Integral Inequalities and Differential Equations via Fractional Calculus

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.


Introduction
During the last few decades, fractional calculus has been extensively developed due to its important applications in many field of research [1][2][3][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][6][7][8][9][10][11][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][14][15][16][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][19][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].
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.

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 a, b ½ is defined as where Γ α ð Þ ≔ Ð ∞ 0 e Àu u αÀ1 du: Note that for α > 0, β > 0, we have and 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 a, b ½ . We recall the ωÀconcepts: Definition 1.2. The fractional ωÀweighted expectation of order α > 0, for a random variable X with a positive p:d:f : f defined on a, b ½ , is given by Definition 1.3. The fractional ωÀweighted variance of order α > 0 for a random variable X having a p:d:f : f on [a, b] is given by Definition 1.4. The fractional ωÀweighted 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: We introduce the covariance of fractional order as follows. Definition 1.5. Let f 1 and f 2 be two continuous on a, b ½ : We define the fractional ωÀweighted covariance of order α > 0 for where μ is the classical expectation of X. It is to note that when ω x ð Þ ¼ 1, x ∈ a, b ½ , then we put Þand n À 1 < α ≤ n, the Caputo fractional derivative of order α is defined by We recall also the following properties. Lemma 1.7. Let n ∈  * , and n À 1 < α < n. The general solution of D α y t ð Þ ¼ 0, t ∈ a, b ½ is given by where c i ∈ , i ¼ 0, 1, 2, ::, n À 1: Lemma 1.8. Let n ∈  * and n À 1 < α < n. Then for some c i ∈ , i ¼ 0, 1, 2, ::, n À 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 0, ∞ , then we have Proof: We use the hypothesis f τ Hence, we have Thus, it yields that In the same manner, we have And then, 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, t ½ : 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, ∞ , Remark 1.12. Taking α ¼ 1 in this second theorem, we obtain Theorem 2.2 in [26] on 0, t ½ : Using the notions of concave and L p Àfunctions, 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, ∞ : The proof of this theorem is based on the following auxiliary result. Lemma 1.14. Let h be a concave function on a, b ½ : Then for any

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 f i À Á i¼1, … n are n positive, continuous, and decreasing functions on a, b ½ : Then, the following inequality for we observe that Hence, we get 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 f i À Á i¼1, … n are positive, continuous, and decreasing functions on a, b ½ : Then for any fixed p in 1, 2, … , n f gand for any Proof: Multiplying both sides of (23) by tÀρ ing the resulting inequality with respect to ρ over a, t ð Þ, a < t ≤ b and using Fubini's theorem, we obtain the desired inequality. Remark 1.18.
ii. Applying Theorem 1.17 for Introducing a positive increasing function g to the family f i À Á i¼1, … n , we establish the following theorem. Theorem 1.19. Let f i À Á i¼1, … n and g be positive continuous functions on a, b ½ , such that g is increasing and f i À Á i¼1, … n are decreasing on a, b ½ : Then, the following inequality holds for any a < t ≤ b, α > 0, δ > 0, β ≥ γ p > 0, where p is a fixed integer in 1, 2, … , n f g : Remark 1.20. Applying Theorem 1.19 for α ¼ 1, t ¼ b, n ¼ 1, we obtain Theorem 4 of [27].

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 αÀversion.
Theorem 1.21. Let X, Y, U, and V be four random variables, such that X $ 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 α, β ð ÞÀversion that generalizes the above result. We have Theorem 1.23. Let X, Y, U, and V be four random variables, such that X Remark 1.24. If α ¼ β ¼ 1, then the above theorem reduces to Theorem 3.1 of [7].

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 a, b ½ ; μ ¼ E X ð Þ. Then, we have We can prove this result by the application of the covariance definition in the case where ω x ð Þ ¼ 1: The following theorem establishes a lower bound for Var α g X ð Þ ð Þof any function g ∈ C 1 a, b ½ ð Þ. We have Theorem 1.26. Let X be a random variable having a p:d:f defined on a, b ½ , such that μ ¼ E X ð Þ. Then, we have for any g ∈ C 1 a, b ½ ð Þ.
To prove this result, we use fractional Cauchy-Schwarz inequality established in [29]. Remark 1.27. Let us consider Ω ∈ C a, b Then, we present the following result. Theorem 1.28. Let X be a random variable having a p:d:f : defined on a, b ½ , such Proof: We have On the other hand, we can see that Thanks to the fractional version of Cauchy Schwarz inequality [29], and using the fact that we obtain This ends the proof. Remark 1.29. Thanks to (30) and (31), we obtain the following fractional covariance identity It generalizes the good standard identity obtained in [28] that corresponds to α ¼ 1 and it is given by 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 a, b ½ , E X ð Þ ¼ μ. Then, for any α ≥ 1, the following general covariance identity holds where g ∈ C 1 a, b ½ ð Þ, with E|Z X ð Þg 0 X ð Þ| < ∞, h x ð Þ is a given function and and The definition of Z X ð Þ implies that Hence, we obtain

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: associated with the conditions where c D α denotes the Caputo fractional derivative of fractional order α, D 2 is the two-order classical derivative, f : 0, 1 ½ Â Â  !  is a given function, and β ∈ , such that βsin λη ð Þ 6 ¼ sin λ ð Þ:

Integral representation
We recall the following result [20]: 9 Integral Inequalities and Differential Equations via Fractional Calculus DOI: http://dx.doi.org/10.5772/intechopen.91140 Lemma 1.32. Let θ be a continuous function on 0, 1 ½ . The unique solution of the problem is given by where c i ∈ , 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: where Δ ≔ λ sinλ À βsinλη ð Þ :

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 : 0, 1 ½ Â Â  !  is continuous, and there exist two constants Λ 1 , Λ 2 > 0, such that for all t ∈ 0, 1 ½ and Then problems (39) and (40) have a unique solution on 0, 1 ½ . Proof: We introduce the space Then, E, ∥:∥ E ð Þis a Banach space. Also, we consider the operator T : E ! E defined by We shall prove that the above operator is contractive over the space E . Let u 1 , u 2 ∈ E. Then, for each t ∈ 0, 1 ½ , we have Hence, it yields that With the same arguments as before, we can write. Again, by (H1), we obtain Consequently, we get This implies that Using (47) and (48), we can state that 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.

Existence of solutions
We prove the following theorem. 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 Ω ð Þ is a bounded set. Let u ∈ Ω: Then, for any t ∈ 0, 1 ½ , we have Using (H4), we get ∥Tu∥ ∞ ≤ ΦM: In the same manner, we find that ∥D δ Tu∥ ∞ ≤ ϒ M: From (49) and (50), we have The operator is thus bounded. Therefore, 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.

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.