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Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations
© Ahmad et al.; licensee Springer 2012
Received: 7 May 2012
Accepted: 2 August 2012
Published: 8 August 2012
In this paper, we study a nonlinear fractional q-difference equation with nonlocal boundary conditions. The existence of solutions for the problem is shown by applying some well-known tools of fixed-point theory such as Banach’s contraction principle, Krasnoselskii’s fixed-point theorem, and the Leray-Schauder nonlinear alternative. Some illustrating examples are also discussed.
MSC:34A08, 39A05, 39A12, 39A13.
In recent years, the topic of fractional differential equations has gained considerable attention and has evolved as an interesting and popular field of research. It is mainly due to the fact that several times the tools of fractional calculus are found to be more practical and effective than the corresponding ones of classical calculus in the mathematical modeling of dynamical systems associated with phenomena such as fractals and chaos. In fact, fractional calculus has numerous applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity and damping, control theory, wave propagation, percolation, identification, fitting of experimental data, etc. [1–4]. The development of fractional calculus ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. For some recent work on fractional differential equations, we refer to [5–13] and the references therein.
The pioneer work on q-difference calculus or quantum calculus dates back to Jackson’s papers [14, 15], while a systematic treatment of the subject can be found in [16, 17]. For some recent existence results on q-difference equations, see [18–20] and the references therein.
Fractional q-difference equations have recently attracted the attention of several researchers. For some earlier work on the topic, we refer to  and , whereas some recent work on the existence theory of fractional q-difference equations can be found in [32–36]. However, the study of boundary value problems of fractional q-difference equations is at its infancy and much of the work on the topic is yet to be done.
where , is the fractional q-derivative of the Caputo type, and ().
2 Preliminaries on fractional q-calculus
In this section, we cite some definitions and fundamental results of the q-calculus as well as of the fractional q-calculus [37, 38]. We also give a lemma that will be used in obtaining the main results of the paper.
and satisfies (see, ).
provided that the series converges.
provided that the series exist. Throughout the paper, we will assume that the series in the q-integrals converge.
Note that for , we have , for some , thus the definite integral is just a finite sum, so no question about convergence is raised.
For more details of the basic material on q-calculus, see the book .
Definition 2.1 ()
Definition 2.2 ()
where is the smallest integer greater than or equal to α.
Definition 2.3 ()
where is the smallest integer greater than or equal to α.
Lemma 2.5 ()
Lemma 2.6 ()
Lemma 2.7 ()
In order to define the solution for the problem (1.1)-(1.2), we need the following lemma.
Substituting the values of , in (2.4), we obtain (2.2). □
Observe that problem (1.1)-(1.2) has a solution if the operator equation has a fixed point, where F is given by (2.5).
3 Main results
Let denote the Banach space of all continuous functions from endowed with the norm defined by .
Theorem 3.1 Assume that is continuous and that there exists a q-integrable function such that
(A1) , , .
where k is given by (3.1).
This shows that .
Since by assumption (3.4), therefore, F is a contraction. Hence, it follows by Banach’s contraction principle that the problem (1.1)-(1.2) has a unique solution. □
In case (L is a constant), the condition (3.4) becomes and Theorem 3.1 takes the form of the following result.
Then the boundary value problem (1.1)-(1.2) has a unique solution.
Our next existence results is based on Krasnoselskii’s fixed-point theorem .
Lemma 3.3 (Krasnoselskii)
A is compact and continuous;
B is a contraction mapping.
Then there exists such that .
Theorem 3.4 Let be a continuous function satisfying (A1). In addition, we assume that
then the boundary value problem (1.1)-(1.2) has at least one solution on .
which is independent of x and tends to zero as . Thus, is equicontinuous. So is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Thus, all the assumptions of Lemma 3.3 are satisfied. So the conclusion of Lemma 3.3 implies that the boundary value problem (1.1)-(1.2) has at least one solution on . □
In the special case when , we see that there always exists a positive r so that (3.6) holds true, thus we have the following corollary.
If (3.7) holds, then the boundary value problem (1.1)-(1.2) has at least one solution on .
The next existence result is based on Leray-Schauder nonlinear alternative.
Lemma 3.6 (Nonlinear alternative for single valued maps )
F has a fixed point in , or
there is a (the boundary of U in C) and with .
Theorem 3.7 Let be a continuous function. Assume that:
(A4) there exist functions , and a nondecreasing function such that ;
Then the boundary value problem (1.1)-(1.2) has at least one solution on .
which proves our assertion.
Obviously, the right-hand side of the above inequality tends to zero independently of as . Therefore, it follows by the Arzelá-Ascoli theorem that is completely continuous.
Thus, the operator F satisfies all the conditions of Lemma 3.6, and hence by its conclusion, either condition (i) or condition (ii) holds. We show that the conclusion (ii) is not possible.
which is a contradiction. Consequently, by the Leray-Schauder alternative (Lemma 3.6), we deduce that F has a fixed point which is a solution of the problem (1.1)-(1.2). This completes the proof. □
Remark 3.8 If , in (A4) are continuous, then , , where A is defined by (3.5).
all the assumptions of Theorem 3.1 are satisfied. Therefore, by Theorem 3.1, problem (3.9)-(3.10) has a unique solution.
Clearly, , , . Consequently, , , and the condition (3.8) implies that . Thus, all the assumptions of Theorem 3.7 are satisfied. Therefore, the conclusion of Theorem 3.7 applies to the problem (3.11)-(3.12).
The research of B. Ahmad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors thank the reviewers for their useful comments.
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