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On nonlocal boundary value problems of nonlinear q-difference equations
Advances in Difference Equations volume 2012, Article number: 81 (2012)
Abstract
This paper studies a nonlocal boundary value problem of nonlinear third-order q-difference equations. Our results are based on Leray-Schauder degree theory and some standard fixed point theorems.
MSC 2000: 39A05; 39A13.
1 Introduction
In this paper, we study a nonlocal nonlinear boundary value problem (BVP) of third-order q-difference equations given by
where f ∈ C(I q × ℝ,ℝ), I q = {qn: n ∈ ℕ} ∪ {0,1}, q ∈ (0,1) is a fixed constant, η ∈ {qn: n ∈ ℕ} and α ≠ 1/η2 is a real number.
The subject of q-difference equations has evolved into a multidisciplinary subject in the last few decades. In fact, it is a truly operational subject and its operational formulas were often used with great success in the theory of classical orthogonal polynomials and Bessel functions [1, 2]. For some pioneer work on q-difference equations, we refer the reader to [1, 3–5], whereas the recent development of the subject can be found in [6–17] and references therein. However, the theory of boundary value problems for nonlinear q-difference equations is still in the initial stages and many aspects of this theory need to be explored. In particular, the study of nonlocal boundary value problems for nonlinear q-difference equations is yet to be initiated.
The aim of our paper is to present some existence results for the problem (1.1). The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we apply Banach's contraction principle to prove the uniqueness of the solution of the problem, while the third result is based on Krasnoselskii's fixed point theorem. The methods used are standard; however, their exposition in the framework of problem (1.1) is new. In Sect. 2, we present some basic material that we need in the sequel and Sect. 3 contains main results of the paper. Some illustrative examples are also discussed.
2 Preliminaries
Let us recall some basic concepts of q-calculus [8, 9].
For 0 < q < 1, we define the q-derivative of a real-valued function f as
Note that
The higher order q-derivatives are defined inductively as
For example, , where k is a positive integer and the q-bracket [k] q = (qk- 1)/(q - 1). In particular, D q (t2) = (1 + q)t.
For y ≥ 0, let us set and define the definite q-integral of a function by
provided that the series converges. For , we define
Similarly, we have
Observe that
and if f is continuous at x = 0, then
This implies that if D q f(t) = σ(t), then f(t) = I q σ(t) + c, where c is an arbitrary constant.
In q-calculus, the product rule and integration by parts formula are
In the limit q → 1-, the above results correspond to their counterparts in standard calculus.
For , it is possible to introduce an inner product
and the resulting Hilbert space is denoted by .
As argued in [16], we can write the solution of the third-order q-difference equation in the following form:
where a0, a1, a2 are arbitrary constants and α1(q), α2(q), α3(q) can be fixed appropriately.
Choosing α1(q) = 1/(1 + q), α2(q) = -q, α3(q) = q3/(1 + q) and using (2.1) and (2.2), we find that
Thus, the solution (2.4) of takes the form
Lemma 2.1 The BVP (1.1) is equivalent to the integral equation
Proof. In view of (2.5), the solution of can be written as
where a1, a2, a2 are arbitrary constants. Using the boundary conditions of (1.1) in (2.7), we find that a0 = 0, a1 = 0 and
Substituting the values of a0, a1 and a2 in (2.7), we obtain (2.6). This completes the proof.
We define
where
Remark 2.1 For q → 1-, equation (2.6) takes the form
which is equivalent to the solution of a classical third-order nonlocal boundary value problem
3 Existence results
Let denote the Banach space of all continuous functions from I q → ℝ endowed with the norm defined by ∥x∥ = sup{|x(t)| : t ∈ I q }.
Theorem 3.1 Assume that there exist constants M1 ≥ 0 and M2 > 0 such that M1G1 < 1 and |f(t, u)| ≤ M1|u| + M2 for all t ∈ I q , u ∈ ℝ, where G1 is given by (2.8). Then the problem (1.1) has at least one solution.
