Advances in Mathematical Physics
Volume 2018 (2018), Article ID 9089865, 9 pages
https://doi.org/10.1155/2018/9089865

Research Article

On the Second-Order Quantum -Difference Equations with Separated Boundary Conditions

Chanon Promsakon,¹ Nattapong Kamsrisuk,¹ Sotiris K. Ntouyas,2,3 and Jessada Tariboon¹

¹Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
²Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
³Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Jessada Tariboon; jessada.t@sci.kmutnb.ac.th

Received 19 August 2018; Accepted 12 November 2018; Published 2 December 2018

Academic Editor: Antonio Scarfone

Copyright © 2018 Chanon Promsakon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we investigate the existence and uniqueness of solutions for a boundary value problem for second-order quantum -difference equations with separated boundary conditions, by using classical fixed point theorems. Examples illustrating the main results are also presented.

1. Introduction

Quantum calculus or -calculus is known as the study of calculus without limits. The study of -calculus was initiated by Euler on studying infinite series. In 1910, Jackson [1, 2] was the first one, who establised the -derivative or -difference operator for a function on by

and gave its properties. The -integral of a function on is defined by

provided that infinite series converges. The general theory of linear quantum difference equations was published in 1912 by Carmichael [3]. Details of its basic notions, results, and methods can be found in the text [4]. For other papers on the subject, see [5–7]. In recent years, the topic has been attracting the attention of several researchers and a variety of new results can be found in [8–13] and the references therein. In addition, the classical quantum calculus was generalized to -calculus by Tariboon and Ntouyas [14]. For details, we refer to the recent monograph [15].

There are some applications of -calculus and difference equations to molecular problems in physics. In 1967, Finkelstein [16] studied behaviors of hydrogen atoms by using Schrödinger equation and -calculus. In [17], the author investigated the -field theory. The -Coulomb problem and -hydrogen atom were studied by [18–21]. In addition, Yang-Mills theories and also -Yang-Mills equation were developed by [22–24]. The theory of quantum group applied to vibration and rotation molecules with -algebra and -Heisemberg algebra technique was established in [25–27]. The subject of elementary particle physics and chemical physics using -calculus was investigated in [28–31]. The string theory involving -calculus was studied in [32].

The subject of -calculus first appeared in quantum algebras which contained two quantum numbers and introduced by [33]. For some recent results see [34–41] and the references cited therein. Recently in [42], the authors initiated the study on boundary value problems for -difference equations by investigating the existence and uniqueness of solution for first-order quantum -difference equation subject to a nonlocal condition of the form

where , , , are quantum numbers, is -difference operator, , , , , , are given constants, and the points , . They proved existence and uniqueness results for problem (3)-(4) by using the classical fixed point theorems, such as Banach’s fixed point theorem, Boyd and Wong fixed point theorem for nonlinear contractions, and Leray-Schauder nonlinear alternative.

In this paper, we continue the study on boundary value problems, by considering the following second-order quantum -difference equation with separated boundary conditions

where are two given quantum numbers, is the second-order -difference operator, and , , , , , are given real constants. Some new existence results are established by using Schaefer fixed point theorem, Krasnoselskii’s fixed point theorem, and Lelay-Schauder’s nonlinear alternative. In addition the uniqueness of solutions is established via Banach contraction mapping principle.

The paper is organized as follows: In Section 2, we recall some definitions and basic facts from -calculus. The main existence and uniqueness results are given in Section 3. Examples illustrating the obtained results are presented in Section 4.

2. Preliminaries

In this section, we recall some basic concepts of -calculus (see [41]). The -number is defined by

where . For each , , the -factorial and -binomial are defined by

and

respectively.

Let . The -derivative of function is defined as

and . Observe that the function is defined on which is extended from of a function . In addition, we say that is -differentiable on provided that exists for all .

Let a function Then the -integral of is defined by

provided that the right hand side converges. Note that the domain of function is which shrinks from of a function since .

In the following theorems we collect the basic properties of -differentiation and -integration, respectively. See [37].

Theorem 1 (see [37]). Suppose that are -differentiable on . Then
(a) is -differentiable on , and

(b) is -differentiable on for any constant , and

(c) is -differentiable on , and

(d) If , then is -differentiable on with

Theorem 2 (see [37]). Let be continuous functions and constants , . The following formulas hold:
(a) The -integration by parts is given by

(b) .
(c) .
(d) where .

Theorem 3 (see [42]). Let a function and constants . Then for , we have

The second-order -difference of a function can be expressed as

Now, for , we transform the linear second-order -difference equation with separated boundary condition

into an integral equation which will be used to define the solution of the boundary value problem (5). For convenience we put constants ,

Lemma 4. The unique solution of linear problem (18) is given by

Proof. Applying -integration to the first equation of (18), we have

Taking -integration to (21) and using Theorem 3, we have

For convenience, we set constants , and then

In particular for in (23), we have

Using (21) and by replacing , we obtain

From the first boundary condition of (18) and the above setting constants, we get

The second boundary condition of (18) with (24) and (25) yields

Solving (26) and (27) for constants and , it follows that

and

Substituting constants and into (23), we obtain the unique solution in (20) of linear problem (18). The proof is completed.

3. Main Results

Let denote the Banach space of all continuous functions from to endowed with sup-norm In view of Lemma 4 we define an operator by

It should be noticed that problem (5) has solutions if and only if the operator of equation has fixed points. Our first result is an existence theorem for separated boundary value problem of quantum -difference equation (5) by using Schaefer’s fixed point theorem.

Theorem 5. Assume that

(H₁) is a continuous function and there exists a constant such that

 for each and all .

