathbf{1}) = \beta u(\boldsymbol{\eta}), \boldsymbol{\eta} \in (\mathbf{0}, \mathbf{1}),\tag{40}$$

where *<sup>c</sup> D<sup>α</sup>* denotes the Caputo fractional derivative of fractional order *α*, *D*<sup>2</sup> is the two-order classical derivative, *f* : ½ �� 0, 1 � ! is a given function, and *β* ∈ , such that *βsin*ð Þ *λη* 6¼ *sin* ð Þ*λ :*

#### **4.1 Integral representation**

We recall the following result [20]:

Lemma 1.32. Let *θ* be a continuous function on 0, 1 ½ �. The unique solution of the problem

$$\begin{cases} \,^cD^a(D^2 + \lambda^2)u(t) = \theta(t), \\ t \in [0, 1], \quad \lambda \in \mathbb{R}\_+^\* \\ n - 1 < a \le n, \quad n \in \mathbb{N}^\*, \end{cases} \tag{41}$$

Then problem (39) and (40) has a unique solution on 0, 1 ½ �.

*Integral Inequalities and Differential Equations via Fractional Calculus*

*<sup>E</sup>* <sup>¼</sup> *<sup>u</sup>*; *<sup>u</sup>*∈Cð Þ ½ � 0, 1 , *<sup>D</sup><sup>δ</sup>*

*sinλ*ð Þ *t* � *s J*

sin ð Þ *λt* <sup>Δ</sup> *<sup>β</sup>*

*sinλ*ð Þ 1 � *s J*

*α*

j*sinλ η*ð Þj � *s J*

<sup>0</sup>j*f*ð*s; <sup>u</sup>*1ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

<sup>þ</sup> <sup>∣</sup>*β*<sup>∣</sup> ∣Δ∣ *ηα*þ<sup>1</sup>

� �

We shall prove that the above operator is contractive over the space *E*.

<sup>0</sup><sup>∣</sup> *f s; <sup>u</sup>*1ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

*α*

*α*

ð *η*

2 4

0

*α*

*<sup>u</sup>*<sup>∈</sup> Cð Þ ½ � 0, 1 � �,

*u*∥∞*:*

*u s*ð Þ � �*ds*

*α*

*<sup>u</sup>*1ð Þ*<sup>s</sup>* � � � *f s; <sup>u</sup>*2ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

<sup>0</sup><sup>j</sup> *<sup>f</sup>*ð*s; <sup>u</sup>*1ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

*<sup>u</sup>*1ð ÞÞ � *<sup>s</sup> <sup>f</sup>*ð*s; <sup>u</sup>*2ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

<sup>j</sup>*u*<sup>1</sup> � *<sup>u</sup>*2jþj*D<sup>δ</sup>*

∥*Tu*<sup>1</sup> � *Tu*2∥<sup>∞</sup> ≤ ΛΦ∥*u*<sup>1</sup> � *u*2∥*E:* (47)

*<sup>u</sup>*1ð ÞÞ � *<sup>s</sup> <sup>f</sup>*ð*s; <sup>u</sup>*2ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

*<sup>u</sup>*1ð Þ*<sup>s</sup>* � � � *f s; <sup>u</sup>*2ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

<sup>0</sup><sup>j</sup> *f s; <sup>u</sup>*1ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

<sup>0</sup> *<sup>f</sup>*ð*s; u s*ð Þ*; <sup>D</sup><sup>δ</sup>*

3 5

*<sup>u</sup>*2ð Þ*<sup>s</sup>* � �∣*ds*

*<sup>u</sup>*1ð ÞÞ � *<sup>s</sup> <sup>f</sup>*ð*s; <sup>u</sup>*2ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

*u*2ð ÞÞj *s ds*

*<sup>u</sup>*<sup>1</sup> � *<sup>D</sup><sup>δ</sup>*

*<sup>u</sup>*2ð Þ*<sup>s</sup>* � �∣*ds*

*u*2ð ÞÞj *s ds*

*<sup>u</sup>*1ð Þ*<sup>s</sup>* � � � *f s; <sup>u</sup>*2ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

*<sup>u</sup>*2<sup>j</sup> � �*:*

3 5 ≔ A

*<sup>u</sup>*2ð Þ*<sup>s</sup>* � �j*ds*

3 5 ≔ B*:*

*u s*ð ÞÞ*ds*

ð ÞÞ *s ds*

(46)

*u*2ð ÞÞj *s ds*

<sup>0</sup> *f s; u s*ð Þ*; <sup>D</sup><sup>δ</sup>*

*sinλ η*ð Þ � *s J*

<sup>0</sup> *<sup>f</sup>*ð*s; u s*ð Þ*; <sup>D</sup><sup>δ</sup>*

Proof: We introduce the space

*DOI: http://dx.doi.org/10.5772/intechopen.91140*

Then, *E*, ∥*:*∥*<sup>E</sup>* ð Þ is a Banach space.

