Global structure of positive solutions for second-order difference equation with nonlinear boundary value condition
© Lu and Ma; licensee Springer. 2014
Received: 22 March 2014
Accepted: 25 June 2014
Published: 22 July 2014
This paper is devoted to the study of the global structure of the positive solution of a second-order nonlinear difference equation coupled with a nonlinear boundary value condition. The main result is based on Dancer’s bifurcation theorem.
The development in numerical analysis has propelled interest in difference equations and their relationship to their differential counterparts. The theory of discrete nonlinear boundary value problems has often been connected (e.g. Gaines ) to the study of corresponding topics in differential equations and the investigation of the differences between the two approaches. This spirit remains in the recent publications (see e.g. Kelley and Peterson , Agarwal  or Bereanu and Mawhin ). This paper can be seen as a part of this research stream. We investigate the nonlinear discrete Sturm-Liouville problems coupled with a nonlinear boundary value condition, transform it into the equivalent operator equation, and use Dancer’s bifurcation theorem to obtain the existence of a positive solution.
where a and b are integers, , , , , , , is continuously differentiable, , and are all continuous, are continuously Fréchet differentiable; here X is the set of real-valued functions defined on , Y is the set of real-valued functions defined on . Under some hypotheses, they showed that (1.2) has a solution by the Brouwer fixed point theorem.
where are constants, the functions , with on and functions f, g satisfy the following:
Through careful analysis we have found that the boundary condition in (1.3) is nonlinear but it can be linearized and this makes it possible to establish existence results for positive solutions of (1.3) in terms of the principal eigenvalue of the corresponding linearized problem. Notice that this condition is different from those given in [5, 6].
Let , and define to be the space of all maps from into ℝ. Then it is a Banach space with the norm .
Let . Then P is a cone which is normal and has a nonempty interior and .
It is well known (cf. Kelly and Peterson ) that for , is positive and simple, and that it is a unique eigenvalue with positive eigenfunction .
hold. Then (1.3) has at least one positive solution.
has at least one positive solution.
has no positive solution.
The rest of this paper is organized as follows. In Section 2, we state some preliminary results and Dancer’s bifurcation theorem. It is worth to note that the proof of the main result is based upon Dancer’s bifurcation theorem, which is different from the topological degree arguments used in [5, 6, 12, 13]. In Section 3, we reduce (1.3) to a compact operator equation and prove Theorem 1.1 and Corollary 1.2.
2 Preliminaries and Dancer’s global bifurcation theorem
, and ϕ is increasing on ;
, and ψ is decreasing on .
Moreover, if and on I, then on .
Proof It is a direct consequence of Atici [, Section 2], so we omit it. □
Then is a linear bounded function in E.
The following lemma will play a very important role in the proof of our main results, which is essentially a consequence of Dancer [, Theorem 2].
K has a nonempty interior and ;
- (ii)is K-completely continuous and positive, for , for and
where is a strongly positive linear compact operator on E with , satisfies as locally uniformly in λ.
such that .
3 The proof of the main result
To prove Theorem 1.1, we begin with the reduction of (1.3) to a suitable equation for a compact operator.
Taking into account , , one can repeat the argument of the operator T with some minor changes, and it follows that is a linear mapping of E compactly into E and it is strongly positive.
as a bifurcation problem from the trivial solution .
such that .
Then for all since is the only solution for (3.5) (i.e. (3.2), since (3.2) and (3.5) are equivalent to (1.3)) for .
here , . This together with (H2) and [, Lemma 2.2] implies that , which is a contradiction. Therefore, (3.5) with has only a trivial solution.
Case 1 .
We divide the proof into two steps.
Step 2. We show that there exists a constant M such that for all n.
By Lemma 2.2, we only need to show that A has a linear minorant V and there exists a such that and .
By the same method as used for defining and , we may define and as follows:
where , satisfies (2.1) and (2.2) with , , respectively.
Case 2 .
Proof of Corollary 1.2 It is a direct consequence of Theorem 1.1, so we omit it. □
Obviously, the conditions (H1), (H2) are satisfied, furthermore , , , , and , . From Theorem 1.1, the problem (3.9) has at least one positive solution u on if .
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11361054, No. 11201378, No. 11061030), Gansu provincial National Science Foundation of China (No. 1208RJZA258), SRFDP (No. 20126203110004).
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