S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems

Abstract By using the bifurcation method, we study the existence of an S-shaped connected component in the set of positive solutions for discrete second-order Neumann boundary value problem. By figuring the shape of unbounded connected component of positive solutions, we show that the Neumann boundary value problem has three positive solutions suggesting suitable conditions on the weight function and nonlinearity.


Introduction
In this paper, we are concerned with the existence and multiplicity of positive solutions for the following second-order Neumann boundary value problem for some ∈ ( +∞) * * m m , 0, . Recently, second-order Neumann boundary value problems have attracted the attention of many specialists both in differential equations and difference equations because of their interesting applications, see [1][2][3][4][5][6][7][8][9][10] and references therein. In [1], Feltrin and Sovrano studied the second-order Neumann boundary value problem where the weight function (⋅) a changes its sign. Based upon a shooting method, they showed that (1.2) has three positive solutions suggesting suitable conditions on the weight function and nonlinearity. In another paper, the same authors [2] successfully established some further multiplicity results for the positive solutions of problem (1.2). More precisely, they obtained at least eight positive solutions via the shooting method. Moreover, Boscaggin [3], Boscaggin and Zanolin [4] and Boscaggin and Garrione [6] also studied the existence and multiplicity of positive solutions for problem (1.2). For more details, we refer the reader to [11] and references therein.
On the other hand, in high-dimensional case, several results of existence and multiplicity can be found for Neumann boundary value problems associated with where ( ) > k x 0 (see [12,13] for some classical results in this direction). Clearly, (1.1) is the one-dimensional discrete case of (1.3).
However, for problem (1.1), although it is quite relevant from the point of view of applications, not sufficiently developed yet. Recently, Long and Chen [8] established the existence of multiple solutions to problem (1.1). In fact, they considered the problem . By using the invariant set of descending flow and variational method, they proved that problem (1.4) has at least three nontrivial solutions: one is positive, one is negative and one is sign-changing.
Motivated by the aforementioned studies, in this paper, we employ a bifurcation technique of Sim and Tanaka [14] and prove that an unbounded subcontinuum of positive solutions of (1.1) bifurcates from the trivial solution and grows to the right from the initial point, to the left at some point and to the right near = +∞ λ . Roughly speaking, we obtain that there exists an S-shaped connected component in the positive solution set of problem (1.1). As a by-product, we assert further that (1.1) has one, two or three positive solutions for λ lying in various intervals in .
Let Let E be an n-dimensional Hilbert space equipped with the usual inner product (⋅ ⋅) , and the usual norm ∥⋅∥, then W is isomorphic to E, ∥⋅∥ 1 and ∥⋅∥ are equivalent.
We first study the following eigenvalue problem: . For simplicity, denote ( ) μ q 1 to be μ 1 , and the associated eigenfunction ϕ is positive.
Remark 1.1. The eigenvalue μ 1 is the minimum of the "Rayleigh quotient," that is Let η 1 be the principal eigenvalue of the eigenvalue problem and the corresponding eigenfunction It is easy to find that if (H1) holds, then Our main results are as follows.  For other results concerning the existence of an S-shaped connected component in the set of solutions for diverse boundary value problems, see [16][17][18].
The rest of this paper is arranged as follows. In Section 2, we state some notations and preliminary results and obtain a global bifurcation phenomenon from the trivial branch with the rightward direction. In Section 3, we show the change of direction of bifurcation and complete the proof of Theorem 1.1.

Preliminaries and rightward bifurcation
Define by setting Next, we consider the following boundary value problem: . Then (2.1) and the system of linear algebra equations ( + ) = P Q u h are equivalent. Therefore, the unique solution of (2.1) is In addition, we have the following lemma. where Obviously, < < m M 0 and < < σ 0 1 (see [7]).
We extend f to a continuous function f˜defined by Obviously, within the context of positive solutions, problem (1.1) is equivalent to the same problem with f replaced by f˜. Furthermore, f˜is an odd function for ∈ s . In the sequel of the proof we shall replace f with f˜. However, for the sake of simplicity, the modified function f˜will still be denoted by f. Then it is well known that L and × → H E E : are completely continuous (see [8]). By a simple calculation, we have   . It follows that ≠ / λ μ f 1 0 . Lemma 1.1 follows v 0 must change its sign, and as a consequence for some n large enough, u n must change sign. This is a contradiction. Dividing the both sides of (2.4) by ∥ ∥ u n , we get Lebesgue's dominated convergence theorem shows that 1 , which means that v is a nontrivial solution of (1.5) with = λ μ 1 , and hence ≡ v ϕ. □   In this section, we show that there is a direction turn of the bifurcation under condition (H3) and accordingly we finish the proof of Theorem 1.1.