ð Þ *Tu* ð Þ*<sup>t</sup>* <sup>≔</sup> <sup>1</sup>

endowed with the norm <sup>∥</sup>*u*∥*<sup>E</sup>* ≔ ∥*u*∥<sup>∞</sup> <sup>þ</sup> <sup>∥</sup>*D<sup>δ</sup>*

Also, we consider the operator *T* : *E* ! *E* defined by

*λ* ð*t*

0

þ

� ð 1

Let *u*1, *u*<sup>2</sup> ∈*E*. Then, for each *t*∈ ½ � 0, 1 , we have

∣*sinλ*ð Þ *t* � *s* ∣ *J*

2 4

j*sinλ*ð Þj 1 � *s J*

1 *λ* þ 1 ∣Δ∣

With the same arguments as before, we can write.

2 4 *α*

ð *η*

0

*α*

<sup>0</sup><sup>∣</sup> *f s; <sup>u</sup>*1ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

j*sinλ η*ð Þj � *s J*

<sup>0</sup><sup>j</sup> *<sup>f</sup>*ð*s; <sup>u</sup>*1ð Þ*<sup>s</sup> ; <sup>D</sup><sup>δ</sup>*

*α*

∣*sinλ*ð Þ *t* � *s* ∣*J*

j*sinλ*ð Þj 1 � *s J*

<sup>þ</sup> <sup>∣</sup>*λcos*ð Þ *<sup>λ</sup><sup>t</sup>* <sup>∣</sup> <sup>∣</sup>Δ<sup>∣</sup> <sup>j</sup>*<sup>β</sup>*

ð *η*

0

*α*

<sup>∣</sup>*Tu*1ðÞ�*<sup>t</sup> Tu*2ð Þ*<sup>t</sup>* ∣ ≤ <sup>1</sup>

*λ* ð*t*

þ ð1

By (H1), we have

A ≤

Hence, it yields that

*<sup>u</sup>*2ð Þ*<sup>t</sup>* ∣ ≤ <sup>1</sup> *λ*

> þ ð1

> > 0

∣*T*<sup>0</sup>

**159**

*u*1ðÞ�*t T*<sup>0</sup>

0

Λ Γð Þ *α* þ 2

0

<sup>þ</sup> <sup>∣</sup>sin ð Þ *<sup>λ</sup><sup>t</sup>* <sup>∣</sup> <sup>∣</sup>Δ<sup>∣</sup> <sup>j</sup>*<sup>β</sup>*

0

is given by

$$u(t) = \frac{1}{\lambda} \Big| \int\_0^t \sin \lambda (t - s) \left( \int\_0^t \frac{(s - \tau)^{\alpha - 1}}{\Gamma(\alpha)} \theta(\tau) d\tau + \sum\_{i = 1}^{n - 1} c\_i s^i \right) ds + c\_n \cos(\lambda t) + c\_{n + 1} \sin(\lambda t), \tag{42}$$

where *ci* ∈ , *i* ¼ 1 … *n* þ 1.

Thanks to the above lemma, we can state that

The class of Langevin equations (39) and (40) has the following integral representation:

$$\begin{split} u(t) &= \frac{1}{\lambda} \Big[ \sin \lambda(t-s) \Big( \int\_{0}^{t} \frac{(s-\tau)^{a-1}}{\Gamma(a)} f(\tau, u(\tau), D^{\delta}(\tau)) d\tau \Big) ds \\ &+ \frac{\sin(\lambda t)}{\Delta} \Big[ \beta \Big[ \sin \lambda(\eta-s) \Big( \int\_{0}^{t} \frac{(s-\tau)^{a-1}}{\Gamma(a)} f(\tau, u(\tau), D^{\delta}(\tau)) d\tau \Big) ds \\ &- \int\_{0}^{1} \sin \lambda(1-s) \Big( \int\_{0}^{t} \frac{(s-\tau)^{a-1}}{\Gamma(a)} f(\tau, u(\tau), D^{\delta}(\tau)) d\tau \Big) ds \Big], \end{split} \tag{43}$$

where

$$
\Delta := \lambda (\sin \lambda - \beta \sin \lambda \eta). \tag{44}
$$

#### **4.2 Existence and uniqueness of solutions**

Using the above integral representation (43), we can prove the following existence and uniqueness theorem.

