Eigenvalue of boundary value problem for nonlinear singular third-order q-difference equations
© Yu and Wang; licensee Springer. 2014
Received: 4 October 2013
Accepted: 26 December 2013
Published: 16 January 2014
In this paper, we establish the existence of positive solutions of a boundary value problem for nonlinear singular third-order q-difference equations , , , , , by using Krasnoselskii’s fixed-point theorem on a cone, where λ is a positive parameter. Finally, we give an example to demonstrate the use of the main result of this paper. The conclusions in this paper essentially extend and improve known results.
The q-difference equations initiated in the beginning of the 20th 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 , and nuclear and high-energy physics . For some recent work on q-difference equations, we refer the reader to [8–12]. However, the theory of boundary value problems (BVPs) for nonlinear q-difference equations is still in an early stage and many aspects of this theory need to be explored. To the best of our knowledge, for the BVPs of nonlinear third-order q-difference equations, a few works were done, see [13, 14] and the references therein.
by Krasnoselskii’s fixed-point theorem on a cone.
by Leray-Schauder degree theory and some standard fixed-point theorems.
where is a positive parameter, is continuous and , f is a continuous function, , is a fixed constant, and , .
Obviously, when , BVP (1.1) reduces to the standard BVP in .
Throughout this paper, we always suppose the following conditions to hold:
(C2) , and .
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 2.1 In the limit , the above results correspond to their counterparts in standard calculus.
, implies ;
, implies .
Theorem 2.1 (Krasnoselskii) 
Let E be a Banach space and let be a cone in E. Assume that and are open subsets of E with and . Let be a completely continuous operator. In addition, suppose either
(H1) , and , or
(H2) , and ,
holds. Then T has a fixed point in .
This completes the proof. □
It is obvious that, when (C2) holds, , and , .
Lemma 2.2 Let (C2) hold, then for , where .
The proof is complete. □
It is easy to see that if , then .
Obviously, T is well defined and is a solution of BVP (1.1) if and only if u is a fixed point of T.
We adopt the following assumption:
(C3) may be singular at , , and .
Lemma 2.3 Assume (C1), (C2), and (C3) hold, then is completely continuous.
where we have used the fact that , and , . Hence, converges uniformly to T as , and therefore T is completely continuous also. This completes the proof. □
3 Main results
Here we assume that if and if and if and if .
The main result of this paper is the following.
Theorem 3.1 Suppose that (C1), (C2) and (C3) hold and . Then for each , BVP (1.1) has at least one positive solution.
Choose sufficiently small such that . Then we obtain . Thus if we let , then for .
Choose sufficiently small such that . Then we have . Let , then and for .
Condition (H1) of Krasnoselskii’s fixed-point theorem is satisfied. Hence, by Theorem 2.1, the result of Theorem 3.1 holds. This completes the proof of Theorem 3.1. □
Theorem 3.2 Suppose that (C1), (C2) and (C3) hold and . Then for each , BVP (1.1) has at least one positive solution.
Proof It is similar to the proof of Theorem 3.1. □
Theorem 3.3 Suppose that (C1), (C2) and (C3) hold and for . Then BVP (1.1) has no positive solution.
This is a contradiction and completes the proof. □
Theorem 3.4 Suppose that (C1), (C2) and (C3) hold and for . Then BVP (1.1) has no positive solution.
Proof It is similar to the proof of Theorem 3.3. □
Then , , , , and . By direct calculations, we obtain and . From Theorem 3.1 we see that if then the problem (4.1) has a positive solution. From Theorem 3.3 we see that if then the problem (4.1) has no positive solution. By Theorem 3.4 we see that if then the problem (4.1) has no positive solution.
This work was supported by the Natural Science Foundation of China (10901045), (11201112) and (61304106), the Natural Science Foundation of Hebei Province (A2013208147) and (A2011208012).
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