Integral Inequalities and Differential Equations via Fractional Calculus

*Zoubir Dahmani and Meriem Mansouria Belhamiti*

## **Abstract**

In this chapter, fractional calculus is used to develop some results on integral inequalities and differential equations. We develop some results related to the Hermite-Hadamard inequality. Then, we establish other integral results related to the Minkowski inequality. We continue to present our results by establishing new classes of fractional integral inequalities using a family of positive functions; these classes of inequalities can be considered as generalizations of order *n* for some other classical/fractional integral results published recently. As applications on inequalities, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to the classical case, are generalised for any *α* ≥1, *β* ≥1. For the part of differential equations, we present a contribution that allow us to develop a class of fractional chaotic electrical circuit. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equation. Then, by establishing some sufficient conditions, another result for the existence of at least one solution is also discussed.

**Keywords:** fractional calculus, fixed point, Riemann-Liouville integral, Caputo derivative, integral inequality

#### **1. Introduction**

During the last few decades, fractional calculus has been extensively developed due to its important applications in many field of research [1–4]. On the other hand, the integral inequalities are very important in probability theory and in applied sciences. For more details, we refer the reader to [5–12] and the references therein. Moreover, the study of integral inequalities using fractional integration theory is also of great importance; we refer to [1, 13–17] for some applications.

Also, boundary value problems of fractional differential equations have occupied an important area in the fractional calculus domain, since these problems appear in several applications of sciences and engineering, like mechanics, chemistry, electricity, chemistry, biology, finance, and control theory. For more details, we refer the reader to [3, 18–20].

In this chapter, we use the Riemann-Liouville integrals to present some results related to Minkowski and Hermite-Hadamard inequalities [21]. We continue to present our results by establishing several classes of fractional integral inequalities using a family of positive functions; these classes of inequalities can be considered as generalizations for some other fractional and classical integral results published recently [22]. Then, as applications, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to *α* ¼ 1, are generalized for any *α*≥ 1 and *β* ≥1; see [23].

**Definition 1.3.** The fractional *ω*�weighted variance of order *α* >0 for a random

**Definition 1.4.** The fractional *ω*�weighted moment of orders *r* >0, *α* >0 for a continuous random variable *X* having a *p:d:f: f* defined on [a, b] is defined by the

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

**Definition 1.5.** Let *f* <sup>1</sup> and *f* <sup>2</sup> be two continuous on ½ � *a*, *b :* We define the fractional

*Varα*,*<sup>ω</sup>*ð Þ *X* ≔ *Varα*ð Þ *X* ,*Covα*,*<sup>ω</sup>*ð Þ *X* ≔ *Covα*ð Þ *X* , *Eα*,*<sup>ω</sup>*ð Þ *X* ≔ *Eα*ð Þ *X*

**Definition 1.6.** For a function *<sup>K</sup>* <sup>∈</sup>*C<sup>n</sup>*ð Þ ½ � *<sup>a</sup>*, *<sup>b</sup>* , and *<sup>n</sup>* � <sup>1</sup><sup>&</sup>lt; *<sup>α</sup>* <sup>≤</sup> *<sup>n</sup>*, the Caputo

*dtn* ð Þ *K t*ð Þ

ð*t*

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>n</sup>*�*α*�<sup>1</sup>

*ci*ð Þ *t* � *a i*

> *ci*ð Þ *t* � *a i*

*a*

Lemma 1.7. Let *<sup>n</sup>* <sup>∈</sup> <sup>∗</sup> , and *<sup>n</sup>* � <sup>1</sup><sup>&</sup>lt; *<sup>α</sup>*<sup>&</sup>lt; *<sup>n</sup>*. The general solution of *<sup>D</sup><sup>α</sup>y t*ðÞ¼ 0,

*y t*ðÞ¼ <sup>X</sup>*<sup>n</sup>*�<sup>1</sup>

*i*¼0

*i*¼0

*<sup>n</sup>*�*<sup>α</sup> <sup>d</sup><sup>n</sup>*

<sup>¼</sup> <sup>1</sup> Γð Þ *n* � *α* *τr*

ð Þ *<sup>τ</sup>* � *E X*ð Þ <sup>2</sup>

*ω τ*ð Þ*f*ð Þ*τ dτ*, *α* >0*:* (5)

*ω τ*ð Þ*f*ð Þ*τ dτ*, *α* >0*:* (6)

(7)