Theorem 1.33. Assume that the following hypotheses are valid:

(H1): The function *f* : ½ �� 0, 1 � ! is continuous, and there exist two constants Λ1,Λ<sup>2</sup> >0, such that for all *t*∈ ½ � 0, 1 and *ui*, *vi* ∈ , *i* ¼ 1, 2,

$$|f(t, \mu\_1, \mu\_2) - f(t, \nu\_1, \nu\_1)| \le \Lambda\_1 |\mu\_1 - \nu\_1| + \Lambda\_2 |\mu\_2 - \nu\_2|.\tag{45}$$

(H2): Suppose that Λ≤ <sup>1</sup> ð Þ <sup>Φ</sup>þ<sup>ϒ</sup> ,

where

$$\Phi \coloneqq \frac{\Delta\_1 + \lambda + \beta\_1 \lambda \eta^{a+1}}{\Gamma(a+2)\lambda \Delta\_1}, \Psi \coloneqq \frac{\Delta\_1 s^a (a+1) + \lambda^2 + \beta\_1 \lambda^2 \eta^{a+1}}{\Gamma(a+2)\lambda \Delta\_1}, \Upsilon \coloneqq \frac{\Psi}{\Gamma(2-\delta)},$$

$$\Lambda \coloneqq \max\left(\Lambda\_1, \Lambda\_2\right), \quad \Delta\_1 = |\Delta|, \quad \beta\_1 = |\beta|.$$

*Integral Inequalities and Differential Equations via Fractional Calculus DOI: http://dx.doi.org/10.5772/intechopen.91140*

Then problem (39) and (40) has a unique solution on 0, 1 ½ �. Proof: We introduce the space

$$E = \left\{ u; u \in \mathcal{C}([0, 1]), D^\delta u \in \mathcal{C}([0, 1]) \right\},$$

endowed with the norm <sup>∥</sup>*u*∥*<sup>E</sup>* ≔ ∥*u*∥<sup>∞</sup> <sup>þ</sup> <sup>∥</sup>*D<sup>δ</sup> u*∥∞*:* Then, *E*, ∥*:*∥*<sup>E</sup>* ð Þ is a Banach space. Also, we consider the operator *T* : *E* ! *E* defined by

$$\begin{aligned}(Tu)(t) & \coloneqq \frac{1}{\lambda} \Big[ \sin \lambda (t - s) f\_0^\alpha f \left( s, u(s), D^\delta u(s) \right) ds \\\\ & \quad + \frac{\sin \left( \lambda t \right)}{\Delta} \Bigg[ \beta \Big[ \sin \lambda (\eta - s) f\_0^\alpha f (s, u(s), D^\delta (s)) ds \\\\ & \quad - \int\_0^1 \sin \lambda (1 - s) f\_0^\alpha f (s, u(s), D^\delta u(s)) ds \Bigg] \end{aligned} \tag{46}$$

We shall prove that the above operator is contractive over the space *E*. Let *u*1, *u*<sup>2</sup> ∈*E*. Then, for each *t*∈ ½ � 0, 1 , we have

$$\begin{split} |T u\_1(t) - T u\_2(t)| &\le \frac{1}{\lambda} \left| \left| \sin \lambda(t - s) \right| \right| \left. f\_0^u(\boldsymbol{\delta}, u\_1(s), D^\delta u\_1(s)) \right| - f(s, u\_2(s), D^\delta u\_2(s)) |ds \\ &\quad + \frac{|\sin(\lambda t)|}{|\Delta|} \left[ |\beta| \left| \sin \lambda(\eta - s) \right| \right. \left. f\_0^u[f(s, u\_1(s), D^\delta u\_1(s)) - f(s, u\_2(s), D^\delta u\_2(s))] ds \right. \\ &\left. + \left[ \left| \sin \lambda(1 - s) \right| \left. f\_0^u[f(s, u\_1(s), D^\delta u\_1(s)) - f(s, u\_2(s), D^\delta u\_2(s))] ds \right. \right] := \mathcal{A} \end{split}$$

By (H1), we have

Lemma 1.32. Let *θ* be a continuous function on 0, 1 ½ �. The unique solution of the

*<sup>D</sup><sup>α</sup> <sup>D</sup>*<sup>2</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>2</sup> � �*u t*ðÞ¼ *<sup>θ</sup>*ð Þ*<sup>t</sup>* ,

*<sup>n</sup>* � <sup>1</sup><sup>&</sup>lt; *<sup>α</sup>*<sup>≤</sup> *<sup>n</sup>*, *<sup>n</sup>*<sup>∈</sup> <sup>∗</sup> ,

