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Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives
Advances in Difference Equations volume 2013, Article number: 124 (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.
In this paper, we study the existence of solutions for a boundary value problem with nonlinear second-order q-difference equations
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
Definition 2.1 For , we define the q-derivative of a real-value function f as
Note that .
Definition 2.2 The higher-order q-derivatives are defined inductively as
For example, , where k is a positive integer and the bracket . In particular, .
Definition 2.3 The q-integral of a function f defined in the interval is given by
and for , we denote
Similarly, we have
and if f is continuous at , then .
In q-calculus, the product rule and integration by parts formula are
Remark 2.4 In the limit , the above results correspond to their counterparts in standard calculus.
Definition 2.5 is called an S-Carathéodory function if and only if
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 )
Let E be a Banach space, let C be a closed and convex subset of E, and let U be an open subset of C and . 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 .
Lemma 2.7 Let , then the BVP
has a unique solution
Proof Integrating the q-difference equation from 0 to t, we get
Integrating (2.6) from 0 to t and changing the order of integration, we have
where , are arbitrary constants. Using the boundary conditions in (2.7), we find that , and
Substituting the values of and in (2.7), we obtain
This completes the proof. □
Remark 2.8 For , equation (2.4) takes the form
which is the solution of a classical second-order ordinary differential equation and the associated form of Green’s function for the classical case is
We consider the Banach space equipped with the standard norm , and , .
Define an integral operator by
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.
Theorem 3.1 Let be a continuous function, and there exists such that
In addition, suppose either
() for , or
holds, where . Then BVP (1.1) has a unique solution.
Proof Case 1: . Let us set and choose
where δ is such that . Now we show that , where . For each , we have
Hence, we obtain that , so .
Now, for and for each , we have
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. □
Corollary 3.2 Assume that is a continuous function and there exist two positive constants , such that
In addition, suppose either
() for , or
holds. Then BVP (1.1) has a unique solution.
Corollary 3.3 Assume that is a continuous function and there exist two functions such that
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.
Proof The proof consists of several steps.
T maps bounded sets into bounded sets in .
Let be a bounded set in and . Then we have
T maps bounded sets into equicontinuous sets of .
Let , , and let be a bounded set of as before. Then, for , we have
As a consequence of the Arzelá-Ascoli theorem, we can conclude that is completely continuous. This proof is completed. □
Theorem 3.5 Let be an S-Carathéodory function. Suppose further that there exists a real number such that
Then BVP (1.1) has at least one solution.
Proof In view of Lemma 3.4, we obtain that is completely continuous. Let and . Then, for , we have
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.
Theorem 3.6 Let be an S-Carathéodory function. Suppose further that there exist functions with such that
Then BVP (1.1) has at least one solution provided , where .
Proof We consider the space
and define the operator by
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.
Next we show that the fixed point of has a priori bound M, which is independent of λ. Assume that , and set
By (2.5), it is clear that for each . For any , we have
and so it holds that
At the same time, we have
Set , which is independent of λ. So, BVP (1.1) has at least one solution. This completes the proof. □
Example 4.1 Consider the following BVP:
Here, , , . Clearly, . Then , and . By Corollary 3.2, we obtain that BVP (4.1) has a unique solution.
Example 4.2 Consider the following BVP:
Here, , . It is obvious that , where , , . Then , , , , so . By Theorem 3.6, we obtain that BVP (4.2) has at least one solution.
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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).
The authors declare that they have no competing interests.
Each of the authors, CLY and JFW contributed to each part of this work equally and read and approved the final version of the manuscript.
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Yu, C., Wang, J. Existence of solutions for nonlinear second-order q-difference equations with first-order q-derivatives. Adv Differ Equ 2013, 124 (2013). https://doi.org/10.1186/1687-1847-2013-124
- q-difference equations
- Leray-Schauder nonlinear alternative
- boundary value problem
- fixed point theorem