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Eigenvalue of boundary value problem for nonlinear singular third-order q-difference equations
Advances in Difference Equations volume 2014, Article number: 21 (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.
Recently, in , El-Shahed has studied the existence of positive solutions for the following nonlinear singular third-order BVP:
by Krasnoselskii’s fixed-point theorem on a cone.
More recently, in  Ahmad has studied the existence of positive solutions for the following nonlinear BVP of third-order q-difference equations:
by Leray-Schauder degree theory and some standard fixed-point theorems.
Motivated by the work above, in this paper, we will study the following BVP of nonlinear singular third-order q-difference equations:
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
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 2.1 In the limit , the above results correspond to their counterparts in standard calculus.
Definition 2.4 Let E be a real Banach space. A nonempty closed convex set is called a cone if it satisfies the following two conditions:
, 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 .
Lemma 2.1 Let , then the BVP
has a unique solution
Proof Integrate the q-difference equation from 0 to t, we get
Integrate (2.4) from 0 to t, and change the order of integration, we have
Integrating (2.5) from 0 to t, and changing the order of integration, we obtain
where , , are arbitrary constants. Using the boundary conditions , , in (2.6), we find that , and
Substituting the values of , , and in (2.6), we obtain
This completes the proof. □
Remark 2.2 For , equation (2.6) takes the form
which is the solution of a classical third-order ordinary differential equation and the associated form of Green’s function for the classical case is
It is obvious that, when (C2) holds, , and , .
Lemma 2.2 Let (C2) hold, then for , where .
Proof If , then
If , then
The proof is complete. □
We consider the Banach space equipped with standard norm , . Define a cone P by
It is easy to see that if , then .
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.
Remark 2.3 By Lemma 2.2, we obtain, for , on and
We adopt the following assumption:
(C3) may be singular at , , and .
Lemma 2.3 Assume (C1), (C2), and (C3) hold, then is completely continuous.
Proof Define the functions for by
Next, for , we define the operator by
Obviously, is completely continuous on P for any by an application of the Ascoli-Arzelá theorem. Denote . Then converges uniformly to T as . In fact, for any , for each fixed and , from (C1), we obtain
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
In this section, we will apply Krasnoselskii’s fixed-point theorem to the eigenvalue problem (1.1). First, we define some important constants:
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.
Proof By the definition of , we see that there exists an , such that for . If with , we have
Choose sufficiently small such that . Then we obtain . Thus if we let , then for .
From the definition of , we see that there exist an and , such that for . Let , if with we have
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.
Proof Assume to the contrary that u is a positive solution of BVP (1.1). Then
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. □
Consider the following BVP:
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.
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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).
The authors declare that they have no competing interests.
Each of the authors, CY and JW 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. Eigenvalue of boundary value problem for nonlinear singular third-order q-difference equations. Adv Differ Equ 2014, 21 (2014). https://doi.org/10.1186/1687-1847-2014-21
- q-difference equations
- positive solutions
- singular boundary value problem
- Krasnoselskii’s fixed-point theorem