þ

*i*¼1 *cis i*

The class of Langevin equations (39) and (40) has the following integral repre-

ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>α</sup>*�<sup>1</sup>

ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>α</sup>*�<sup>1</sup>

Using the above integral representation (43), we can prove the following exis-

(H1): The function *f* : ½ �� 0, 1 � ! is continuous, and there exist two

Λ ≔ *max* ð Þ Λ1,Λ<sup>2</sup> , Δ<sup>1</sup> ¼ ∣Δ∣, *β*<sup>1</sup> ¼ ∣*β*∣*:*

∣ *f t*ð Þ� , *u*1, *u*<sup>2</sup> *f t*ð Þ , *v*1, *v*<sup>1</sup> ∣ ≤ Λ1∣*u*<sup>1</sup> � *v*1∣ þ Λ2∣*u*<sup>2</sup> � *v*2∣*:* (45)

*<sup>α</sup>*ð Þþ *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>λ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*1*λ*<sup>2</sup>

Γð Þ *α* þ 2 *λ*Δ<sup>1</sup>

*sinλ η*ð Þ � *s*

<sup>Γ</sup>ð Þ *<sup>α</sup> <sup>f</sup>*ð*τ; <sup>u</sup>*ð Þ*<sup>τ</sup> ; <sup>D</sup><sup>δ</sup>*

<sup>Γ</sup>ð Þ *<sup>α</sup> <sup>f</sup>*ð*τ; <sup>u</sup>*ð Þ*<sup>τ</sup> ; <sup>D</sup><sup>δ</sup>*

!

ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>α</sup>*�<sup>1</sup>

Δ ≔ *λ*ð Þ *sinλ* � *βsinλη :* (44)

*ηα*þ<sup>1</sup>

, <sup>ϒ</sup> <sup>≔</sup> <sup>Ψ</sup>

<sup>Γ</sup>ð Þ <sup>2</sup> � *<sup>δ</sup>* ,

ð*s* 0

!

1

A*ds* þ *cn cos*ð Þþ *λt cn*þ<sup>1</sup> *sin*ð Þ *λt* ,

ð ÞÞ *τ dτ*

<sup>Γ</sup>ð Þ *<sup>α</sup> <sup>f</sup>*ð*τ; <sup>u</sup>*ð Þ*<sup>τ</sup> ; <sup>D</sup><sup>δ</sup>*

ð ÞÞ *τ dτ*

!

*ds*

*ds* 3 5,

ð ÞÞ *τ dτ*

*ds*

(43)

(41)

(42)

*<sup>t</sup>* <sup>∈</sup>½ � 0, 1 , *<sup>λ</sup>*<sup>∈</sup> <sup>∗</sup>

<sup>Γ</sup>ð Þ *<sup>α</sup> θ τ*ð Þ*d<sup>τ</sup>* <sup>þ</sup>X*n*�<sup>1</sup>

ð*s* 0

ð *η*

2 4

0

Theorem 1.33. Assume that the following hypotheses are valid:

constants Λ1,Λ<sup>2</sup> >0, such that for all *t*∈ ½ � 0, 1 and *ui*, *vi* ∈ , *i* ¼ 1, 2,

ð Þ <sup>Φ</sup>þ<sup>ϒ</sup> ,

, <sup>Ψ</sup> <sup>≔</sup> <sup>Δ</sup>1*<sup>s</sup>*

ð*s* 0

*c*

ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>α</sup>*�<sup>1</sup>

problem

is given by

*Functional Calculus*

0

*sinλ*ð Þ *t* � *s*

where *ci* ∈ , *i* ¼ 1 … *n* þ 1.

*u t*ðÞ¼ <sup>1</sup> *λ* ð*t*

0

þ

� ð 1

**4.2 Existence and uniqueness of solutions**

tence and uniqueness theorem.

(H2): Suppose that Λ≤ <sup>1</sup>

<sup>Φ</sup> <sup>≔</sup> <sup>Δ</sup><sup>1</sup> <sup>þ</sup> *<sup>λ</sup>* <sup>þ</sup> *<sup>β</sup>*1*λη<sup>α</sup>*þ<sup>1</sup> Γð Þ *α* þ 2 *λ*Δ<sup>1</sup>

0

ð*s*

0 @

0

Thanks to the above lemma, we can state that

*sinλ*ð Þ *t* � *s*

sinð Þ *λt* <sup>Δ</sup> *<sup>β</sup>*

*sinλ*ð Þ 1 � *s*

*u t*ðÞ¼ <sup>1</sup> *λ* ð*t*

sentation:

where

where

**158**

$$\mathcal{A} \le \frac{\Lambda}{\Gamma(\alpha + 2)} \left( \frac{1}{\lambda} + \frac{1}{|\Delta|} + \frac{|\beta|}{|\Delta|} \eta^{\alpha + 1} \right) \left( |u\_1 - u\_2| + |D^\delta u\_1 - D^\delta u\_2| \right).$$

Hence, it yields that

$$\|Tu\_1 - Tu\_2\|\_\infty \le \Lambda \Phi \|u\_1 - u\_2\|\_E. \tag{47}$$

With the same arguments as before, we can write.

