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A study of secondorder qdifference equations with boundary conditions
Advances in Difference Equations volume 2012, Article number: 35 (2012)
Abstract
This paper studies a boundary value problem of nonlinear secondorder qdifference equations with nonseparated boundary conditions. As a first step, the given boundary value problem is converted to an equivalent integral operator equation by using the qdifference calculus. Then the existence and uniqueness of solutions of the problem is proved via the resulting integral operator equation by means of LeraySchauder nonlinear alternative and some standard fixed point theorems. Our approach is simpler than the one involving the typical series solution form of qdifference equations. The results corresponding to a secondorder qdifference equation with antiperiodic boundary conditions appear as a special case. Furthermore, our results reduce to the corresponding results for classical secondorder boundary value problems with nonseparated boundary conditions in the limit q → 1, which provides a useful check.
2010 Mathematics Subject Classification. 39A05, 39A13.
1 Introduction
In this paper, we discuss the existence of solutions for the secondorder qdifference equation with nonseparated boundary conditions
where f\in C\left(I\times \mathbb{R},\mathbb{R}\right),I=\left[0,T\right]\cap {q}^{\stackrel{\u0304}{\mathbb{N}}},{q}^{\stackrel{\u0304}{\mathbb{N}}}:=\left\{{q}^{n}:n\in \mathbb{N}\right\}\cup \left\{0\right\} and T\in {q}^{\stackrel{\u0304}{\mathbb{N}}} is a fixed constant, and η ≠ 1 is a fixed real number.
The study of qdifference equations, initiated in the beginning of the 20th century ([1–4]), and, up to date, it has evolved into a multidisciplinary subject, (for example, see ([5–15]) and references therein). For some recent work on qdifference equations, we refer the reader to the papers ([16–23]). However, the theory of boundary value problems for nonlinear qdifference equations is still in the initial stage and many aspects of this theory need to be explored.
The main objective of this paper is to develop some existence and uniqueness results for the boundary value problem (1.1). Our results are based on a variety of fixed point theorems such as Banach's contraction principle, LeraySchauder nonlinear alternative and Krasnoselskii's fixed point theorem. Some illustrative examples and special cases are also discussed.
2 Preliminaries
Let us recall some basic concepts of qcalculus [14, 15].
For 0 < q < 1, we define the qderivative of a real valued function f as
The higher order qderivatives are given by
The qintegral of a function f defined in the interval {J}_{a}=\left[a,b\right]\cap {q}^{\stackrel{\u0304}{\mathbb{N}}} is given by
and for a = 0, we denote
provided the series converges. If a\in {J}_{0}=\left[0,b\right]\cap {q}^{\stackrel{\u0304}{\mathbb{N}}} and f is defined in the interval J_{0},
then
Similarly, we have
Observe that
and if f is continuous at x = 0, then
In qcalculus, the product rule and integration by parts formula are
In the limit q → 1 the above results correspond to their counterparts in standard calculus.
Lemma 2.1 The unique solution of (1.1) is given by
Proof. Integrating the equation {D}_{q}^{2}u=f\left(t,u\right), we get
Integrating (2.5), we obtain
which, on changing the order of integration, takes the form
Alternatively, (2.7) can be written as
Here a_{1},a_{2} are arbitrary constants. Conversely, differentiating (2.8) by applying the formulae (2.1) and (2.2) of qcalculus, one can easily obtain {D}_{q}^{2}u=f\left(t,u\right).
Now, using the boundary conditions of (1.1) in (2.8), we find that
Substituting the values of a_{1} and a_{2} in (2.8), we obtain (2.4). This completes the proof.
For the forthcoming analysis, let \mathcal{C}=C\left(I,\mathbb{R}\right) denotes the Banach space of all continuous functions from I to ℝ endowed with the norm defined by ║x║ = sup{x(t), t ∈ I}.
Furthermore, we set
Theorem 2.1 Let f: I × ℝ → ℝ be a continuous function satisfying the condition
where L is a Lipschitz constant. Then the boundary value problem (1.1) has a unique solution, provided Λ = L Λ _{1} < 1, where Λ_{1}is given by (2.9).
Proof. We define \u03dc:\mathcal{C}\to \mathcal{C} by
for u\in \mathcal{C} and t ∈ I. Let us set sup_{t∈I}f(t, 0) = M_{0} and choose
where δ is such that Λ ≤ δ < 1. Now we show that \u03dc{B}_{r}\subset {B}_{r}, where {B}_{r}=\left\{u\in \mathcal{C}:\parallel u\parallel \le r\right\}. For u ∈ B_{ r } , we have
Now, for u,v\in \mathcal{C} and for each t∈I, we obtain
where
which depends only on the parameters involved in the problem. As Λ < 1, therefore Ϝ is a contraction. Thus, the conclusion of the theorem follows by Banach's contraction mapping principle. This completes the proof.
