Existence of Positive Solutions for a Discrete Three-Point Boundary Value Problem

A discrete three-point boundary value problem Δxk−1 + λ fk(xk)= 0, k = 1,2, . . . ,n, x0 = 0, axl = xn+1, is considered, where 1≤ l ≤ n is a fixed integer, a is a real constant number, and λ is a positive parameter. A characterization of the values of λ is carried out so that the boundary value problem has the positive solutions. Particularly, in this paper the constant a can be negative numbers. The similar results are not valid for the three-point boundary value problem of differential equations.


Introduction
It is of interest to note here that the three-point or multipoint boundary value problems in the continuous case have been studied in great detail in the recent papers [1][2][3][4][5][6][7][8][9][10][11] since the early 1980s.In numerical integration of differential equations, it is taken by granted that their difference approximations retain the same existence and uniqueness property of solutions.However, in the case of boundary value problems, a number of examples can be cited where this assumption fails.This has led a number of recent investigations providing necessary and/or sufficient conditions for the existence and uniqueness of the solutions of discrete boundary value problems (see [12][13][14][15][16][17][18]).Then, how do we consider the three-point or multipoint boundary value problems of difference equations?
Recently, in [19] we have considered the existence of positive solutions for the nonlinear discrete three-point boundary value problem where n∈{2,3,...}, l ∈ [1,n]={1,2,...,n}, α is a positive number, and f ∈ C(R + ,R + ).For (1.1), the existence of one or two positive solutions was established when f is superliner or sublinear.In fact, system (1.1) can be regarded as a discrete reality model.A horizontal string of negligible mass is stretched between the points x = 0 and x = n + 1, and concentrated forces with magnitudes f (x 1 ), f (x 2 ),..., f (x n ) and downward directions are applied at the points x = 1,2,...,n, respectively.Suppose an end of the string is fixed, and another end of the string has some relation with the points 1,2,...,n − 1, or n.For example, we can suppose that x n+1 = ax l for some l ∈ [1,n].By the Hooker law, we can obtain the discrete three-point boundary value problem (1.1).
In this paper, we will consider a more general nonlinear discrete three-point boundary problem of the form where λ is a positive parameter, a is a real constant number, n is a positive integer, and Δ denotes the forward difference operator defined by 3) is called to be positive if x k > 0 for k ∈ [1,n].If, for a particular λ the boundary value problem (1.2)-(1.3)has a positive solution x, then λ is called an eigenvalue and x a corresponding eigenfunction of (1.2)-(1.3).We let 3) has a positive solution (1.4) be the set of eigenvalues of (1.2)- (1.3).Further, we introduce the notations The following is the plan of this paper.It is well known that Green's functions are important for the boundary value problems.Thus, in the next section we will give the Green function of (1.2)-(1.3)by considering the inversive matrix of corresponding vector matrix equation.The properties of the Green function are also considered in this section.In this paper, we will be concerned with several eigenvalue characterizations of (1.2)-(1.3).Such problems have been extensively studied for discrete or continuous two-point boundary value problems.In Section 3, we will consider the eigenvalue characterizations of existence of one positive solution.The existence of triple solutions will be established in Section 4. In Section 5, we will give some remarks which explain some difference between differential equation and difference equation for the corresponding problem.
When |a| < (n + 1)/l, we can get that A −1 exists.In fact, let B = (b i j ) n×n (where b ii = 2 (i = 1,2,...,n), b j, j+1 = b j+1, j = −1 (j = 1,2,...,n − 1), and other entries are zero) and H = (h i j ) n×n (where h nl = −a and other entries are zero), we have A = B + H and Note that B −1 = (g i j ) n×n , where ( Thus, we have where D = (d i j ) n×n , d il = i for i = 1,2,...,n and other entries are zero.It is well known that it is easy to get and furthermore, Thus, we have (2.8) Lemma 2.1.When |a| < (n + 1)/l, the matrix A is invertible and its inversion is (2.9) In view of Lemma 2.1, system (1.2)-(1.3)can be rewritten by x = λA −1 F(x) or Naturally, we can call that is the Green function of problem (1.2)-(1.3),where 1 ≤ l ≤ n is a fixed integer, a is a real constant, and they satisfy the condition |a| < (n + 1)/l.For 1 ≤ i, j ≤ n, g i j is defined by (2.3).
In the following, we will discuss the properties of G(i, j).In this paper, we will be concerned with the existence of positive solutions for (1.2)-(1.3).Thus, we ask G(i, j) > 0 for 1 ≤ i, j ≤ n.When 0 ≤ a < (n + 1)/l, it easily follows that G(i, j) > 0 for 1 ≤ i, j ≤ n.In the following, we assume that a < 0. Note that min 1≤i, j≤n Thus, we only need to consider the sign of (2.13) By the definition of g l j , we have which implies that holds, then the Green function G(i, j) is positive for Huting Yuan et al. 5 In the following, we will give the estimation of G(i, j).When 0 ≤ a < (n + 1)/l, we have where (n + 1)/2 is the largest integer part of (n + 1)/2.When a is a negative number and the condition (2.15) holds, we have known that and we have also

