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.
- fractional calculus
- fixed point
- Riemann-Liouville integral
- Caputo derivative
- integral inequality
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 . 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 . 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 , are generalized for any and ; see .
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 .
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
Definition 1.1. The Riemann-Liouville fractional integral operator of order , for a continuous function f on is defined as
Note that for , we have
In the rest of this chapter, for short, we note a probability density function by . So, let us consider a positive continuous function defined on . We recall the concepts:
Definition 1.2. The fractional weighted expectation of order , for a random variable with a positive defined on , is given by
Definition 1.3. The fractional weighted variance of order for a random variable having a on [a, b] is given by
Definition 1.4. The fractional weighted moment of orders for a continuous random variable having a defined on [a, b] is defined by the quantity:
We introduce the covariance of fractional order as follows.
Definition 1.5. Let and be two continuous on We define the fractional weighted covariance of order for by
where is the classical expectation of .
It is to note that when , then we put
Definition 1.6. For a function and , the Caputo fractional derivative of order is defined by
We recall also the following properties.
Lemma 1.7. Let , and . The general solution of is given by
Lemma 1.8. Let and . Then
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 .
Theorem 1.9. Let and let be two positive functions on such that for all If then we have
Proof: We use the hypothesis We can write
Hence, we have
Thus, it yields that
In the same manner, we have
Remark 1.10. Applying the above theorem for we obtain Theorem 1.2 of  on
With the same arguments as before, we present the following theorem.
Theorem 1.11. Let and let be two positive functions on such that for all If then we have
Remark 1.12. Taking in this second theorem, we obtain Theorem 2.2 in  on
Using the notions of concave and functions, we present to the reader the following result.
Theorem 1.13. Suppose that and let be two positive functions on If are two concave functions on then we have
The proof of this theorem is based on the following auxiliary result.
Lemma 1.14. Let be a concave function on Then for any we have
3.2 A family of fractional integral inequalities
Theorem 1.15. Suppose that are positive, continuous, and decreasing functions on Then, the following inequality
holds for any where is a fixed integer in .
Proof: It is clear that
for any fixed and for any
we observe that
Also, we have
Hence, we get
The proof is thus achieved.
Remark 1.16. Applying Theorem 1.15 for we obtain Theorem 3 in .
Using other sufficient conditions, we prove the following generalization.
Theorem 1.17. Suppose that are positive, continuous, and decreasing functions on Then for any fixed in and for any we have
Proof: Multiplying both sides of (23) by then integrating the resulting inequality with respect to over and using Fubini’s theorem, we obtain the desired inequality.
Applying Theorem 1.17 for we obtain Theorem 1.15.
Applying Theorem 1.17 for we obtain Theorem 3 of .
Introducing a positive increasing function to the family , we establish the following theorem.
Theorem 1.19. Let and be positive continuous functions on such that is increasing and are decreasing on Then, the following inequality
holds for any where is a fixed integer in
Remark 1.20. Applying Theorem 1.19 for we obtain Theorem 4 of .
3.3 Some estimations on random variables
3.3.1 Bounds for fractional moments of beta distribution
In what follows, we present some fractional results on the beta distribution . So let us prove the following version.
Theorem 1.21. Let , and be four random variables, such that , and . If , then
For the proof of this result, we can apply a weighted version of the fractional Chebyshev inequality as is mentioned in .
Remark 1.22. The above theorem generalizes Theorem 3.1 of .
We propose also the following version that generalizes the above result. We have
Theorem 1.23. Let , and be four random variables, such that , and . If , then
Remark 1.24. If , then the above theorem reduces to Theorem 3.1 of .
3.3.2 Identities and lower bounds
In the following theorem, the fractional covariance of and is expressed with the derivative of . It can be considered as a generalization of a covariance identity established by the authors of . So, we prove the result:
Theorem 1.25. Let be a random variable having a defined on . Then, we have
We can prove this result by the application of the covariance definition in the case where
The following theorem establishes a lower bound for of any function . We have
Theorem 1.26. Let be a random variable having a defined on , such that . Then, we have
for any .
To prove this result, we use fractional Cauchy-Schwarz inequality established in .
Remark 1.27. Let us consider that satisfies . Then, we present the following result.
Theorem 1.28. Let be a random variable having a defined on such that and . Then, we have
Proof: We have
On the other hand, we can see that
Thanks to the fractional version of Cauchy Schwarz inequality , and using the fact that
This ends the proof.
It generalizes the good standard identity obtained in  that corresponds to 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 be a continuous random variable with a having a support an interval , . Then, for any , the following general covariance identity holds
where , with is a given function and
Proof: We have
The definition of implies that
Hence, we obtain
Remark 1.31. Taking , in the above theorem, we obtain Theorem 2.2 of .
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 So, let us consider the following problem:
associated with the conditions
where denotes the Caputo fractional derivative of fractional order , is the two-order classical derivative, is a given function, and , such that
4.1 Integral representation
We recall the following result :
Lemma 1.32. Let be a continuous function on . The unique solution of the problem
is given by
Thanks to the above lemma, we can state that
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 is continuous, and there exist two constants such that for all and
(H2): Suppose that
Proof: We introduce the space
endowed with the norm
Then, is a Banach space.
Also, we consider the operator defined by
We shall prove that the above operator is contractive over the space .
Let . Then, for each , we have
By (H1), 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
Thanks to (H2), we can say that the operator 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 is jointly continuous.
(H4): There exists a positive constant for any
Proof: We use Schaefer fixed point theorem to prove this result. So we proceed into three steps.
Step 1: We prove that is continuous and bounded.
Since the function is continuous by (H3), then the operator is also continuous; this proof is trivial and hence it is omitted.
Let be a bounded set. We need to prove that is a bounded set.
Let Then, for any we have
Using (H4), we get
In the same manner, we find that
The operator is thus bounded.
Step 2: Equicontinuity.
Let Then, for each , we have
Analogously, we can obtain
Consequently, we can write
The operator is thus equicontinuous.
As a consequence of Step 1 and Step 2 and thanks to Arzela-Ascoli theorem, we conclude that is completely continuous.
Step 3: We prove that is a bounded set.
Let . Then, for each , the following two inequalities are valid:
Thanks to steps 1, 2, and 3 and by Schaefer fixed point theorem, the operator has at least one fixed point. This ends the proof of the above theorem.
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.