A New Result for -Hilfer Fractional Pantograph-Type Langevin Equation and Inclusions

Abstract

In this paper, we deal with the existence and uniqueness of solution for -Hilfer Langevin fractional pantograph differential equation and inclusion; these types of pantograph equations are a special class of delay differential equations. The existence and uniqueness results are obtained by making use of the Krasnoselskii fixed-point theorem and Banach contraction principle, and for the inclusion version, we use the Martelli fixed-point theorem to get the existence result. In the end, we are giving an example to illustrate our results.

1. Introduction

Over the past few years, fractional differential equations have attracted the interest of many mathematicians due to their ability to describe several complex problems in different scientific and engineering fields such as physics, biology, chemistry, and control theory (for more details, see [1–14]).

With the recent outstanding development in fractional differential equations, the Langevin equation has been considered a part of fractional calculus, and its form is , introduced by Paul Langevin in 1908. The Langevin equation is an effective tool that can describe processes as regards time evolution of the velocity of the Brownian motion [15–19] and also describe the evolution of physical phenomena in fluctuating environments [8] (for more details see, for example, [2, 20, 21]).

There are diverse definitions of fractional integrals and derivatives, the famous definitions are the Riemann-Liouville and the Caputo fractional derivatives. Hilfer [3] introduced the generalization of these derivatives under the name of Hilfer fractional derivative of order and parameter .

The authors in [14] have investigated the existence and uniqueness of solution for an initial value problem of Langevin equation involving two fractional orders, as follows: where is is the Caputo fractional derivative of order , is a given continuous function, and is a real number.

Motivated by the mentioned work and the new research going on in this direction, we study a new and a challenging case of fractional derivative called the -Hilfer derivative [22]; this brand of fractional derivative generalizes the well-known fractional derivatives (Riemman-Liouville, Caputo, -Riemman-Liouville, Hilfer-Hadamard, and Katugampola derivative), for different values of function and parameter ; these values are described in what follows.

In this paper, we study the existence and uniqueness results of the solutions for the following problem: where are the -Hilfer fractional derivative of order , and parameter , , , , , , , are the -Riemann-Liouville fractional integral of order , , , , and is a continuous function.

We also will study the multivalued version of the problem (2) by considering the following problem: where is a multivalued map ( is the family of all nonempty subjects of ).

The novelty of this paper and its challenges are that it collects and generalizes the types of fractional derivative, for different values of function and parameter such as follows: (i)If and , then the problems (2) and (3) reduce to the Caputo-type(ii)If and , then the problems (2) and (3) reduce to the Riemman-Liouville-type(iii)If , then the problems (2) and (3) reduce to the -Riemman-Liouville-type(iv)If , then the problems (2) and (3) reduce to the Hilfer-type(v)If , then the problems (2) and (3) reduce to the Hilfer-Hadamard-type(vi)If , then the problems (2) and (3) reduce to Katugampola-type

This paper is structured as follows: In the second section, we will present some auxiliary lemmas, some basic definitions, and theorems which are needed throughout this paper. In the third section, we discuss the existence and uniqueness results for the first problem, by using Krasnoselskii’s fixed-point theorem and Banach’s contraction principle. In the fourth section, we deal with the existence results for the inclusion version, by making use of Martelli’s fixed-point theorem, which is applicable to completely continuous operators. Finally, in the last part, we give an example to support our study.

2. Preliminaries and Notations

2.1. Fractional Calculus

In this section, we introduce some definitions, lemmas, and useful notations that will be used throughout this paper.

Let denote the Banach space of all continuous functions from into with the norm defined by . We denote by the -times absolutely continuous functions given by

Definition 2.1 [23]. Let , , be a finite or infinite interval of the half-axis and . In addition, let be a positive increasing function on , which has a continuous derivative on . The -Riemann-Liouville fractional integral of a function with respect to another function on is defined by where represents the Gamma function.

Definition 2.2 [23]. Let and , . The Riemann-Liouville derivative of a function with respect to another function of order , correspondent to the Riemann-Liouville is defined by where and denote the integer part of the real number .

Definition 2.3 [23]. Let with , is the interval such that and are two functions such that is increasing and for all . The -Hilfer fractional derivative of a function of order and type is defined by where and denote the integer part of the real number , with .

Lemma 2.4 [23]. Let . Then we have the following semigroup property given by

Proposition 2.5 [23, 24]. Let , , and . Then, -fractional integral and derivative of a power function are given by (i)(ii),

Lemma 2.6 [23]. If , , and , then for all , where

Lemma 2.7. Let , , , and . The function is a solution of the problem: if and only if where

Proof. The problem (10) can be written as Applying the -Riemann-Liouville fractional integral of order to both sides, we obtain the following by using Lemma 2.6where is constant and . Next, applying the -Riemann-Liouville fractional integral of order to both sides of (14), we get by using Lemma 2.6: From using the boundary condition in (15), we obtain that . We get From using the boundary condition , in (15), we find Substituting the value of in (16), we obtain the solution (11). The converse follows by direct computation.

2.2. Multivalued Analysis

For a normed space , we define

For more details of multivalued analysis, see ([2, 3]).

Definition 2.8. A multivalued map is said to be Carathéodory if (i) is measurable for each (ii) is upper semicontinuous for almost all Furthermore, a Carathéodory function is called -Carathéodory if: (iii)For each , there exists such thatfor all with and for a.e. .