The next existence result is based on LeraySchauder alternative.
Theorem 2.2 (Nonlinear alternative for single valued maps)[24]. Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and 0∈ U. Suppose that F:Ū →C is a continuous, compact (that is, F(Ū) is a relatively compact subset of C) map. Then either
(i) F has a fixed point in Ū, or
(ii) there is a u ∈ ∂U (the boundary of U in C) and λ ∈ (0,1) with u = λF(u).
Theorem 2.3 Assume that:
(H_{1}) there exists a continuous nondecreasing function ψ: [0,∞) → (0,∞) and a function p ∈ L^{1}(I,ℝ_{+}) such that
(H_{2}) there exists a number M < 0 such that
where
Then the BVP (1.1) has at least one solution.
Proof. We define \u03dc:\mathcal{C}\to \mathcal{C} as in (2.10). The proof consists of several steps.

(i)
F maps bounded sets into bounded sets in C(I,ℝ).
Let B_{ k } = {u∈ C(I,ℝ): ║u║ ≤ k} be a bounded set in C(I,ℝ) and u ∈ B_{ k } . Then we have
Thus

(ii)
F maps bounded sets into equicontinuous sets of C(I,ℝ).
Let r_{1}, r_{2}∈ I, r_{1} < r_{2} and B_{ k } be a bounded set of C(I, ℝ) as before. Then for u ∈ B_{ k } we have
The right hand side tends to zero as r_{2} r_{1}→ 0. As a consequence of ArzeláAscoli theorem, we can conclude that \u03dc:C\left(I,\mathbb{R}\right)\to C\left(I,\mathbb{R}\right) is completely continuous.

(iii)
Let λ ∈ (0,1) and let u=\lambda \u03dcu. Then, for t ∈ I, we have
\begin{array}{ll}\hfill \leftu\left(t\right)\right& =\left\lambda \u03dcu\left(t\right)\right\le {\int}_{0}^{t}\lefttqs\right\leftf\left(s,u\left(s\right)\right)\right{d}_{q}s\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\frac{\left\eta \right}{{\left(\eta 1\right)}^{2}}{\int}_{0}^{T}\leftT+\left(1\eta \right)\left(tqs\right)\right\leftf\left(s,u\left(s\right)\right)\right{d}_{q}s\phantom{\rule{2em}{0ex}}\\ \le T\left(1+\frac{\left\eta \right\left(1+\left1\eta \right\right)}{{\left(\eta 1\right)}^{2}}\right)\psi \left(\u2225u\u2225\right){\int}_{0}^{T}p\left(s\right)d,\phantom{\rule{2em}{0ex}}\end{array}
and consequently
In view of (H_{2}), there exists M such that ║u║ ≠ M. Let us set
Note that the operator \u03dc:\overline{U}\to C\left(I,\mathbb{R}\right) is continuous and completely continuous (which is well known to be compact restricted to bounded sets). From the choice of U, there is no u ∈ ∂U such that u=\lambda \u03dc\left(u\right) for some λ ∈ (0,1). Consequently, by Theorem 2.2, we deduce that Ϝ has a fixed point u ∈ Ū which is a solution of the problem (1.1). This completes the proof.
Theorem 2.4 Assume that there exist constants 0 ≤ c < 1/Δ_{2}and N > 0 such that\leftf\left(t,u\right)\right\le \frac{c}{{T}^{2}}\leftu\right+Nfor all t ∈ I, u ∈ C(I,ℝ), where
Then the BVP (1.1) has at least one solution.
Proof. We define \u03dc:\mathcal{C}\to \mathcal{C} as in (2.10). As in Theorem 2.3, using the assumption \leftf\left(t,u\right)\right\le \frac{c}{{T}^{2}}\leftu\right+N (a special form of the condition (H_{1}) with p(t) = 1, \psi (\leftu\right)=\frac{c}{{T}^{2}}\leftu\right+N), we can prove that Ϝ is completely continuous.
Assume that u=\lambda \u03dcu for some λ∈ [0, 1]. For all t ∈ I we have
which implies that
Letting
we have ║u║ ≤ R. Consequently, by the nonlinear alternative of LeraySchauder type [24], this completes the proof.
For next theorem, we need the following fixed point theorem [25].
Theorem 2.5 Let Ω be an open bounded subset of a Banach space E with 0∈ Ω and B:\stackrel{\u0304}{\Omega}\to E be a compact operator. Then B has a fixed point in \stackrel{\u0304}{\Omega} provided║Buu║^{2} ≥ ║Bu║^{2}║u║^{2}, u∈∂ Ω.
Theorem 2.6 Assume that there exists a positive constant M _{1} such that
where Λ_{1}is given by (2.9). Then the BVP (1.1) has at least one solution.