Then the separated boundary value problem (5) has at least one solution on

Proof. Schaefer’s fixed point theorem is used to prove that the operator defined by (30) has at least one fixed point. So, we divide the proof into four steps.
Step 1 (continuity of ). Let be a sequence of function such that on . Given that is continuous function on , we have

Therefore, we obtain

which implies that as . Hence the operator is continuous.
Step 2 ( maps bounded sets into bounded sets in ). Choosing , we define a bounded ball as . Then, for any , we have

This means that . Therefore, the set is uniformly bounded.
Step 3 ( maps bounded sets into equicontinuous sets of ). Let with be two points and be a bounded ball in . Then for any , we get

As , the right-hand side of the above inequality (which is independent of ) tends to zero. This shows that the set is equicontinuous set. From a consequence of Steps 1 to 3, together with the Arzelá-Ascoli theorem, we deduce that the operator is completely continuous.
Step 4. Finally, we show that the set

is bounded.
Let be a solution of problem (5). Then for some . Hence, for each , by the method of computation in Step 2, we obtain

This gives that the set is bounded. By applying Schaefer’s fixed point theorem, we get that has at least one fixed point which is a solution of the second-order quantum -difference equation with separated boundary value problem (5) on . The proof is completed.

The second existence of a unique solution is based on Banach’s contraction mapping principle. For convenience we set a positive constant

Theorem 6. Assume that the function is continuous and satisfying the assumption:
(H₂) There exists constant such that for each and .
If

where is defined by (38), then the boundary value problem of quantum -difference equation (5) has a unique solution on .

Proof. The existence of a unique fixed point of operator equation , where the operator is defined by (30), is proved by using Banach’s contraction mapping principle. Let us define a ball with the value satisfying

where . Note that the above inequality is well defined as . Next, we will show that . For any , we have

which implies that . Therefore, we have . Finally we shall show that is a contraction. For any and for each , we obtain

Therefore, we have . As , then the operator is a contraction. Hence, by the Banach contraction mapping principle, it follows that the operator has a unique fixed point which is the unique solution of problem (5) on . This completes the proof.

The third existence theorem is based on Lelay-Schauder’s nonlinear alternative.

Lemma 7 (nonlinear alternative for single-value maps [43]). Let be a Banach space, be a closed, convex subset of , be an open subset of , and . Suppose that is continuous and 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 8. Assume that is a continuous function. In addition we suppose that

(H₃)there exist a continuous nondecreasing function and a function such that

(H₄)there exists a constant such that

Then the separated boundary value problem (5) has at least one solution on .

Proof. As in the proof of Theorem 5, the operator is completely continuous. The result will follow from the Lelay-Schauder nonlinear alternative (Lemma 7) once we have proved the boundedness of the set of all solutions to equations for .
Let be a solution of problem (5). Then, from , we have

Thus, we obtain

In view of , there exists a positive constant such that . Let us define the set

Note that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Therefore, by applying the nonlinear alternative of Leray-Schauder type, we can conclude that the operator has at least one fixed point in , which is a solution of the quantum -difference boundary value problem (5) on . The proof is completed.

The final existence theorem is established by using Krasoselskii’s fixed point theorem.

Lemma 9 (Krasnoselskii’s fixed point theorem [44]). Let be a closed, bounded, convex, and nonemtry subset of a Banach space . Let be the operators such that whenever ; is compact and continuous; is a contraction mapping. Then there exists such that .

Theorem 10. Assume that is a continuous function satisfying the assumption . In addition we suppose that
(H₅) and .
If

then the separated boundary value problem of quantum -difference equation (5) has at least one solution on .

Proof. Let us define and choose a suitable constant as

where is defined by (38). Furthermore, we define the operators and on by

Observe that . For , we have

which yields . Therefore, the condition of Lemma 9 is satisfied. It follows from the assumption together with (48) that is a contraction mapping. Hence the condition of Lemma 9 is fulfilled. Now we will show that the condition of Lemma 9 is satisfied. Given that function is continuous, we get that the operator is continuous. It is easy to verify that

Thus, the set is uniformly bounded. Next we prove the compactness of the operator . Give and let with . Then we get

which is independent of and tends to zero as . So, the set is equicontinuous. By the Arzelá-Ascoli theorem, is compact on . Thus all the assumptions of Lemma 9 are satisfied. Therefore the separated boundary value problem of quantum -difference equation (5) has at least one solution on . The proof is completed.

4. Examples

Example 1. Consider the following separated boundary value problem for second-order quantum -difference equation of the form

Set constants , , , , , , , , and . Then we can find constants as , , , and .
(i) Let by

It is easy to see that which satisfies . Thus, by Theorem 5, the separated boundary value problem (54) with given by (55) has at least one solution on
(ii) Given with

it follows that

Therefore, the condition is satisfied with . Indeed, we have . Hence, by Theorem 6, the separated boundary value problem of quantum -difference equation (54) has a unique solution on .

Example 2. Consider the following separated boundary value problem for second-order quantum -difference equation of the form

Set constants , , , , , , , , and . Then we can compute that , , , and .
(i) Let be a nonlinear function defined by

Choosing and , we notice that the condition of Theorem 8 is satisfied. Furthermore, we can find that . Then there exists a constant such that . Therefore, the separated boundary value problem (58) with given by (59) has at least one solution on .
(ii) Let a function be defined by

It follows that

Then the function defined in (60) satisfies condition with In addition, the condition is satisfied with and

Hence, by applying Theorem 10, the separated boundary value problem for second-order quantum -difference equation (58) with defined in (60) has at least one solution on .

Remark 3. In Example 2, Case (ii), although the function satisfies condition , Theorem 6 cannot be applied, since .

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-60-ART-074.

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