$$\begin{split} |T'u\_1(t) - T'u\_2(t)| &\leq \frac{1}{\lambda} |\sin \lambda(t-s)| f\_0^u |f\left(s, u\_1(s), D^\delta u\_1(s)\right) - f\left(s, u\_2(s), D^\delta u\_2(s)\right)| ds \\ &+ \frac{|\lambda \cos(\lambda t)|}{|\Delta|} \left[ |\partial \right]^\eta |\sin \lambda(\eta-s)| f\_0^u |f\left(s, u\_1(s), D^\delta u\_1(s)\right) - f\left(s, u\_2(s), D^\delta u\_2(s)\right)| ds \\ &+ \begin{bmatrix} 1 \\ |\sin \lambda(1-s)| f\_0^u |f(s, u\_1(s), D^\delta u\_1(s)) - f(s, u\_2(s), D^\delta u\_2(s))| ds \\ 0 \end{bmatrix} = \mathcal{B}. \end{split}$$

**159**

Again, by (H1), we obtain

$$\mathcal{B} \le \Lambda \left( \frac{\Delta\_1 s^a (a+1) + \lambda^2 + \beta\_1 \lambda^2 \eta^{a+1}}{\Gamma(a+2)\lambda \Delta\_1} \right) \left( |u\_1 - u\_2| + |D^\delta u\_1 - D^\delta u\_2| \right).$$

Consequently, we get

$$\|\|T'u\_1 - T'u\_2\|\|\_\infty \le \Lambda \Psi \|\|u\_1 - u\_2\|\|\_E.$$

This implies that

$$\|\|D^{\delta}Tu\_{1} - D^{\delta}Tu\_{2}\|\|\_{\infty} \leq \Lambda \Upsilon \|\|u\_{1} - u\_{2}\|\_{E}.\tag{48}$$

Step 2: Equicontinuity.

*λ* ð*t*2

þ

þ ð1

0

<sup>≤</sup> <sup>∣</sup>Θ<sup>∣</sup> ∣Δ∣

þ 1 *λ* ð*t*2

∣*sinλ η*ð Þ � *s J*

Analogously, we can obtain

Consequently, we can write

∣Θ∣ ∣Δ∣

∣*D<sup>δ</sup>*

The operator *T* is thus equicontinuous.

∣*D<sup>δ</sup>*

*u t*ð Þ<sup>∣</sup> <sup>¼</sup> <sup>∣</sup>*λD<sup>δ</sup>*

conclude that *T* is completely continuous.

*u t*ð Þ<sup>1</sup> ∣ ≤ *λ*

where

Θ ≔ *β* ð *η*

*u t*ð Þ� <sup>2</sup> *T*<sup>0</sup>

Therefore,

and

**161**

∣*T*<sup>0</sup>

0

0

*α*

<sup>0</sup><sup>∣</sup> *f s*, *u s*ð Þ, *<sup>D</sup><sup>δ</sup>*

*Tu t*ð Þ� <sup>2</sup> *<sup>D</sup><sup>δ</sup>*

∣*cos*ð Þ� *λt*<sup>2</sup> *cos*ð Þ *λt*<sup>1</sup> ∣ þ

As *t*<sup>1</sup> ! *t*2, the right-hand sides of (51) and (52) tend to zero.