Proof. Let be a suitable ball with radius R > 0. Define an operator as
In view of Lemma 2.1, we just need to prove the existence of at least one solution such that . Thus, it is sufficient to show that the operator satisfies
Let us define
Then, by Arzela-Ascoli theorem, is completely continuous. If (3.1) is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that
where denotes the unit operator. By the nonzero property of Leray-Schauder degree, for at least one u ∈ B R . Let us set
where R will be fixed later. In order to prove (3.1), we assume that for some λ ∈ [0,1] and for all t ∈ I q so that
which implies that
Letting , (3.1) holds. This completes the proof.
Theorem 3.2 Let f : I q × ℝ → ℝ be a jointly continuous function satisfying the Lipschitz condition
where L is a Lipschitz constant. Then the boundary value problem (1.1) has a unique solution provided L < 1/G1, where G1 is given by (2.8).
Proof. Let us define an operator by
Let us set and choose
Then we show that , where . For u ∈ B r , we have
where we have used (3.2).
Now, for u, v ∈ ℝ, we obtain
As L < 1/G1, therefore is a contraction. Thus, the conclusion of the theorem follows by Banach's contraction mapping principle. This completes the proof.
To prove the next existence result, we need the following known fixed point theorem due to Krasnoselskii [18].
Theorem 3.3 Let be a closed convex and nonempty subset of a Banach space X. Let A, B be the operators such that (i) whenever ; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists such that z = Az + Bz.
Theorem 3.4 Assume that f : I q × ℝ → ℝ is a continuous function such that
Furthermore, |f(t, u)| ≤ μ(t), ∀(t, u) ∈ I q × ℝ, with μ ∈ C(I q , ℝ+). Then the boundary value problem (1.1) has at least one solution on I q if
Proof. Letting , we fix (G1 is given by (2.8) and consider . We define the operators and on as
For , we find that
Thus, . It follows from (3.3) and (3.4) that is a contraction mapping. Continuity of f implies that the operator is continuous. Also, is uniformly bounded on as
Now we prove the compactness of the operator .
In view of (H1), we define , and consequently we have
which is independent of u and tends to zero as t2 → t1. So is relatively compact on . Hence, by the Arzelá-Ascoli Theorem, is compact on . Thus all the assump tions of Theorem 3.3 are satisfied. So the conclusion of Theorem 3.3 implies that (1.1) has at least one solution on I q . This completes the proof.
Remark 3.1 In the limit q → 1-, our results reduce to the ones for a classical third-order nonlocal nonlinear boundary value problem (2.9).
Example 3.1. Consider the following problem
Here q = 1/2 and M1 will be fixed later. Observe that
and
Clearly M2 = 2 and we can choose . Thus, Theorem 3.1 applies to the problem (3.5).
Example 3.2. Consider the following problem with unbounded nonlinearity
Clearly
with M1 = 5 < 1/G1 = 21/4 (G1 is given in Example 3.1) and M2 = 2. Thus, by the conclusion of Theorem 3.1, the problem (3.6) has a solution.
Example 3.3. Consider
With f(t, u) = L (cos t + tan-1 u), we find that
and
Fixing , it follows by Theorem 3.2 that the problem (3.7) has a unique solution.
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Acknowledgements
The research of Bashir Ahmad was 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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Each of the authors, BA and JJN contributed to each part of this study equally and read and approved the final version of the manuscript.
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Ahmad, B., Nieto, J.J. On nonlocal boundary value problems of nonlinear q-difference equations. Adv Differ Equ 2012, 81 (2012). https://doi.org/10.1186/1687-1847-2012-81
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DOI: https://doi.org/10.1186/1687-1847-2012-81
Keywords
- q-difference equations
- nonlocal boundary conditions
- Leray-Schauder degree theory
- fixed point theorems