ð Þ *<sup>b</sup>* � *<sup>τ</sup> <sup>α</sup>*�<sup>1</sup> *<sup>f</sup>* <sup>1</sup>ð Þ� *<sup>τ</sup> <sup>f</sup>* <sup>1</sup>ð Þ *<sup>μ</sup>* � � *<sup>f</sup>* <sup>2</sup>ð Þ� *<sup>τ</sup> <sup>f</sup>* <sup>2</sup>ð Þ *<sup>μ</sup>* � �*ω τ*ð Þ*f*ð Þ*<sup>τ</sup> <sup>d</sup>τ*, *<sup>α</sup>* <sup>&</sup>gt;0,

*<sup>K</sup>*ð Þ *<sup>n</sup>* ð Þ*<sup>s</sup> ds:*

, (8)

, *t*∈½ � *a*, *b* , (9)

variable *X* having a *p:d:f: f* on [a, b] is given by

*<sup>E</sup><sup>α</sup>*,*<sup>ω</sup> <sup>X</sup><sup>r</sup>* ð Þ <sup>≔</sup> <sup>1</sup>

Γð Þ *α* ð *b*

where *μ* is the classical expectation of *X*.

fractional derivative of order *α* is defined by

We recall also the following properties.

where *ci* ∈ , *i* ¼ 0, 1, 2, *::*, *n* � 1*:*

*J*

for some *ci* ∈ , *i* ¼ 0, 1, 2, *::*, *n* � 1*:*

*<sup>D</sup>αK t*ðÞ¼ *<sup>J</sup>*

Lemma 1.8. Let *<sup>n</sup>* <sup>∈</sup> <sup>∗</sup> and *<sup>n</sup>* � <sup>1</sup><sup>&</sup>lt; *<sup>α</sup>*<sup>&</sup>lt; *<sup>n</sup>*. Then

*<sup>α</sup>D<sup>α</sup>y t*ðÞ¼ *y t*ðÞþX*<sup>n</sup>*�<sup>1</sup>

*a*

Γð Þ *α*

ð *b*

*Integral Inequalities and Differential Equations via Fractional Calculus*

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

*a*

Γð Þ *α*

We introduce the covariance of fractional order as follows.

*<sup>ω</sup>*�weighted covariance of order *<sup>α</sup>* <sup>&</sup>gt;0 for *<sup>f</sup>* <sup>1</sup>ð Þ *<sup>X</sup>* , *<sup>f</sup>* <sup>2</sup>ð Þ *<sup>X</sup>* � � by

It is to note that when *ω*ð Þ¼ *x* 1, *x*∈½ � *a*, *b* , then we put

ð *b*

*a*

*<sup>α</sup>*,*<sup>ω</sup>*ð Þ¼ *<sup>X</sup> <sup>V</sup><sup>α</sup>*,*<sup>ω</sup>*ð Þ *<sup>X</sup>* <sup>≔</sup> <sup>1</sup>

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

*σ*2

quantity:

*Cov<sup>α</sup>*,*<sup>ω</sup> <sup>f</sup>* <sup>1</sup>ð Þ *<sup>X</sup>* , *<sup>f</sup>* <sup>2</sup>ð Þ *<sup>X</sup>* � � <sup>≔</sup> <sup>1</sup>

*t*∈½ � *a*, *b* is given by

**151**

For the part of differential equations, with my coauthor, we present a contribution that allows us to develop a class of fractional differential equations generalizing the chaotic electrical circuit model. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equations. Then, by establishing some sufficient conditions on the data of the problem, another result for the existence of at least one solution is also discussed. The considered class has some relationship with the good paper in [20].

The chapter is structured as follows: In Section 2, we recall some preliminaries on fractional calculus that will be used in the chapter. Section 3 is devoted to the main results on integral inequalities as well as to some estimates on continuous random variables. The Section 4 deals with the class of differential equations of Langevin type: we study the existence and uniqueness of solutions for the considered class by means of Banach contraction principle, and then using Schaefer fixed point theorem, an existence result is discussed. At the end, the Conclusion follows.

## **2. Preliminaries on fractional calculus**

In this section, we present some definitions and lemmas that will be used in this chapter. For more details, we refer the reader to [2, 13, 15, 24].

