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Global structure of positive solutions for second-order difference equation with nonlinear boundary value condition
Advances in Difference Equations volume 2014, Article number: 188 (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.
It is well known that the discrete Sturm-Liouville boundary value problem
In 1998, Agarwal and O’Regan  studied the existence of solutions of (1.1) by fixed point theorem whenever , . In 2000, Atici  obtained the existence of positive solutions of (1.1) by the fixed point theorem in cones. Cabada and Otero-Espinar , Rodríguez [8, 9], Rodríguez and Abernathy , Ma , Henderson et al. , and Anderson et al.  also studied the discrete Sturm-Liouville problems by various methods. It is worth to point out that Rodríguez and Abernathy  studied the existence of solutions of the following boundary value problem of the difference equation:
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.
However, as far as we know, there is very little work to study the existence of positive solutions of second-order difference equation with nonlinear boundary value condition. Motivated by the above works [5–12], we study the global structure of positive solutions of the following discrete boundary value problem:
where are constants, the functions , with on and functions f, g satisfy the following:
(H1) with for and there exist constants and functions such that
(H2) with for and there exist constants and functions , such that
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 .
By the constant we denote the first eigenvalue of the eigenvalue problem,
By a constant we denote the first eigenvalue of the eigenvalue problem,
It is well known (cf. Kelly and Peterson ) that for , is positive and simple, and that it is a unique eigenvalue with positive eigenfunction .
Let be the closure of the set
Theorem 1.1 Let (H1)-(H2) hold. Then there exists an unbounded, closed, and connected component in , which joins with . Moreover, if
hold. Then (1.3) has at least one positive solution.
Corollary 1.2 Let (H1)-(H2) hold. If
hold. Then the problem
has at least one positive solution.
which satisfies , then the nonlinear boundary value problem
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
Let , be the solution of the initial value problem
respectively, where . It is easy to compute and show that
, and ϕ is increasing on ;
, and ψ is decreasing on .
Lemma 2.1 Let . Then the linear boundary value problem
has a solution
Moreover, if and on I, then on .
Proof It is a direct consequence of Atici [, Section 2], so we omit it. □
Let . Then the linear boundary value problem
has a solution
From the properties of , , it follows that
Let be defined as follows:
Let be defined as
Then is a linear bounded function in E.
Suppose that E is a real Banach space with norm . Let K be a cone in E. A nonlinear mapping is said to be positive if . It is said to be K-completely continuous if A is continuous and maps bounded subsets of to a precompact subset of E. Finally, a positive linear operator V on E is said to be a linear minorant for A if for . If B is a continuous linear operator on E, denote by the spectrum radius of B. Define
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].
Lemma 2.2 Assume that
K has a nonempty interior and ;
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.
Then there exists an unbounded connected subset of
such that .
Moreover, if A has a linear minorant V and there exists a such that and , then can be chosen in .
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.
From Lemma 2.1 and the compactness of T, let denote the inverse operator of the linear boundary value problem
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.
Let be the solution of the linear boundary value problem
Repeating the argument of with some minor changes, it follows that is a linear, bounded mapping and
here is the trace operator and , satisfies (2.1) and (2.2) with , , respectively. Then the problem (1.3) is equivalent to the operator equation
Similarly, let denote the inverse operator of the linear boundary value problem
Then is a linear mapping of E compactly into E and it is strongly positive. Let be the solution of the linear boundary value problem
Then is a linear mapping bounded mapping and
here , satisfies (2.1) and (2.2) with , , respectively. Furthermore, the problem (1.3) is also equivalent to the operator equation
From (H1) and (H2), it follows that
Let , . Then and are nondecreasing and satisfy
Let us consider
as a bifurcation problem from the trivial solution .
Define the linear operator B
It is easy to verify that is completely continuous and strongly positive on E. From [, Theorem 19.3], it follows that . Define by
then we have from (3.3)
So, we imply that if with is a nontrivial solution of (3.5), then . Combining this with Lemma 2.2, we conclude that there exists an unbounded connected subset of the set
such that .
Proof of Theorem 1.1 It is clear that any solution of (3.5) of the form yields a solution y of (1.3). We will show that joins to .
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 .
In fact, suppose on the contrary that y is a nontrivial solution of the problem
then y satisfies the linear boundary value problem
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 .
In this case, we show that
We divide the proof into two steps.
Step 1. We show that if there exists a constant number such that
then joins with .
From (3.6), we have as . We divide the equation
by and let . Since is bounded in E, choosing a subsequence and relabeling if necessary, we see that for some with . Moreover, from (3.4) and the fact that and are nondecreasing, we have
since and . Thus
where , again choosing a subsequence and relabeling if necessary. So it follows that
Since , and , the strong positivity of ensures that on . Therefore, , and accordingly, joins to .
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 .
From (H1) and (H2), there exist constants such that
By the same method as used for defining and , we may define and as follows:
Let denote the inverse operator of the linear boundary value problem
Then is a linear mapping of E compactly into E and it is strong positive. Let be the solution of the linear boundary value problem
where . Then is a linear, bounded mapping and
where , satisfies (2.1) and (2.2) with , , respectively.
Moreover, the problem (1.3) can be rewritten as the operator equation
Then V is a linear minorant of A. Let be the eigenvalue of the linear problem
and let be the corresponding eigenfunction. Then
Therefore we have from Lemma 2.2
Case 2 .
In this case, if is such that
and , then
If there exists , such that for all , . Applying a similar argument to that used in Step 1 of Case 1, after taking a subsequence and relabeling if necessary, it follows that
Again joins to and the result follows. □
Proof of Corollary 1.2 It is a direct consequence of Theorem 1.1, so we omit it. □
Example Let us consider the following boundary value problem of the difference equation:
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 .
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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).
The authors declare that they have no competing interests.
YL and RM completed the main study, YL carried out the results of this article and drafted the manuscript and RM checked the proofs and verified the calculation. All the authors read and approved the manuscript.