Existence of one positive solution
In the following, we will ask that the condition (2.16) hold.Let E be the Banach space defined by [1,n] |x k |, and let where δ = m/M when 0 ≤ al < n + 1 and δ = m /M when a is a negative number and (2.15) holds.Define an operator T : P → E by Then for x ∈ P, we have which imply that Tx i ≥ δ Tx , that is, TP ⊂ C.
Proof.Let L > 0 be given and denote . (3.5) Then we have for λ ∈ (0,c] and x ∈ C(L).By Schauder fixed point theorem, T has a fixed point in C(L).
The proof is complete.
The following theorem is immediately obtained by using Theorem 3.1.
Huting Yuan et al. 7 By Theorems 3.1-3.3and the definition of G(i, j), we can also obtain the following theorem.3) is strict increasing and satisfies lim k→∞ λ k = c, the sequence {x (k) } ∞ k=1 is the corresponding solutions sequence of {λ k } ∞ k=1 .Then, we have which implies that we get that { x (k) } is bounded.For every i ∈ [1,n], choosing the subsequence {x (ki)   i } such that lim ki→∞ x (ki) i = limsup k→∞ x (k) i = x (0) i , then x (0) is a positive solution of the equation The proof is complete.
In the following, we do not require the monotonicity of f k for k ∈ [1,n], but a fixed point theorem will be used.It can be seen in [20,21].Lemma 3.5.Let E be a Banach space, and let For the sake of convenience, we set Theorem 3.6.Suppose that there exist two positive numbers a and b such that a = b and b B 0 min k∈ [1,n]
Proof.Assume that a < b and choose x ∈ ∂C a .Then we have which implies that Tx ≤ x for x ∈ ∂C a .Let x ∈ ∂C b , then we get that which implies that Tx ≤ x for x ∈ ∂C b .The proof is complete by Lemma 3.5.When a > b, the proof is similar.
Huting Yuan et al. 9 The following theorem is immediately obtained by using Theorem 3.6.
Theorem 3.7.Assume that there exists k 0 ∈ [1,n] such that f k0 (u) > 0 for u > 0 and that f k∞ and f 0k are finite for any k ∈ [1,n].Then for each λ satisfying the boundary value problem (1.2)-( 1.3) has a positive solution.
Proof.If the condition (3.24) is a fact, we can choose ε > 0 such that 1 δB 0 min k∈ [1,n] which implies that Tx ≤ x for x ∈ ∂C H1 .
By Theorem 3.6, we can similarly obtain the following results.Their proofs will be omitted.
The proof is similar to Theorem 4.2, and hence is omitted.

( 4 . 18 )
Similarly, we can prove that the conditions (H1) and (H3) of Lemma 4.1 hold.An application of Lemma 4.1 now yields our proof.Huting Yuan et al.13