Theorem 2.9 (Krasnoselskii fixed-point theorem [25]). Let be a closed, bounded, convex, and nonempty subset of a Banach space. Let be the operators such that (i) whenever (ii) is compact and continuous(iii) is contraction mappingThen there exists such that .

Theorem 2.10 (Martelli fixed-point theorem [26]). Let be a Banach space and be a completely continuous multivalued map. If the set is bounded, then has a fixed point.

3. Existence and Uniqueness Results for Problem (2)

In this section, we investigate the existence and uniqueness results for the problem (2), for this to simplify the computations, we use the following notations:

In view of Lemma 2.7, we define the operator by where denotes the Banach space of all continuous functions from into with the norm .

3.1. Existence Result via Krasnoselskii’s Fixed-Point Theorem

Theorem 3.1. Assume that
is a continuous function such that , , with
, where is given by (21)
Then there exists at least one solution for the problem (2) on .

Proof. Let and , where , we will show that the operator defined by (22) satisfies the conditions of Krasnoselskii’s fixed-point theorem, for that we split the operator into the sum of two operators and defined, on the closed ball, by For every , we have then , which implies that .
Since is continuous, then the operator is continuous, and it is uniformly bounded on as Next, we prove that the operator is compact, for that setting , and let , ; we obtain The right-hand side tends to zero as , independently of . Then, is equicontinuous; hence, is relatively compact on . By the Arzelà-Ascoli theorem, it implies that is compact on .
In the next step, we will show that is a contraction mapping; for that, let , and for , we have This implies that , by using ; we deduce that is a contraction mapping. It follows by using Krasnoselskii’s fixed-point theorem, the problem (2) has at least one solution on .

3.2. Uniqueness Result via Banach’s Fixed-Point Theorem

To deal with the uniqueness of solution for our problem (2), we use Banach’s fixed-point theorem.

Theorem 3.2. Assume that ; , for each and .
If , where are, respectively, given by (20) and (21); then the problem (2) has a unique solution on .

Proof. By considering the operator defined in (22), we transform the problem (2) into a fixed-point problem . By using Banach contraction principle, we will show that has a unique fixed point.
We set and choose such that , where are, respectively, given by (20) and (21).
Step 1: we show that .
For any we have Then we have which implies that .
Step 2: next we show that the operator is a contraction.
For any , and for , we have which implies . As , then is a contraction, and by applying Banach’s fixed-point theorem, we get that the operator has a unique fixed point which is the unique solution of our problem (2).

4. Existence Results for the Inclusion Version

In this section, we will investigate the existence result for the inclusion version defined as problem (3).

Definition 4.1. A continuous function is said to be a solution of problem (3) if , and there exists a function with , i.e., on such that for each , define the set of selections of by

Lemma 4.2 ([5]). Let be a Banach space and be a -Carathéodory multivalued map. And let be a linear continuous mapping from to . Then the operator is a closed graph operator in .

In what follows, we deal with the upper semicontinuous case, and for the existence results, we use Martelli’s fixed-point theorem.

Theorem 4.3. Suppose that holds and the following assumptions hold:
is -Carathéodory and has nonempty convex values, and for each fixed , the set is nonempty and convex.
for all and all , where and is continuous and nondecreasing function.
Then the problem (3) has at least one solution on .

Proof. In order to transform the problem (3) into a fixed-point problem. Let be defined by We will show that satisfies the conditions of the Theorem 2.10; for the poof, we give it in steps:
Step 1: is convex for each .
If belong to , then there exist such that for each we have: for . Let ; then, for each , we have Thus, (because is convex); then is convex for each
Step 2: is bounded.
For a positive number , let be bounded ball in ; then for each and , there exists , such that then for every , we have Then where are, respectively, given by (20) and (21).
Step 3: is equicontinuous.
Let , and where , as above then for each and ; there exist ; then we obtain As , the right-hand side of the above inequality tends to zero, implying that is equicontinuous. By using the Arzelà-Ascoli theorem, we get that is relatively compact; then is completely continuous.
To prove that the operator is upper semicontinuous, it is enough to show that has a closed graph.
Step 4: has a closed graph.
Let , and ; we will prove that .
For , then there exists such that for each : We should prove that such that for each We have that as .
Consider the operator defined by with By using Lemma 4.2, is a closed graph operator; then we get Since , and , then It follows that such that We deduce that is an upper semicontinuous multivalued map, with convex closed values.
Step 5: is bounded.
Let , and then for some ; thus, there exists a function such that From Step 2, and for every , we have Then, by using , we get Finally the set is bounded; then from Theorem 2.10, we deduce that the problem (3) has at least one solution.

5. Example

Consider the following -Hilfer fractional pantograph Langevin equation given by where , , , , , , , , , , , , , , , and . With this given data, we get , , , and .

Set , , .

Let and ; then we get

This implies that the assumption of Theorem 3.2 holds with .

It follows that ; then by applying Theorem 3.2, our problem has a unique solution on .

6. Conclusion

The present paper examined the -Hilfer fractional pantograph Langevin equation and inclusion. The challenges and the novelty of this work generalize the types of fractional derivatives. With the assistance of Krasnoselskii and Banach fixed-point theorems, we investigate the existence and uniqueness results for the single valued problem, and by making use of the Martelli fixed-point theorem, we study the existence result for the multivalued problem. In the end, we illustrate our result with an example.

Disclosure

No potential conflict of interest was reported by the authors.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the referees for the valuable comments and suggestions that improve the quality of our paper.