On the Second-Order Quantum (p, q)-Difference Equations with Separated Boundary Conditions

1 Intelligent 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


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 [0, ∞) by and gave its properties. The -integral of a function on [0, ∞) 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][6][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][9][10][11][12][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][19][20][21]. In addition, Yang-Mills theories and also -Yang-Mills equation were developed by [22][23][24]. The theory of quantum group applied to vibration and rotation molecules with -algebra and -Heisemberg algebra technique was established in [25][26][27]. The subject of elementary particle physics and chemical physics using -calculus was investigated in [28][29][30][31]. The string theory involving -calculus was studied in [32].
In this paper, we continue the study on ( , ) boundary value problems, by considering the following second-order quantum ( , )-difference equation with separated boundary conditions 2 , where 0 < < ≤ 1 are two given quantum numbers, 2 , is the second-order ( , )-difference operator, and ∈ ([0, / 2 ] × R, R), > 0, , , = 1, 2, 3, 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.

Advances in Mathematical Physics
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.

Main Results
It should be noticed that problem (5) has solutions if and only if the operator A of equation = A has fixed points. Our first result is an existence theorem for separated boundary value problem of quantum ( , )-difference equation (5) for each ∈ [0, / 2 ] and all ∈ R.
Then the separated boundary value problem (5) has at least one solution on [0, ].
Proof. Schaefer's fixed point theorem is used to prove that the operator A defined by (30) has at least one fixed point. So, we divide the proof into four steps.
Therefore, we obtain which implies that ‖A − A ‖ → 0 as → ∞. Hence the operator A is continuous.

Step 2 (A maps bounded sets into bounded sets in [0, ]).
Choosing > 0, we define a bounded ball as = { ∈ C : ‖ ‖ ≤ }. Then, for any ∈ , we have This means that ‖A ‖ ≤ . Therefore, the set A is uniformly bounded.

5
Step 3 (A maps bounded sets into equicontinuous sets of [0, ]). Let 1 , 2 ∈ [0, ] with 1 < 2 be two points and be a bounded ball in C. Then for any ∈ , we get As 1 → 2 , the right-hand side of the above inequality (which is independent of ) tends to zero. This shows that the set A is equicontinuous set. From a consequence of Steps 1 to 3, together with the Arzelá-Ascoli theorem, we deduce that the operator A : C → C is completely continuous.
Step 4. Finally, we show that the set is bounded. Let ∈ be a solution of problem (5). Then ( ) = (A )( ) for some 0 < < 1. Hence, for each ∈ [0, ], 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 A has at least one fixed point which is a solution of the second-order quantum ( , )difference equation with separated boundary value problem (5) on [0, ]. The proof is completed.
The third existence theorem is based on Lelay-Schauder's nonlinear alternative.
Lemma (nonlinear alternative for single-value maps [43] (H 4 ) there exists a constant > 0 such that Then the separated boundary value problem (5) has at least one solution on [0, ].
Proof. As in the proof of Theorem 5, the operator A : C → C 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 = A for ∈ [0, 1].
Let be a solution of problem (5). Then, from ( 3 ), we have Thus, we obtain In view of ( 4 ), there exists a positive constant such that ‖ ‖ ̸ = . Let us define the set Note that the operator A : → C is continuous and completely continuous. From the choice of , there is no ∈ such that = A for some ∈ (0,1). Therefore, by applying the nonlinear alternative of Leray-Schauder type, we can conclude that the operator A has at least one fixed point in , which is a solution of the quantum ( , )-difference boundary value problem (5) on [0, ]. The proof is completed.
The final existence theorem is established by using Krasoselskii's fixed point theorem.