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Existence of Solutions for Fractional q-Integrodifference Equations with Nonlocal Fractional q-Integral Conditions
We study a class of fractional q-integrodifference equations with nonlocal fractional q-integral boundary conditions which have different quantum numbers. By applying the Banach contraction principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative, the existence and uniqueness of solutions are obtained. In addition, some examples to illustrate our results are given.
In this paper, we deal with the following nonlocal fractional -integral boundary value problem of nonlinear fractional -integrodifference equation: where , , , are given constants, is the fractional -derivative of Riemann-Liouville type of order , is the fractional -integral of order with , and , is a continuous function.
The early work on -difference calculus or quantum calculus dates back to Jackson’s paper . Basic definitions and properties of quantum calculus can be found in the book . The fractional -difference calculus had its origin in the works by Al-Salam  and Agarwal . Motivated by recent interest in the study of fractional-order differential equations, the topic of -fractional equations has attracted the attention of many researchers. The details of some recent development of the subject can be found in ([5–17]) and the references cited therein, whereas the background material on -fractional calculus can be found in a recent book .
In this paper, we will study the existence and uniqueness of solutions of a class of boundary value problems for fractional -integrodifference equations with nonlocal fractional -integral conditions which have different quantum numbers. So, the novelty of this paper lies in the fact that there are four different quantum numbers. In addition, the boundary condition of (1) does not contain the value of unknown function at the right side of boundary point . One may interpret the -integral boundary condition in (1) as the -integrals with different quantum numbers are related through a real number .
The paper is organized as follows. In Section 2, for the convenience of the reader, we cite some definitions and fundamental results on -calculus as well as the fractional -calculus. Some auxiliary lemmas, needed in the proofs of our main results, are presented in Section 3. Section 4 contains the existence and uniqueness results for problem (1) which are shown by applying Banach’s contraction principle, Krasnoselskii’s fixed point theorem, and Leray-Schauder’s nonlinear alternative. Finally, some examples illustrating the applicability of our results are presented in Section 5.
For , define
The -analogue of the power function with is More generally, if , then
We use the notation for . The -gamma function is defined by Obviously, .
The -derivative of a function is defined by and -derivatives of higher order are given by The -integral of a function defined on the interval is given by If and is defined in the interval , then its integral from to is defined by Similar to derivatives, an operator is given by The fundamental theorem of calculus applies to operators and ; that is, and if is continuous at . Then
Definition 1. Let and be a function defined on . The fractional -integral of Riemann-Liouville type is given by and
Definition 2. The fractional -derivative of Riemann-Liouville type of order is defined by and where is the smallest integer greater than or equal to .
Definition 3. For any , is called the -beta function.
The expression of -beta function in terms of the -gamma function can be written as
Lemma 4 (see ). Let , and be a function defined in . Then, the following formulas hold: (1);(2).
Lemma 5 (see ). Let and be a positive integer. Then, the following equality holds:
3. Some Auxiliary Lemmas
Lemma 6. Let , and . Then one has
Proof. Using the definitions of -analogue of power function and -beta function, we have The proof is complete.
Lemma 7. Let , and . Then one has
Proof. Taking into account Lemma 6, we have This completes the proof.
Lemma 8. Let , , and . Then, for , the unique solution of boundary value problem, subject to the nonlocal fractional condition, is given by where
Proof. From , we let . Using the Definition 2 and Lemma 4, (22) can be expressed as From Lemma 5, we have for some constants . It follows from the first condition of (23) that . Applying the Riemann-Liouville fractional -integral of order for (27) with and taking into account of Lemma 6, we have In particular, we have Using the Riemann-Liouville fractional -integral of order and repeating the above process, we get The second nonlocal condition of (23) implies Substituting the values of and in (27), we get the desired result in (24).
4. Main Results
In this section, we denote as the Banach space of all continuous functions from to endowed with the norm defined by . In view of Lemma 8, we define an operator by where is defined by (25). It should be noticed that problem (1) has solutions if and only if the operator has fixed points.
For the sake of convenience of proving the results, we set
The first result on the existence and uniqueness of solutions is based on the Banach contraction mapping principle.
Theorem 9. Let be a continuous function satisfying the assumption:
there exist constants such that for each and .
If where is given by (33), then the boundary value problem (1) has a unique solution on .
Proof. We transform problem (1) into a fixed point problem, , where the operator is defined by (32). By applying the Banach contraction mapping principle, we will show that has a fixed point which is the unique solution of problem (1).
Setting and choosing where , and the constant defined by (34), we will show that , where . For any , we have The assumption implies that for all and .
Then, by using Lemmas 6 and 7, we have Then, we have which yields .
Next, for any and for each , we have The above result implies that . As , is a contraction. Hence, by the Banach contraction mapping principle, we deduce that has a fixed point which is the unique solution of problem (1).
The second existence result is based on Krasnoselskii’s fixed point theorem.
Lemma 10 (Krasnoselskii’s fixed point theorem ). Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (a) whenever ; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists such that .
Theorem 11. Assume that is a continuous function satisfying the assumption . In addition one supposes that
, for all and .
If then the boundary value problem (1) has at least one solution on .
Proof. Let us set and choose a suitable constant as
where is defined by (34). Now, we define the operators and on the set as
Firstly, we will show that the operators and satisfy condition (a) of Lemma 10. For , we have
Therefore . Further, condition coupled with (42) implies that is contraction mapping. Therefore, condition (c) of Lemma 10 is satisfied.
Finally, we will show that is compact and continuous. Using the continuity of and , we deduce 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 independently of . Therefore, is relatively compact on . Applying the Arzelá-Ascoli theorem, we get that is compact on . Thus all assumptions of Lemma 10 are satisfied. Therefore, the boundary value problem (1) has at least one solution on . The proof is complete.
Using the Leray-Schauder nonlinear alternative, we give the third result.
Lemma 12 (nonlinear alternative for single-valued maps ). Let be a Banach space, let be a closed, convex subset of , let be an open subset of , and let . Suppose that is a continuous, compact (i.e., 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 .
For the sake of convenience of proving the last result, we set
Theorem 13. Assume that is a continuous function. In addition one supposes that
there exist a continuous nondecreasing function and a function such that
there exists a constant such that where and are defined by (47) and (48), respectively, and Then the boundary value problem (1) has at least one solution on .
Proof. Firstly, we will show that the operator , defined by (32), maps bounded sets (balls) into bounded sets in . For a positive number , we set a bounded ball in as . Then, for , we have
Therefore, we conclude that .
Secondly, we will show that maps bounded sets into equicontinuous sets of . Let with and be a bounded set of as in the previous step, and let . Then we have Obviously the right-hand side of the above inequality tends to zero independently of as . Therefore, by applying the Arzelá-Ascoli theorem, we deduce that is completely continuous.
The result will follow from the Leray-Schauder nonlinear alternative once we have proved the boundedness of the set of all solutions to the equation for some . Let be a solution. Then, for , we have