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Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations
Advances in Difference Equations volume 2012, Article number: 140 (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.
In this paper, we discuss the existence and uniqueness of solutions for the nonlocal boundary value problem of fractional q-difference equations given by
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.
Let and define
The q analogue of the power is
The q-gamma function is defined by
and satisfies (see, ).
For , we define the q-derivative of a real valued function f as
The higher order q-derivatives are given by
For , we set and define the definite q-integral of a function by
provided that the series converges.
For , we set
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.
We note that
while if f is continuous at , then
For more details of the basic material on q-calculus, see the book .
Definition 2.1 ()
Let and f be a function defined on . The fractional q-integral of the Riemann-Liouville type is and
Definition 2.2 ()
The fractional q-derivative of the Riemann-Liouville type of order is defined by and
where is the smallest integer greater than or equal to α.
Definition 2.3 ()
The fractional q-derivative of the Caputo type of order is defined by
where is the smallest integer greater than or equal to α.
Lemma 2.4 Let and let f be a function defined on . Then the next formulas hold:
Lemma 2.5 ()
Let and . Then the following equality holds:
Lemma 2.6 ()
Let . Then the following equality holds:
Lemma 2.7 ()
For , , the following is valid:
In particular, for , , using q-integration by parts, we have
In order to define the solution for the problem (1.1)-(1.2), we need the following lemma.
Lemma 2.8 For a given the unique solution of the boundary value problem
is given by
Proof In view of Lemmas 2.4 and 2.6, integrating equation in (2.1), we have
Using the boundary conditions of (2.1) in (2.4), we have
Solving the above system of equations for , , we get
Substituting the values of , in (2.4), we obtain (2.2). □
In view of Lemma 2.8, we define an operator as
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 .
For the sake of convenience, we set
Theorem 3.1 Assume that is continuous and that there exists a q-integrable function such that
(A1) , , .
Then the boundary value problem (1.1)-(1.2) has a unique solution provided
where k is given by (3.1).
Proof Let us fix and choose
We define and show that , where F is defined by (2.5). For , observe that
Then , , we have
which, in view of (3.1) and (3.5), implies that
This shows that .
Now, for , we obtain
which, in view of (3.1), yields
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.
Corollary 3.2 Assume that is a continuous function and that
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)
Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be two operators such that:
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
(A2) there exists a function and a nondecreasing function with
(A3) there exists a constant with
then the boundary value problem (1.1)-(1.2) has at least one solution on .
Proof Consider the set , where is given in (3.6) and define the operators and on as
For , we find that
Thus, . From (A1) and (3.7) it follows that is a contraction mapping. Continuity of f implies that the operator is continuous. Also, is uniformly bounded on as
Now, for any , and with , we have
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.
Corollary 3.5 Let be a continuous function satisfying (A1). In addition, we assume that
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 )
Let E be a Banach space, C a closed, convex subset of E, U an open subset of C with . Suppose that is a continuous, compact (that is, is a relatively compact subset of C) map. Then either
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 ;
(A5) there exists a number such that
Then the boundary value problem (1.1)-(1.2) has at least one solution on .
Proof Consider the operator defined by (2.5). It is easy to show that F is continuous. Next, we show that F maps bounded sets into bounded sets in . For a positive number ρ, let be a bounded set in . Then we have
Thus, for any it holds
which proves our assertion.
Now we show that F maps bounded sets into equicontinuous sets of . Let with and , where is a bounded set of . Then taking into consideration the inequality for (see, , p.4]), we have
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.
Let with (by (3.8)). Then it can be shown that . Indeed, in view of (A4), we have
Suppose there exists a and a such that . Then for such a choice of x and λ, we have
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).
Example 3.9 Consider the fractional q-difference nonlocal boundary value problem
Here, , , , , , , , and L is a constant to be fixed later on. Moreover, , , , , and
all the assumptions of Theorem 3.1 are satisfied. Therefore, by Theorem 3.1, problem (3.9)-(3.10) has a unique solution.
Example 3.10 Consider the problem
where , , , , , , . In a straightforward manner, it can be found that , , f, and
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).
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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.
The authors declare that they have no competing interests.
Each of the authors, BA, SKN, and IKP contributed to each part of this study equally and read and approved the final version of the manuscript.