Nonlinear Fractional -Difference Equation with Fractional Hadamard and Quantum Integral Nonlocal Conditions

Abstract

In this paper, we establish existence and uniqueness results for a boundary value problem consisting by a nonlinear fractional -difference equation subject to a new type of boundary condition, combining the fractional Hadamard and quantum integrals. Our analysis is based on Banach’s fixed point theorem, a fixed point theorem for nonlinear contractions, Krasnosel’ski ’s fixed point theorem, and Leray-Schauder nonlinear alternative. Examples are given to illustrate our results.

1. Introduction

The aim of this paper is to investigate the existence and uniqueness of solutions for a nonlinear fractional -difference equation subject to fractional Hadamard and quantum integral condition of the form: where is the fractional -derivative of order , with a quantum number , is a nonlinear continuous function, denotes the fractional quantum integral of order , with quantum number , is the Hadamard fractional integral of order , and are given constants, and are fixed points, for and .

The subject of fractional differential equations has recently evolved into an interesting subject for many researchers due to its multiple applications in economics, engineering, physics, chemistry, signal analysis, etc. Various types of fractional derivative and integral operator were studied: Riemann-Liouville, conformable fractional integral operators, Caputo, Hadamard, Erdelyi-Kober, Grünwald-Letnikov, Marchaud, and Riesz are just a few to name. The Hadamard-type fractional derivative differs from the preceding ones in the sense that the kernel of the integral and derivative contain logarithmic function of arbitrary exponent. Details and properties of Hadamard fractional derivatives and integrals can be found in Kilbas et al. [1]. Recently, there were some results on Hadamard-type fractional differential equations, see [2–11] and references cited therein.

Nonlinear fractional -difference equations appear in the mathematical modeling of many phenomena in engineering and science and have attracted much attention by many researchers, see for example [12–21] and references therein.

In the present paper, the novelty lies in the fact that we combine in boundary conditions both Hadamard and quantum integrals. To the best of our knowledge, this type of boundary condition appears for the first time in the literature. It is important to notice that we are combining in our work, fractional calculus, and quantum calculus. The key tool for this combination is the Property 2.25 of [1].

Some special cases of the second condition of (1) can be seen by reducing as which is mixed quantum and Hadamard calculus. If , then we have which is also mixed Riemann-Liouville and Hadamard fractional integral condition. If , we have integral condition of the form: which is a variety used in physical boundary value problems.

We establish existence and uniqueness results by using standard fixed point theorems. We prove two existence and uniqueness results with the help of the Banach contraction mapping principle and a fixed point theorem on nonlinear contractions due to Boyd and Wong. Moreover, we prove two existence results, one via Leray-Schauder nonlinear alternative and another one via Krasnosel’ski ’s fixed point theorem.

The paper is organized as follows: in Section 2, we recall some preliminary facts that we need in the sequel. In Section 3, we prove our main results. Some examples to illustrate our results are presented in Section 4.

2. Preliminaries

To present the preliminary, we suggest the basic quantum calculus in the book of Kac and Cheung [22], fractional quantum calculus in [23–25], and the Hadamard fractional calculus in [1]. Let a fixed constant be a quantum number. The -number is defined by

For example, . The -power function for any , , is defined as

If , then and . For example, . The notation of -power function is appeared in kernels of fractional -calculus as Definitions 1 and 2. Now, the -gamma function is defined by

Now, we observe that . Next, we discuss about the -derivative of a function which is defined by

If exists, then . The -integral formula can be presented as

The higher order of -derivative and -integral operators is with and . Next, the fundamental theorem of calculus for operators and can be stated as formulas and if is continuous at the point then

Let us give the definitions of fractional quantum calculus of the Riemann-Liouville type fractional derivative and also integral operators.

Definition 1 [24]. Let a constant and be the function on . The Riemann-Liouville fractional -integral of order is defined by and .

Definition 2 [24]. The Riemann-Liouville fractional -derivative of order of a function is given by and , where is the smallest integer greater than or equal to .

Now, for , the -beta function is presented by which is related to the -gamma function by

The fundamental formulas for fractional quantum calculus are in the following lemma.

Lemma 3 [24, 26]. Let , be a positive integer and be a function defined in . Then, the following formulas hold

The fractional -integration of the two deferent quantum numbers is given by lemma.