Proof. Let us define Γ = {u ∈ C(I,ℝ): ║u║ < M_{1}}. As in Theorem 2.3, using the assumption on f(t, u), it can be shown that Ϝ is completely continuous. Thus, in view of Theorem 2.5, we just need to show that
For all t ∈ I, u∈ ∂ Γ, we have
which yields \parallel \u03dcu\parallel \le {M}_{1}. Since (2.13) holds, therefore, we obtain the result.
Remark 2.1 In view of the assumption \leftf\left(t,u\right)\right\le \frac{c}{{T}^{2}}\u2225u\u2225+N,u\in C\left(I,\mathbb{R}\right),t\in I of Theorem 24, we see that it suffices to take {M}_{1}\ge \frac{N{\Delta}_{2}{T}^{2}}{1c{\Delta}_{2}}.
Before presenting the last result, we state a fixed point theorem due to Krasnoselskii [26] which is needed to prove the existence of solutions of (1.1).
Theorem 2.7 (Krasnoselskii) Let Ω be a closed convex, bounded and nonempty subset of a Banach space X. Let{\u03dc}_{1},{\u03dc}_{2}be the operators such that: (i){\u03dc}_{1}x+{\u03dc}_{2}y\in \Omegawhenever x, y ∈ Ω; (ii){\u03dc}_{1}is compact and continuous; (iii){\u03dc}_{2}is a contraction mapping. Then there exists z ∈ Ω such thatz={\u03dc}_{1}z+{\u03dc}_{2}z.
Theorem 2.8 Assume that f: I × ℝ → ℝ is a continuous function satisfying (H_{0}) and the following assumption holds:
(H_{3}) f(t, u) ≤ µ(t), ∀(t, u) ∈ I × ℝ, and µ ∈C(I, ℝ_{+}).
If
then the boundary value problem (1.1) has at least one solution on I.
Proof. With max_{t∈I}µ(t) = ║µ║, let us fix r ≥ ║µ║Λ_{1} (Λ_{1} is given by (2.9)) and define {B}_{r}=\left\{u\in \mathcal{C}:\parallel u\parallel \le r\right\}.
Define the operators {\u03dc}_{1}u and {\u03dc}_{2}u on the set B_{ r } as
For u, υ ∈ B_{ r } , we have
which implies that {\u03dc}_{1}u+{\u03dc}_{2}v\in {B}_{r}. In view of the condition (2.14), it follows that {\u03dc}_{2} is a contraction mapping. Continuity of f together with the assumption (H_{3}) implies that the operator {\u03dc}_{1} is continuous and uniformly bounded on B_{ r } . Let us define sup_{(t,u) ∈ I × Br}f(t,u) = f_{max} < ∞. Then, for t_{1}, t_{2} ∈ I with t_{2} < t_{1} and u ∈ B_{ r } , we have
which is independent of u and tends to zero as t_{2} t_{1} → 0. So {\u03dc}_{1} is relatively compact on B_{ r } . Hence, by the ArzeláAscoli Theorem, {\u03dc}_{1} is compact on B_{ r } . Thus all the assumptions of Theorem 2.7 are satisfied and the conclusion of Theorem 2.7 implies that the boundary value problem (1.1) has at least one solution on I. This completes the proof.
3 Examples
Example 3.1. Consider
Here, f(t,u) = L(t^{2} + cost + 1 + tan^{−1}u), L to be fixed later, q = 1/2, η = 1/3 and T = 1. Clearly
and
For L<\frac{1}{{\Lambda}_{1}}=\frac{12}{21}, it follows by Theorem 2.1 that the problem (3.1) has a unique solution.
Example 3.2. Consider the following problem
Here q = 1/2, η = 1/3, T = 1, and c will be fixed later. Observe that
and {\Delta}_{2}=\frac{21}{12}. Obviously N = 1 and we can choose c<\frac{1}{{\Delta}_{2}}=\frac{12}{21}. Thus, the conclusion of Theorem 2.4 applies to the problem (3.2).
Remark. By fixing η in the results of this paper, we can obtain some special new results for qdifference equations. For instance, our results correspond to the ones for secondorder qdifference equations with antiperiodic boundary conditions (u(0) = u(T), D_{ q }u(0) = D_{ q }u(T)) for η = 1, and the results for an initial value problem of secondorder qdifference equations follow for η = 0. Furthermore, the classical results for second order boundary value problems with nonseparated boundary conditions follow in the limit q → 1^{−}.
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Acknowledgements
The authors are grateful to the reviewers for their useful comments. The research of B. Ahmad and A. Alsaedi was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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Each of the authors, BA, AA and SKN contributed to each part of this work equally and read and approved the final version of the manuscript.
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Ahmad, B., Alsaedi, A. & Ntouyas, S.K. A study of secondorder qdifference equations with boundary conditions. Adv Differ Equ 2012, 35 (2012). https://doi.org/10.1186/16871847201235
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DOI: https://doi.org/10.1186/16871847201235