*u s*ð Þ � �∣*ds* <sup>þ</sup>

2 4

*DOI: http://dx.doi.org/10.5772/intechopen.91140*

0

<sup>∣</sup>*Tu t*ð Þ� <sup>2</sup> *Tu t*ð Þ<sup>1</sup> ∣≤∣ <sup>1</sup>

Let *u*∈ *E:* Then, for each *t*1, *t*<sup>2</sup> ∈½ � 0, 1 , we have

*sinλ*ð Þ *t*<sup>2</sup> � *s J*

sin ð Þ� *λt*<sup>2</sup> sin ð Þ *λt*<sup>1</sup>

*sinλ*ð Þ 1 � *s J*

*α*

*Integral Inequalities and Differential Equations via Fractional Calculus*

<sup>Δ</sup> *<sup>β</sup>*

*α*

∣sin ð Þ� *λt*<sup>2</sup> sin ð Þ *λt*<sup>1</sup> ∣ þ

∣ð*sinλ*ð Þ� *t*<sup>2</sup> � *s sinλ*ð Þ *t*<sup>1</sup> � *s* ∣*J*

<sup>0</sup> *<sup>f</sup>*ð*s; u s*ð Þ*; <sup>D</sup><sup>δ</sup>*

ð *η*

2 4

<sup>0</sup> *<sup>f</sup>*ð*s; u s*ð Þ*; <sup>D</sup><sup>δ</sup>*

*sinλ η*ð Þ � *s J*

*u s*ð ÞÞ*ds*

1 *λ* ð*t*2

*t*1

*α*

ð 1

0

1�*δ* ∣*T*<sup>0</sup>

1 *λ*

*Tu t*ð Þ<sup>1</sup> ∣ ≤ *J*

∥*Tu t*ð Þ� <sup>2</sup> *Tu t*ð Þ<sup>1</sup> ∥*<sup>E</sup>* ! 0*:*

As a consequence of Step 1 and Step 2 and thanks to Arzela-Ascoli theorem, we

∣*u t*ð Þ∣ ¼ ∣*λTu t*ð Þ∣≤∣*Tu t*ð Þ∣ ≤ **M***Φ*

*Tu t*ð Þ∣≤∣*D<sup>δ</sup>*

Step 3: We prove that Σ ≔ f g *u*∈*E*; *u* ¼ *λTu*, 0 < *λ*< 1 is a bounded set. Let *u*∈ Σ. Then, for each *t* ∈½ � 0, 1 , the following two inequalities are valid:

0

*u s*ð ÞÞ*ds* �

*α*

*sinλ*ð Þ *t*<sup>2</sup> � *s J*

<sup>0</sup><sup>∣</sup> *f s; u s*ð Þ*; <sup>D</sup><sup>δ</sup>*

∣*sinλ*ð Þ 1 � *s* ∣*J*

∣*sinλ*ð Þ� *t*<sup>2</sup> � *s sinλ*ð Þ *t*<sup>1</sup> � *s* ∣*J*

*u t*ð Þ� <sup>2</sup> *T*<sup>0</sup>

*Tu t*ð Þ∣ ≤ **M***ϒ:*

3 5∣ ð*t*1

*sinλ*ð Þ *t*<sup>1</sup> � *s J*

*α*

*u s*ð ÞÞ*ds*

<sup>0</sup> *<sup>f</sup>*ð*s; u s*ð Þ*; <sup>D</sup><sup>δ</sup>*

*u s*ð ÞÞ*ds*

(51)

3 5

0

<sup>0</sup> *<sup>f</sup>*ð*s; u s*ð Þ*; <sup>D</sup><sup>δ</sup>*

*α*

*u s*ð Þ � �∣*ds*,

<sup>0</sup><sup>∣</sup> *f s; u s*ð Þ*; <sup>D</sup><sup>δ</sup>*

*α*

<sup>0</sup><sup>∣</sup> *f s*, *u s*ð Þ, *<sup>D</sup><sup>δ</sup>*

*α*

*u s*ð Þ � �∣*ds:*

<sup>0</sup><sup>∣</sup> *f s; u s*ð Þ*; <sup>D</sup><sup>δ</sup> u s*ð Þ � �∣*:*

*u t*ð Þ<sup>1</sup> ∣ (52)

*u s*ð Þ � �∣*ds*

Using (47) and (48), we can state that

$$\|Tu\_1 - Tu\_2\|\_E \le \Lambda(\Phi + \Upsilon) \|u\_1 - u\_2\|\_E.$$

Thanks to (H2), we can say that the operator *T* is contractive.

Hence, by Banach fixed point theorem, the operator has a unique fixed point which corresponds to the unique solution of our Langevin problem.

#### **4.3 Existence of solutions**

We prove the following theorem.

Theorem 1.34. Assume that the following conditions are satisfied:

(H3): The function *f* : ½ �� 0, 1 � ! is jointly continuous.

(H4): There exists a positive constant **M**; ∣*f t*ð Þ , *u*, *v* ∣ ≤ **M** for any *t*∈½ � 0, 1 , *u*, *v*∈ *:*

Then the problem (39), (40) has at least one solution on 0, 1 ½ �.

Proof: We use Schaefer fixed point theorem to prove this result. So we proceed into three steps.

Step 1: We prove that *T* is continuous and bounded.

Since the function *f* is continuous by (H3), then the operator is also continuous; this proof is trivial and hence it is omitted.

Let Ω ⊂ *E* be a bounded set. We need to prove that *T*ð Þ Ω is a bounded set. Let *u*∈ Ω*:* Then, for any *t* ∈½ � 0, 1 , we have

$$|(Tu)(t)| \leq \left(\frac{1}{\lambda} + \frac{1}{|\Delta|}\right) \int\_0^1 f\_0^a |f^\cdot(s, u(s), D^\delta u(s))| ds + \frac{|\beta|}{|\Delta|} \int\_0^\eta f\_0^a |f^\cdot(s, u(s), D^\delta(s))| ds := \mathcal{O}$$

Using (H4), we get

$$\|Tu\|\_{\infty} \le \phi \mathcal{M}.\tag{49}$$

In the same manner, we find that

$$\|D^\delta Tu\|\_\infty \le \mathcal{YM}.\tag{50}$$

From (49) and (50), we have

$$\|Tu\|\_{E} \le (\Phi + \Upsilon)\mathcal{M}.$$

The operator is thus bounded.