Lemma 4 [27]. Let constants and be quantum numbers. Then, for , we have

The Hadamard fractional calculus is the subject of fractional derivative and integral which have logarithm kernels inside the singular integral formulas as in the definitions.

Definition 5 [1]. The Hadamard derivative of fractional order for a function is defined as where the notation denotes the integer part of the real number , , and is the usual Gamma function.

Definition 6 [1]. The Hadamard fractional integral of order for a function is defined by provided the integral in right hand side exists.

The key tool for combining the two type of fractional calculus in our work is the following lemma.

Lemma 7 ([1], Property 2.25). Let and The following formulas hold

To accomplish our main purpose, we will use the fixed point theory for considering an operator equation . For finding the operator , let us see the following lemma.

Lemma 8. Suppose that the points , and the constant where , , , , , , , and are defined in problem (1). Then, the linear fractional -difference equation where , and subject to mixed fractional integrals of Hadamard and quantum boundary conditions is equivalent to the linear integral equation

Proof. Since , then (23) can be written as Applying the fractional -integral of order and using Lemma 3, we obtain which yields where . The first boundary condition of (24) implies that Then, (28) is reduced to Now, we apply the fractional quantum integral of Riemann-Liouville of order with quantum number to (29) as Using Lemma 7 for taking the Hadamard fractional integral of order to (29), we get From the second boundary condition of (24) and above two equations, it follows that and consequently where the nonzero constant is defined by (22). Substituting the constant in (29), then, we obtain (25), which is the solution of BVP (23) and (24). The converse can be obtained by a direct computation. The proof is completed.

3. Main Results

At first, we denote by the Banach space of all continuous functions from to endowed with the sup norm as . In view of Lemma 8 and replacing the function by , we define the operator by where is denoted by while and are the Hadamard and quantum fractional integrals of a function as respectively. Now, we are going to prove the main results which are the existence criteria of solution for nonlocal mixed fractional integrals boundary value problem (1). The first, an existence and uniqueness result for (1), is given by using Banach’s fixed point theorem.

Theorem 9. Let be a nonlinear continuous function satisfying the assumption.

There exists a positive constant such that , for each and .

If where is given by then the boundary value problem (1) has a unique solution on .

Proof. The result allows from the operator equation , where the operator is defined by (34). The Banach fixed point theorem is used to show that has a fixed point which is the unique solution of problem (1). Since the function is continuous, then, we can set . After that, we define the radius satisfying of a ball For any , we see that in which we used the following fact: where . By applying Lemmas 4 and 2.3, we have Then, we obtain From this, we conclude that which yields
Next, we will prove that the operator is a contraction. Let , and for each then, we have Hence, we get the result that As from (37), the operator is a contraction. Applying the well known Banach fixed point theorem, it follows that has a fixed point which is the unique solution of the boundary value problem (1). This completes the proof.

Next, the nonlinear contraction theorem will be used to prove a second existence and uniqueness result.

Definition 10. Let be a Banach space and let be a mapping. The operator is said to be a nonlinear contraction if there exists a continuous nondecreasing function such that and for all with the property:

Lemma 11 (see [28]). Let be a Banach space and let be a nonlinear contraction. Then, has a unique fixed point in

Theorem 12. Suppose that a continuous function satisfies the condition:
, , where the function is continuous, and a positive constant is defined by

Then, the mixed fractional Hadamard and quantum integrals nonlocal problem (1) has a unique solution on .

Proof. Let us consider the operator defined in (34) and define a continuous nondecreasing function by Then, we see that the function satisfies and for all .
Next, for any and for each we obtain which implies that and also satisfies Definition 10. Therefore, is a nonlinear contraction. Thus, by applying Lemma 11, the operator has a unique fixed point which is the unique solution of the boundary value problem (1). The proof is finished.

Next, the first existence result will be obtained by applying the following theorem.

