Existence of Positive Solutions of a Class of Discrete Difference Systems

In this paper, we consider the existence of positive solutions of a class of discrete difference systems − Δu(t − 1) = λf(v(t)), t ∈ [1, T ] , − Δv(t − 1) = λg(u(t)), t ∈ [1, T ] , u(0) = u(T + 1) = 0, v(0) = v(T + 1) = 0 where f, g ∈ C([0,∞), R), λ is a positive parameter. We prove the existence of a large positive solution for λ large enough under suitable assumptions on f and g. The proof of our main result is based upon the Schauder’s fixed point theorem. Mathematics Subject Classification: 34B18


Introduction
In this papers, we consider the existence of positive solutions of a class of discrete systems where f, g ∈ C([0, ∞), R), T > 1 is a fixed positive integer, [1, T ] Many problems in applied mathematics lead to the study of systems, see [1][2][3][4][5][6][7] and the references therein.Recently, there has been considerable interest in the existence of positive solutions of the discrete systems [6,7].
In [6,7], Sun and Li concerned with discrete system boundary value problems and gave some sufficient conditions for the existence of one or two positive solutions by using a nonlinear alternative of Leray-Schauder type and Krasnosel ' skii ' s fixed point theorem in a cone.
However, to the best of our knowledge, there are very few works have been done to the existence of positive solutions of the discrete systems (1.1).
Our main features are as follows.First, (1.1) may be a semipositone problem, that is, f and g may take negative values.Second, we will study the systems (1.1) without imposing any sign conditions on f (0), g(0) nor any monotonicity assumptions on f, g.Third, we will prove the existence of a large positive solution of (1.1) for λ large.The arguments used were suggested by [9].
In order to prove our main results, the following well-known fixed point theorems are needed.Lemma 1.1.[10,11] Let C be a closed convex subset of the Banach space E. Suppose that F : C → C and F is compact (i.e., bounded sets in C are mapped into relatively compact sets).Then, F has a fixed point in C.
Throughout the paper, we make the following assumptions: By using Schauder's fixed point theorem, our main result is as follows.
Remark 1.1.As an example, let Then it is easily seen that f, g satisfy assumptions (H1), (H2).does not imply that = 0 for every M > 0, as the following example shows.
Let a k be an increasing sequence of positive numbers such that Remark 1.3.In the single-equation case, and if h is superlinear at ∞, then there are no positive solutions for λ large unless h(0) = h(1) = 0.However, here in the systems, even if one of the nonlinearities is superlinear at infinity, if the combined effect is sublinear , then positive solutions do exist for λ large.

Proof of Theorem 1.1
Let f (x) = f (0) and g(x) = g(0) for x < 0. Let ω 0 be the solution of then X is a Banach space with the norm u = max |u(t)|, and A : X × X → X × X be defined by where Then A is completely continuous and fixed points of A are solutions to problem (1.1).Let ψ = (ω λ , ω λ ), where ω λ = 1 2 λLω 0 , and φ = C λ ω 0 , λg(C λ δ)ω 0 , where δ = ω 0 , g(x) = sup y≤x g(y) and C λ is large enough so that ψ ≤ φ.We claim that A : [ψ, φ] → [ψ, φ] for λ is sufficiently large.Once the claim is proved, it then follows from the Schauder's fixed point theorem that A has a fixed point (u, v) with u ≥ w λ and v ≥ w λ .Clearly, u(t), v(t) → ∞ uniformly on [0, T + 1] as λ → ∞.To prove the claim, we will verify the following: By using the same argument, we obtain and Now, by (H2), it is an interesting exercise in elementary analysis to show that lim x→∞ f(M g(x))/x = 0 for each M > 0. Hence we have which is (B2), and the claim is proved.This completes the proof of theorem1.1.2 Remark 2.1.Using similar arguments as in the proof of theorem 1.1, it can be shown that the system Note that these conditions are satisfied if f (u, v) = u α +v β − 1 and g(u, v) = u γ + v σ − 2 , where α, γ, σ, βσ, βγ ∈ (0, 1).In particular, β could be more than 1.

Remark 1 . 2 .
It is an interesting exercise in elementary analysis to show that assumption (H2) implies that lim x→∞ g(Mf (x))x = 0 for everyM > 0,