Research Article | Open Access
Three-Point Boundary Value Problems of Nonlinear Second-Order -Difference Equations Involving Different Numbers of
We study a new class of three-point boundary value problems of nonlinear second-order q-difference equations. Our problems contain different numbers of q in derivatives and integrals. By using a variety of fixed point theorems (such as Banach’s contraction principle, Boyd and Wong fixed point theorem for nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative) and Leray-Schauder degree theory, some new existence and uniqueness results are obtained. Illustrative examples are also presented.
The -difference calculus or quantum calculus is an old subject that was initially developed by Jackson , Carmichael , Mason , and Adams , in the first quarter of 20th century, has been developed over the years, for instance, see [5–14] and the references therein. In fact, -calculus has a rich history, and the details of its basic notions, results, and methods can be found in the text . In recent years, the topic has attracted the attention of several researchers, and a variety of new results can be found in the papers [16–28] and the references cited therein.
In , Ahmad et al. studied a boundary value problem of nonlinear -difference equations with nonlocal boundary conditions given by where , , and is a fixed constant. The existence of solutions for problem (1) is shown by means of a variety of fixed point theorems such as Banach's contraction principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative.
Yu and Wang  considered a boundary value problem with the nonlinear second-order -difference equation, where and is a fixed number. Existence and uniqueness of the solutions are obtained by means of Banach’s contraction principle, Leray-Schauder nonlinear alternative, and Leray-Schauder continuation theorem.
Pongarm et al.  considered sequential derivative of nonlinear -difference equation with three-point boundary conditions, where , , , and , are given constants. Existence results are proved based on Banach’s contraction mapping principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder degree theory.
We note that in the above-mentioned papers [24, 28] the -numbers in the equation and the boundary conditions are the same. As far as we know the paper by Pongarm et al.  is the first paper which has different values of the -numbers in -derivative and -integral.
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 constant, and . Also, , and , are given constants such that .
It is noteworthy that, in the above problem (4), we have three different values of the -numbers, in -derivatives and the -integral. Moreover, we emphasize the fact that, instead the value is usually used in the literature, we use the values of the function and its derivative in an intermediate point .
The aim of this paper is to prove some existence and uniqueness results for the boundary value problem (4). Our results are based on Banach’s contraction mapping principle, nonlinear contraction, Krasnoselskii’s fixed point theorem, Leray-Schauder nonlinear alternative, and Leray-Schauder degree theory.
The rest of the paper is organized as follows. In Section 2, we provide some basic definitions, preliminaries facts, and a lemma, which are used later. The main results are given in Section 3. In the end, Section 4, some results illustrating the results established in this paper are also presented.
Definition 1. For , one defines the -derivative of a real valued function as
The higher-order -derivatives are given by
For one sets and define, the definite -integral of a function by provided that the series converges.
For , one sets
Note that for , one has , for some ; thus, the definite integral is just a finite sum, so no question about convergence is raised.
One notes that while if is continuous at , then In -calculus, the product rule and integration by parts formula are Further, reversing the order of integration is given by In the limit , the above results correspond to their counterparts in standard calculus.
Lemma 2. Let and . Then, for any , the boundary value problem, is equivalent to the integral equation where
Proof. Taking double -integral for (13), we have By changing the order of -integration, we have In particular, for , we get Taking -derivative for (18), for , we obtain For , we have Therefore, Now, using the first condition of (14) with (19), (22), we have Taking the -integral for (18) from to , we obtain Substituting in (24) and using the second condition of (14), we get Solving the system of linear equations (23) and (25) for the unknown constants and , we have where is defined by (16). Substituting the values of and in (18), we obtain (15). This completes the proof.
Let denotes the Banach space of all the continuous functions from to endowed with the norm defined by . Define an operator by Observe that the problem (4) has solutions if and only if the operator has fixed points.
For the sake of convenience, we set a constant as
3. Main Results
Now, we are in the position to establish the main results. Our first result is based on Banach’s fixed point theorem.
Theorem 3. Assume that is a continuous function satisfying the conditions(H1), for all and , (H2),
where is a Lipschitz constant, and is defined by (28).
Then, the boundary value problem (4) has a unique solution.
Proof. We transform the boundary value problem (4) into a fixed point problem , where is defined by (27). Assume that , and choose a constant satisfying
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.
Next, we can still deduce the existence and uniqueness of a solution to the boundary value problem (4). We will use nonlinear contraction to accomplish this.
Definition 4. Let be a Banach space and let be a mapping. is said to be a nonlinear contraction if there exists a continuous nondecreasing function such that and for all with the following property:
Lemma 5 (Boyd and Wong ). Let be a Banach space and let be a nonlinear contraction. Then, has a unique fixed point in .
Proof. Let the operator be defined as (27). We define a continuous nondecreasing function by
such that and , for all .
Let . Then, we get Thus, This implies that . Hence, is a nonlinear contraction. Therefore, by Lemma 5, the operator has a unique fixed point in , which is a unique solution of problem (4).
The third result is based on the following Krasnoselskii fixed point theorem .
Theorem 7. 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 .
Proof. Setting and choosing a constant
we consider .
In view of Lemma 2, we define the operators and on the ball as For , by computing directly, we have Therefore, . Condition (38) 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 be zero. So, is relatively compact on . Hence, by the Arzelá-Ascoli Theorem, is compact on . Therefore, all the assumptions of Theorem 7 are satisfied, and the conclusion of Theorem 7 implies that the boundary value problem (4) has at least one solution on . This completes the proof.
As the fourth result, we prove the existence of solutions of (4) by using Leray-Schauder nonlinear alternative.
Theorem 9 (Nonlinear Alternative for Single Valued Maps ). Let be a Banach space, a closed convex subset of , 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 10. Assume that: () there exists a continuous nondecreasing function and a function such that () there exists a constant such that Then, the boundary value problem (4) has at least one solution on .
Proof. We will show that maps bounded sets (balls) into bounded sets in . For a positive number , let be a bounded ball in . Then, for , we have
Next, we will show that maps bounded sets into equicontinuous sets of . Let with and . Then, we have As , the right-hand side of the above inequality tends to zero independently of . As satisfies the above assumptions; therefore, it follows by the Arzelá-Ascoli theorem that is completely continuous.
Let be a solution. Then, for and following the similar computations as in the first step, we have
Consequently, we have In view of , there exists such that . Let us set Note that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Theorem 9), we deduce that has a fixed point which is a solution of the problem (4). This completes the proof.
Finally, we prove that problem (4) has at least one solution on by using Leray-Schauder degree theory.
Proof. Let us define an operator as (27). We wish to 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 conclude that a continuous map defined by is completely continuous. If (53) 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 it for , this yields Let , then (53) holds. This completes the proof
In this section, we illustrate our main results with some examples. Let us consider the following boundary value problem of nonlinear second-order -difference equations with three-point boundary conditions Here, we have , , , , , , and . We find that (a) Let be a continuous function given by Since, , then is satisfied with . We can find that Hence, by Theorem 3, problem (58) with given by (60) has a unique solution on .(b) If is a continuous function given by Choosing , we find that Here, Therefore, by Theorem 6, the problem (58) with given by (62) has a unique solution on .(c) Consider a continuous function given by We can show that with and . By Theorem 11, the problem (58) with the given by (65) has at least one solution on .
This research of T. Sitthiwirattham and J. Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a Member of Nonlinear Analysis and Applied Mathematics (NAAM), Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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