Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives
© Yu and Wang; licensee Springer 2013
Received: 16 October 2012
Accepted: 15 April 2013
Published: 2 May 2013
In this paper, we establish the existence of solutions for a boundary value problem with the nonlinear second-order q-difference equation
The existence and uniqueness of solutions for the problem are proved by means of the Leray-Schauder nonlinear alternative and some standard fixed point theorems. Finally, we give two examples to demonstrate the use of the main results. The nonlinear team f contains in the equation.
where , , , and is a fixed real number.
The q-difference equations initiated at the beginning of the twentieth century [1–4] is a very interesting field in difference equations. In the last few decades, it has evolved into a multidisciplinary subject and plays an important role in several fields of physics such as cosmic strings and black holes , conformal quantum mechanics , nuclear and high energy physics . However, the theory of boundary value problems (BVPs) for nonlinear q-difference equations is still in the initial stages and many aspects of this theory need to be explored. To the best of our knowledge, for the BVPs of nonlinear q-difference equations, a few works were done; see [8–13] and the references therein. In particular, the study of BVPs for nonlinear q-difference equation with first-order q-difference is yet to be initiated.
The main aim of this paper is to develop some existence and uniqueness results for BVP (1.1). Our results are based on a variety of fixed point theorems such as the Banach contraction mapping principle, the Leray-Schauder nonlinear alternative and the Leray-Schauder continuous theorem. Some examples and special cases are also discussed.
2 Preliminary results
Note that .
For example, , where k is a positive integer and the bracket . In particular, .
and if f is continuous at , then .
Remark 2.4 In the limit , the above results correspond to their counterparts in standard calculus.
for each , is measurable on I;
for a.e. , is continuous on ;
for each , there exists with on I such that implies , for a.e. I, where , and normed by for all .
Theorem 2.6 (Nonlinear alternative for single-valued maps )
F has a fixed point in , or
there is a (the boundary of U in C) and with .
This completes the proof. □
We consider the Banach space equipped with the standard norm , and , .
Obviously, T is well defined and is a solution of BVP (1.1) if and only if u is a fixed point of T.
3 Existence and uniqueness results
In this section, we apply various fixed point theorems to BVP (1.1). First, we give the uniqueness result based on Banach’s contraction principle.
In addition, suppose either
() for , or
holds, where . Then BVP (1.1) has a unique solution.
Hence, we obtain that , so .
Therefore, we obtain that , so T is a contraction. Thus, the conclusion of the theorem follows by Banach’s contraction mapping principle.
Case 2: . It is similar to the proof of case 1. This completes the proof of Theorem 3.1. □
In addition, suppose either
() for , or
holds. Then BVP (1.1) has a unique solution.
In addition, suppose either
() for , or
Then BVP (1.1) has a unique solution.
Proof It is similar to the proof of Theorem 3.1. □
The next existence result is based on the Leray-Schauder nonlinear alternative theorem.
Lemma 3.4 Let be an S-Carathéodory function. Then is completely continuous.
T maps bounded sets into bounded sets in .
T maps bounded sets into equicontinuous sets of .
As a consequence of the Arzelá-Ascoli theorem, we can conclude that is completely continuous. This proof is completed. □
Then BVP (1.1) has at least one solution.
Therefore, there exists such that . Let us set . Note that the operator is completely continuous (which is known to be compact restricted to bounded sets). From the choice of U, there is no such that for some . Consequently, by Theorem 2.6, we deduce that T has a fixed point which is a solution of problem (1.1). This completes the proof. □
The next existence result is based on the Leray-Schauder continuation theorem.
Then BVP (1.1) has at least one solution provided , where .
Obviously, we can see that . In view of Lemma 3.4, it is easy to know that for each , is completely continuous in P. It is clear that is a solution of BVP (1.1) if and only if u is a fixed point of . Clearly, for each . If for each the fixed points of in P belong to a closed ball of P independent of λ, then the Leray-Schauder continuation theorem completes the proof.
Set , which is independent of λ. So, BVP (1.1) has at least one solution. This completes the proof. □
Here, , , . Clearly, . Then , and . By Corollary 3.2, we obtain that BVP (4.1) has a unique solution.
Here, , . It is obvious that , where , , . Then , , , , so . By Theorem 3.6, we obtain that BVP (4.2) has at least one solution.
Research supported by the Natural Science Foundation of China (10901045), (11201112), the Natural Science Foundation of Hebei Province (A2009000664), (A2011208012) and the Foundation of Hebei University of Science and Technology (XL201047), (XL200757).
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