Theorem 13 (Nonlinear alternative for single valued maps) [29]. Let be a Banach space, a closed, convex subset of be an open subset of , and Suppose that is a continuous, compact (that is, is a relatively compact subset of) map. Then, either (i) has a fixed point in or(ii)There is a (the boundary of in ) and with

Theorem 14. Suppose that is a nonlinear continuous function which satisfies the following conditions:
there exists a continuous nondecreasing function and also a function such that there exists a positive constant such that where defined by (38). Then, the problem (1) has at least one solution on

Proof. For a positive number , we let be a bounded ball in . Now, we will prove that the set is uniformly bounded. For , we can compute that which can be deduced that Then, the set is uniformly bounded. Next, we will show that the set is equicontinuous set of For any two points with and , we have As the right hand side of the above inequality converses to zero, independently of . Then, the set is equicontinuous. Thus, we conclude that the set is relatively compact. Therefore, by the Arzel -Ascoli theorem, the operator is completely continuous.
Finally, we show that the operator cannot be fulfilled the condition in Theorem 13. Then, we have to claim that there exists an open set with for and Then, for each , we apply the computation in the first step, that is which yields inequality The condition implies that there exists a constant such that Now, we define the set From the previous results, we obtain that the operator is continuous and completely continuous. Then, there is no such that for some By applying the nonlinear alternative of the Leray-Schauder type, we get that the operator has a fixed point which is a solution of the nonlinear fractional -difference equation with fractional Hadamard and quantum integral nonlocal conditions. This finishes the proof.

The next existence result is based on Krasnosel’ski ’s fixed point theorem which can be used to relax the condition in Theorem 9.

Theorem 15 (Krasnosel’ski ’s fixed point theorem) [30]. Let be a closed, bounded, convex, and nonempty subset of a Banach space Let be the operators such that () whenever ; (b) is compact and continuous; () is a contraction mapping. Then, there exists such that

Theorem 16. Assume that a continuous function is satisfied condition in Theorem 9 and is bounded as the following condition: (i), and

If inequality holds, then the nonlocal problem (1) has at least one solution on

Proof. Now, we define and choose a positive constant such that where is defined by (38), to be a radius of the ball . Furthermore, we set the operators and on as and in Theorem 15, respectively, by The combination of two operators shows . We have Therefore, we have , and thus condition of Theorem 15 is satisfied. Since the function is fulfilled by condition in Theorem 9, then the operator is a contraction mapping with inequality (57).
Finally, we will show that the operator should satisfy condition in Theorem 15. Using the continuity of , we can show that the operator is continuous. The uniformly boundedness of the set can be shown by To prove is equicontinuous set, we let two points , . For any we have which converses to zero independently of as . So, is an equicontinuous set. Therefore, is a relative compact and by the Arzelá-Ascoli theorem, is compact on . Thus, the assumptions , and of Krasnosel’ski ’s fixed point theorem are satisfied. Then, the nonlinear fractional -difference equation with fractional Hadamard and quantum integral nonlocal conditions (1) has at least one solution on The proof is completed.

Remark 17. The interchanging of operators and gives another result by replacing inequality (57) by the following condition:

4. Examples

Example 18. Consider the nonlinear fractional -difference equation with fractional Hadamard and quantum integral nonlocal conditions of the form:

Here, , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Then, we can compute constants as and .

Let the nonlinear function be defined by

Then, by direct computation, we get , which satisfies condition in Theorem 9 with . Therefore, we have

By the conclusion of Theorem 9, the boundary value problem (64) with (65) has a unique solution on .

Consider now the function by

Then, we can see that

Setting and , we have , and there exists a constant satisfying inequality in . Hence, all assumptions in Theorem 14 are completed. Thus, the problem (64) with (67) has at least one solution on .

If the function is then, we have with . If , then Theorem 9 cannot be used to apply for the problem (64) with (69). For example, if , then . But the inequality in Remark 17 is satisfied as

Hence, by applying Theorem 16 and Remark 17, the problem (64) with (69) has at least one solution on .

5. Conclusion

We investigated the existence and uniqueness of solutions for a nonlocal boundary value problem involving a -difference equation, supplemented with a new type of boundary condition, including both Hadamard fractional and quantum integrals. In our first two results, we establish the existence and uniqueness of solutions by using Banach’s fixed point theorem and a fixed point theorem for nonlinear contractions due to Boyd and Wong. Then, we used the Leray-Schauder nonlinear alternative and Krasnosel’ski ’s fixed point theorem to derive two existence results. Examples are also presented to illustrate our results. It is worthwhile to point out that the results presented in this paper are new and significantly contribute to the existing literature on the topic.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-61-KNOW-031.