*Integral Inequalities and Differential Equations via Fractional Calculus DOI: http://dx.doi.org/10.5772/intechopen.91140*

#### Step 2: Equicontinuity.

Again, by (H1), we obtain

Δ1*s*

*<sup>α</sup>*ð Þþ *<sup>α</sup>* <sup>þ</sup> <sup>1</sup> *<sup>λ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*1*λ*<sup>2</sup>

Γð Þ *α* þ 2 *λ*Δ<sup>1</sup> � �

*u*<sup>1</sup> � *T*<sup>0</sup>

*Tu*<sup>1</sup> � *<sup>D</sup><sup>δ</sup>*

Thanks to (H2), we can say that the operator *T* is contractive.

which corresponds to the unique solution of our Langevin problem.

Theorem 1.34. Assume that the following conditions are satisfied: (H3): The function *f* : ½ �� 0, 1 � ! is jointly continuous. (H4): There exists a positive constant **M**; ∣*f t*ð Þ , *u*, *v* ∣ ≤ **M** for any

Then the problem (39), (40) has at least one solution on 0, 1 ½ �.

<sup>0</sup>∣*f s*, *u s*ð Þ, *<sup>D</sup><sup>δ</sup>*

∥*D<sup>δ</sup>*

Step 1: We prove that *T* is continuous and bounded.

∥*T*<sup>0</sup>

∥*D<sup>δ</sup>*

Using (47) and (48), we can state that

*η<sup>α</sup>*þ<sup>1</sup>

∥*Tu*<sup>1</sup> � *Tu*2∥*<sup>E</sup>* ≤ Λ Φð Þ þ ϒ ∥*u*<sup>1</sup> � *u*2∥*E:*

Hence, by Banach fixed point theorem, the operator has a unique fixed point

Proof: We use Schaefer fixed point theorem to prove this result. So we proceed

Since the function *f* is continuous by (H3), then the operator is also continuous;

Let Ω ⊂ *E* be a bounded set. We need to prove that *T*ð Þ Ω is a bounded set.

*u s*ð Þ � �∣*ds* <sup>þ</sup> <sup>∣</sup>*β*<sup>∣</sup>

∥*Tu*∥*<sup>E</sup>* ≤ ð Þ Φ þ ϒ **M***:*

∣Δ∣ ð*η* 0 *J α*

∥*Tu*∥<sup>∞</sup> ≤ *Φ***M***:* (49)

*Tu*∥<sup>∞</sup> ≤ *ϒ***M***:* (50)

<sup>0</sup>∣*f s*, *u s*ð Þ, *<sup>D</sup><sup>δ</sup>*

ð Þ*<sup>s</sup>* � �∣*ds* <sup>≔</sup> <sup>C</sup>

*u*2∥<sup>∞</sup> ≤ ΛΨ∥*u*<sup>1</sup> � *u*2∥*E:*

<sup>j</sup>*u*<sup>1</sup> � *<sup>u</sup>*2jþj*D<sup>δ</sup>*

*<sup>u</sup>*<sup>1</sup> � *<sup>D</sup><sup>δ</sup>*

*<sup>u</sup>*2<sup>j</sup> � �*:*

*Tu*2∥<sup>∞</sup> ≤ Λϒ∥*u*<sup>1</sup> � *u*2∥*E:* (48)

B ≤ Λ

*Functional Calculus*

Consequently, we get

This implies that

**4.3 Existence of solutions**

*t*∈½ � 0, 1 , *u*, *v*∈ *:*

into three steps.

<sup>∣</sup>ð Þ *Tu* ð Þ*<sup>t</sup>* ∣ ≤ <sup>1</sup>

**160**

Using (H4), we get

*λ* þ 1 ∣Δ∣ � �ð<sup>1</sup>

In the same manner, we find that

From (49) and (50), we have

The operator is thus bounded.

We prove the following theorem.

this proof is trivial and hence it is omitted.

