Research Article | Open Access
Chanon Promsakon, Nattapong Kamsrisuk, Sotiris K. Ntouyas, Jessada Tariboon, "On the Second-Order Quantum -Difference Equations with Separated Boundary Conditions", Advances in Mathematical Physics, vol. 2018, Article ID 9089865, 9 pages, 2018. https://doi.org/10.1155/2018/9089865
On the Second-Order Quantum -Difference Equations with Separated Boundary Conditions
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.
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 byprovided that infinite series converges. The general theory of linear quantum difference equations was published in 1912 by Carmichael . Details of its basic notions, results, and methods can be found in the text . 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 . For details, we refer to the recent monograph .
There are some applications of -calculus and difference equations to molecular problems in physics. In 1967, Finkelstein  studied behaviors of hydrogen atoms by using Schrödinger equation and -calculus. In , 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 .
The subject of -calculus first appeared in quantum algebras which contained two quantum numbers and introduced by . For some recent results see [34–41] and the references cited therein. Recently in , 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 formwhere , , , 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 conditionswhere 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.
In this section, we recall some basic concepts of -calculus (see ). The -number is defined bywhere . For each , , the -factorial and -binomial are defined byandrespectively.
Let . The -derivative of function is defined asand . 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 byprovided 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 .
Theorem 1 (see ). 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 ). Let be continuous functions and constants , . The following formulas hold:
(a) The -integration by parts is given by (b) .
(d) where .
Theorem 3 (see ). 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 haveTaking -integration to (21) and using Theorem 3, we haveFor convenience, we set constants , and thenIn particular for in (23), we haveUsing (21) and by replacing , we obtain From the first boundary condition of (18) and the above setting constants, we getThe second boundary condition of (18) with (24) and (25) yieldsSolving (26) and (27) for constants and , it follows thatandSubstituting 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 byIt 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 .
Ifwhere 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 ). 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 obtainIn view of , there exists a positive constant such that . Let us define the setNote 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 ). 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 .
Ifthen 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