Research Article | Open Access
Sequential Derivatives of Nonlinear -Difference Equations with Three-Point -Integral Boundary Conditions
This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinear -difference equations with three-point -integral boundary conditions. Our results are concerned with several quantum numbers of derivatives and integrals. By using Banach's contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.
The study of -calculus or quantum calculus was initiated by the pioneer works of Jackson , Carmichael , Mason , Adams , Trjitzinsky , and so forth. Since then, in the last few decades, this subject has evolved into a multidisciplinary research area with many applications; for example, see [6–14]. For some recent works, we refer the reader to [15–21] and references therein. However, the theory of boundary value problems for nonlinear -difference equations is still in the beginning stages and it needs to be explored further.
In , Ahmad investigated the existence of solutions for a nonlinear boundary value problem of third-order -difference equation: Using Leray-Schauder degree theory and standard fixed-point theorems, some existence results were obtained. Moreover, he showed that if , then his results corresponded to the classical results. Ahmad et al.  studied a boundary value problem of a nonlinear second-order -difference equation with nonseparated boundary conditions They proved the existence and uniqueness theorems of the problem (2) using the Leray-Schauder nonlinear alternative and some standard fixed-point theorems. For some very recent results on nonlocal boundary value problems of nonlinear -difference equations and inclusions, see [24–26].
In this paper, we discuss the existence of solutions for the following nonlinear -difference equation with three-point integral boundary condition: where , , , , , , is a fixed point, and is a given constant.
The aim of this paper is to prove some existence and uniqueness results for the boundary value problem (3). Our results are based on Banach’s contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory. Since the problem (3) has different values of the quantum numbers of the -derivative and the -integral, the existence results of such problem are also new.
Let us recall some basic concepts of quantum calculus .
For , we define the -derivative of a real-valued function as The higher-order -derivatives are given by The -integral of a function defined on the interval is given by and for = 0, we denote provided the series converges. If and is defined on the interval , then Similarly, we have Observe that and if is continuous at , then In -calculus, the product rule and integration by parts formula are In the limit , the -calculus corresponds to the classical calculus. The above results are also true for quantum numbers , such that and .
Lemma 1. Let , , , , and let be a constant. Then for any , the boundary value problem is equivalent to the integral equation
Proof. For , -integrating (13) from to , we obtain Equation (16) can be written as For , -integrating (17) from to , we have From the first condition of (14), it follows that . For , -integrating equation (18) from to , we get The second boundary condition (14) implies that Therefore, Substituting the values of and in (18), we obtain (15). This completes the proof.
For the forthcoming analysis, let denote the Banach space of all continuous functions from to endowed with the norm defined by .
In the following, for the sake of convenience, we set
3. Main Results
Now, we are in the position to establish the main results. We transform the boundary value problem (3) into a fixed-point problem. In view of Lemma 1, for , , we define the operator as Note that the problem (3) has solutions if and only if the operator equation has fixed points.
Our first result is based on Banach’s fixed-point theorem.
Theorem 2. Assume that is a jointly continuous function satisfying the conditions , for all , ; ,where is a Lipschitz constant, and are defined by (22) and (23), respectively.
Then, the boundary value problem (3) has a unique solution.
Proof. Assume that ; we choose a constant
Now, we will show that , where . For any , we have
Next, we will show that is a contraction. For any and for each , we have Since , is a contraction. Thus, the conclusion of the theorem follows by Banach's contraction mapping principle. This completes the proof.
Our second result is based on the following Krasnoselskii’s fixed-point theorem .
Theorem 3. Let be a bounded closed convex and nonempty subset of a Banach space . Let be operators such that (i) whenever ;(ii) is compact and continuous;(iii) is a contraction mapping.Then, there exists such that .
Theorem 4. Assume that and hold with
, for all , with .
If then the boundary value problem (3) has at least one solution on .
Proof. Setting and choosing a constant
where and are given by (22) and (23), respectively, we consider that .
In view of Lemma 1, we define the operators and on the set as for . By computing directly, we have Therefore, . The condition (28) implies that is a contraction mapping. Next, we will show that is compact and continuous. Continuity of coupled with the assumption implies that the operator is continuous and uniformly bounded on . We define . For with and , we have Actually, as , the right-hand side of the above inequality tends to zero. So is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Therefore, all the assumptions of Theorem 5 are satisfied and the conclusion of Theorem 5 implies that the three-point integral boundary value problem (3) has at least one solution on . This completes the proof.
As the third result, we prove the existence of solutions of (3) by using Leray-Schauder degree theory.
Proof. Let us define an operator as in (24). We will prove that there exists at least one solution of the fixed-point equation We define a ball , with a constant radius , given by Then, it is sufficient to show that satisfies Now, we set Then, by the Arzelá-Ascoli theorem, we get that is completely continuous. If (35) holds, then the following Leray-Schauder degrees are well defined. From the homotopy invariance of topological degree, it follows that where denotes the unit operator. By the nonzero property of Leray-Schauder degree, for at least one . Let us assume that for some . Then, for all , we obtain Taking norm and solving for , this yields Let , then (35) holds. This completes the proof.
In this section, we give two examples to illustrate our results.
Example 6. Consider the following nonlinear -difference equation with boundary value problem:
Set , , , , , , , and . Since , then, and are satisfied with , . Hence, . Therefore, by Theorem 2, the boundary value problem (40) has a unique solution on .
Example 7. Consider the following nonlinear -difference equation with boundary value problem:
Set , , , , , , and . Here, . So, , , and Hence, by Theorem 5, the boundary value problem (42) has at least one solution on .
The authors would like to thank the reviewers for their valuable comments and suggestions on the paper. This research is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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