**Definition 1.1.** The Riemann-Liouville fractional integral operator of order *α* ≥0, for a continuous function f on ½ � *a*, *b* is defined as

$$\begin{aligned} J\_a^a[f(t)] &= \frac{1}{\Gamma(a)} \int\_a^t (t-\tau)^{a-1} f(\tau)d\tau, a>0, \ a$$

where <sup>Γ</sup>ð Þ *<sup>α</sup>* <sup>≔</sup> <sup>Ð</sup> <sup>∞</sup> <sup>0</sup> *<sup>e</sup>*�*uu<sup>α</sup>*�<sup>1</sup>*du:* Note that for *α* >0, *β* > 0, we have

$$J^a J^\beta f(t) = J^{a+\beta} f(t),\tag{2}$$

and

$$J\_a^a J\_a^\beta [f(t)] = J\_a^\beta J\_a^a [f(t)]. \tag{3}$$

In the rest of this chapter, for short, we note a probability density function by *p:d:f*. So, let us consider a positive continuous function *ω* defined on ½ � *a*, *b* . We recall the *ω*�concepts:

**Definition 1.2.** The fractional *ω*�weighted expectation of order *α* > 0, for a random variable *X* with a positive *p:d:f: f* defined on ½ � *a*, *b* , is given by

$$E\_{a,a}(X) \coloneqq f\_a^a[\![taf]\!](b) = \frac{1}{\Gamma(a)} \int\_a^b (b-\tau)^{a-1} \tau a(\tau) f(\tau) d\tau, a>0, a$$

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

**Definition 1.3.** The fractional *ω*�weighted variance of order *α* >0 for a random variable *X* having a *p:d:f: f* on [a, b] is given by

$$\sigma\_{a,\nu}^2(X) = V\_{a,\nu}(X) \coloneqq \frac{1}{\Gamma(a)} \int\_a^b (b-\tau)^{a-1} (\tau - E(X))^2 a(\tau) f(\tau) d\tau, a > 0. \tag{5}$$

**Definition 1.4.** The fractional *ω*�weighted moment of orders *r* >0, *α* >0 for a continuous random variable *X* having a *p:d:f: f* defined on [a, b] is defined by the quantity:

$$E\_{a,a}(X^r) \coloneqq \frac{1}{\Gamma(a)} \int\_a^b (b-\tau)^{a-1} \tau^r \alpha(\tau) f(\tau) d\tau, a > 0. \tag{6}$$

We introduce the covariance of fractional order as follows.

**Definition 1.5.** Let *f* <sup>1</sup> and *f* <sup>2</sup> be two continuous on ½ � *a*, *b :* We define the fractional *<sup>ω</sup>*�weighted covariance of order *<sup>α</sup>* <sup>&</sup>gt;0 for *<sup>f</sup>* <sup>1</sup>ð Þ *<sup>X</sup>* , *<sup>f</sup>* <sup>2</sup>ð Þ *<sup>X</sup>* � � by

$$\text{Cov}\_{a,a}\left(f\_1(X), f\_2(X)\right) \coloneqq \frac{1}{\Gamma(a)} \int\_a^b (b-\tau)^{a-1} \left(f\_1(\tau) - f\_1(\mu)\right) \left(f\_2(\tau) - f\_2(\mu)\right) \omega(\tau) f(\tau) d\tau, a > 0,\tag{7}$$

where *μ* is the classical expectation of *X*. It is to note that when *ω*ð Þ¼ *x* 1, *x*∈½ � *a*, *b* , then we put

$$Var\_{a, \alpha}(X) := Var\_a(X), \\ Cov\_{a, \alpha}(X) := Cov\_a(X), \\ E\_{a, \alpha}(X) := E\_a(X)$$

**Definition 1.6.** For a function *<sup>K</sup>* <sup>∈</sup>*C<sup>n</sup>*ð Þ ½ � *<sup>a</sup>*, *<sup>b</sup>* , and *<sup>n</sup>* � <sup>1</sup><sup>&</sup>lt; *<sup>α</sup>* <sup>≤</sup> *<sup>n</sup>*, the Caputo fractional derivative of order *α* is defined by

$$\begin{aligned} D^a K(t) &= J^{n-a} \frac{d^n}{dt^n} (K(t)) \\ &= \frac{1}{\Gamma(n-a)} \int\_a^t (t-s)^{n-a-1} K^{(n)}(s) ds. \end{aligned}$$

We recall also the following properties.