Let *u*∈ Ω*:* Then, for any *t* ∈½ � 0, 1 , we have

0 *J α* Let *u*∈ *E:* Then, for each *t*1, *t*<sup>2</sup> ∈½ � 0, 1 , we have

$$\begin{aligned} |\operatorname{Tut}(t\_2) - \operatorname{Tut}(t\_1)| &\le \frac{1}{\lambda} \left[ \left| \int\_0^{t\_2} (\sin \lambda(t\_2 - s)I\_0^\alpha f(s, u(s), D^\beta u(s))) ds - \int\_0^{t\_1} \sin \lambda(t\_1 - s)I\_0^\alpha f(s, u(s), D^\beta u(s)) ds \right| \right] \\ &+ \frac{\sin \lambda(\operatorname{d}t\_2) - \sin \lambda(\operatorname{d}t\_1)}{\Delta} \left[ \beta \left| \int\_0^{\eta} \sin \lambda(\eta - s)I\_0^\alpha f(s, u(s), D^\beta u(s)) ds \right. \\ &\left. + \left| \sin \lambda(1 - s)I\_0^\alpha f(s, u(s), D^\beta u(s)) \right| ds \right] \Big| \\ &\le \frac{|\operatorname{G}|}{|\Delta|} |\sin \lambda(\operatorname{d}t\_2) - \sin (\operatorname{d}t\_1)| + \frac{1}{\lambda} \Big| \int\_0^{t\_2} |\sin \lambda(t\_2 - s)I\_0^\alpha| f(s, u(s), D^\beta u(s))| ds \\ &+ \frac{\imath\_2}{\lambda} \Big| |(\sin \lambda(t\_2 - s) - \sin \lambda(t\_1 - s))I\_0^\alpha| f(s, u(s), D^\beta u(s))| ds, \end{aligned} \tag{51}$$

where

$$\Theta := \beta \left\| \int\_0^\eta |\sin \lambda(\eta - s) f\_0^a| f\left(s, u(s), D^\delta u(s)\right)| ds + \int\_0^1 |\sin \lambda(1 - s)| f\_0^a| f\left(s, u(s), D^\delta u(s)\right)| ds \right\|$$

Analogously, we can obtain

$$|T'u(t\_2) - T'u(t\_1)| \leq \lambda \frac{|\Theta|}{|\Delta|} |\cos(\lambda t\_2) - \cos(\lambda t\_1)| + \frac{1}{\lambda} |\sin \lambda(t\_2 - s) - \sin \lambda(t\_1 - s)| f\_0^a[f'(s, u(s), D^\delta u(s))].$$

Consequently, we can write

$$|D^\delta Tu(t\_2) - D^\delta Tu(t\_1)| \le f^{1-\delta} |T'u(t\_2) - T'u(t\_1)|\tag{52}$$

As *t*<sup>1</sup> ! *t*2, the right-hand sides of (51) and (52) tend to zero. Therefore,

$$\|Tu(t\_2) - Tu(t\_1)\|\_E \to 0.$$

The operator *T* is thus equicontinuous.

As a consequence of Step 1 and Step 2 and thanks to Arzela-Ascoli theorem, we conclude that *T* is completely continuous.

Step 3: We prove that Σ ≔ f g *u*∈*E*; *u* ¼ *λTu*, 0 < *λ*< 1 is a bounded set. Let *u*∈ Σ. Then, for each *t* ∈½ � 0, 1 , the following two inequalities are valid:

$$|\mathfrak{u}(t)| = |\lambda T \mathfrak{u}(t)| \le |T \mathfrak{u}(t)| \le \mathcal{M} \mathfrak{G}$$

and

$$|D^\delta u(t)| = |\lambda D^\delta T u(t)| \le |D^\delta T u(t)| \le \mathcal{M} Y.$$

Therefore,

$$\|\mathfrak{u}\|\_{E} \leq \mathcal{M}(\Upsilon + \Phi).$$

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Thanks to steps 1, 2, and 3 and by Schaefer fixed point theorem, the operator *T* has at least one fixed point. This ends the proof of the above theorem.

### **5. Conclusions**

In this chapter, the fractional calculus has been applied for some classes of integral inequalities. In fact, using Riemann-Liouville integral, some Minkowski and Hermite-Hadamard-type inequalities have been established. Several other fractional integral results involving a family of positive functions have been also generated. The obtained results generalizes some classical integral inequalities in the literature. In this chapter, we have also presented some applications on continuous random variables; new identities have been established, and some estimates have been discussed.

The existence and the uniqueness of solutions for nonlocal boundary value problem including the Langevin equations with two fractional parameters have been studied. We have used Caputo approach together with Banach contraction principle to prove the existence and uniqueness result. Then, by application of Schaefer fixed point theorem, another existence result has been also proved. Our approach is simple to apply for a variety of real-world problems.

## **Author details**

Zoubir Dahmani<sup>1</sup> \* and Meriem Mansouria Belhamiti<sup>2</sup>


\*Address all correspondence to: zzdahmani@yahoo.fr

<sup>© 2020</sup> 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.

*Integral Inequalities and Differential Equations via Fractional Calculus DOI: http://dx.doi.org/10.5772/intechopen.91140*