Lemma 1.7. Let *<sup>n</sup>* <sup>∈</sup> <sup>∗</sup> , and *<sup>n</sup>* � <sup>1</sup><sup>&</sup>lt; *<sup>α</sup>*<sup>&</sup>lt; *<sup>n</sup>*. The general solution of *<sup>D</sup><sup>α</sup>y t*ðÞ¼ 0, *t*∈½ � *a*, *b* is given by

$$y(t) = \sum\_{i=0}^{n-1} c\_i (t - a)^i,\tag{8}$$

where *ci* ∈ , *i* ¼ 0, 1, 2, *::*, *n* � 1*:* Lemma 1.8. Let *<sup>n</sup>* <sup>∈</sup> <sup>∗</sup> and *<sup>n</sup>* � <sup>1</sup><sup>&</sup>lt; *<sup>α</sup>*<sup>&</sup>lt; *<sup>n</sup>*. Then

$$J^a D^a \jmath(t) = \jmath(t) + \sum\_{i=0}^{n-1} c\_i (t - a)^i, t \in [a, b], \tag{9}$$

for some *ci* ∈ , *i* ¼ 0, 1, 2, *::*, *n* � 1*:*

using a family of positive functions; these classes of inequalities can be considered as generalizations for some other fractional and classical integral results published recently [22]. Then, as applications, we generate new lower bounds estimating the fractional expectations and variances for the beta random variable. Some classical covariance identities, which correspond to *α* ¼ 1, are generalized for any *α*≥ 1 and

For the part of differential equations, with my coauthor, we present a contribution that allows us to develop a class of fractional differential equations generalizing the chaotic electrical circuit model. We prove recent results for the existence and uniqueness of solutions for a class of Langevin-type equations. Then, by establishing some sufficient conditions on the data of the problem, another result for the existence of at least one solution is also discussed. The considered class has

The chapter is structured as follows: In Section 2, we recall some preliminaries on fractional calculus that will be used in the chapter. Section 3 is devoted to the main results on integral inequalities as well as to some estimates on continuous random variables. The Section 4 deals with the class of differential equations of Langevin type: we study the existence and uniqueness of solutions for the considered class by means of Banach contraction principle, and then using Schaefer fixed point theorem, an existence result is discussed. At the end, the Conclusion follows.

In this section, we present some definitions and lemmas that will be used in this

**Definition 1.1.** The Riemann-Liouville fractional integral operator of order *α* ≥0,

*f*ð Þ*τ dτ*, *α* > 0, *a*< *t*≤ *b*,

*<sup>α</sup>*þ*<sup>β</sup>f t*ð Þ, (2)

*<sup>a</sup>*½ � *f t*ð Þ *:* (3)

*τω τ*ð Þ*f*ð Þ*τ dτ*, *α* > 0, *a*< *t*≤ *b*, (4)

(1)

*β* ≥1; see [23].

*Functional Calculus*

some relationship with the good paper in [20].

**2. Preliminaries on fractional calculus**

for a continuous function f on ½ � *a*, *b* is defined as

*<sup>a</sup>*½ �¼ *f t*ð Þ <sup>1</sup>

*<sup>a</sup>* ½ �¼ *f t*ð Þ *f t*ð Þ,

<sup>0</sup> *<sup>e</sup>*�*uu<sup>α</sup>*�<sup>1</sup>*du:* Note that for *α* >0, *β* > 0, we have

*J α*

*J* 0

where <sup>Γ</sup>ð Þ *<sup>α</sup>* <sup>≔</sup> <sup>Ð</sup> <sup>∞</sup>

and

the *ω*�concepts:

*E<sup>α</sup>*,*<sup>ω</sup>*ð Þ *X* ≔ *J*

**150**

*α*

*<sup>a</sup>*½ � *<sup>t</sup>ω<sup>f</sup>* ð Þ¼ *<sup>b</sup>* <sup>1</sup>

chapter. For more details, we refer the reader to [2, 13, 15, 24].

ð*t*

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

*<sup>β</sup>f t*ðÞ¼ *<sup>J</sup>*

*<sup>a</sup>*½ �¼ *f t*ð Þ *J*

*β aJ α*

In the rest of this chapter, for short, we note a probability density function by *p:d:f*. So, let us consider a positive continuous function *ω* defined on ½ � *a*, *b* . We recall

**Definition 1.2.** The fractional *ω*�weighted expectation of order *α* > 0, for a

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

random variable *X* with a positive *p:d:f: f* defined on ½ � *a*, *b* , is given by

ð *b*

*a*

*a*

*J αJ*

*J α a J β*

Γð Þ *α*

Γð Þ *α*
