Abstract
This paper presents new existence results for the singular discrete boundary value problem −Δ2u(k−1)=g(k,u(k))+λh(k,u(k)), k∈[1,T], u(0)=0=u(T+1). In particular, our nonlinearity may be singular in its dependent variable and is allowed to change sign.
Highlights
Let a,b (b > a) be nonnegative integers
This paper presents new existence results for the singular discrete boundary value problem −Δ2u(k − 1) = g(k, u(k)) + λh(k, u(k)), k ∈ [1, T], u(0) = 0 = u(T + 1)
Main results The main result of the paper is the following
Summary
Let a,b (b > a) be nonnegative integers. We define the discrete interval [a,b] = {a,a + 1,. . . ,b}. We will study positive solutions of the second-order discrete boundary value problem. Suppose the following conditions hold: (G) there exist gi : [1, T] × (0, ∞) → (0, ∞) (i = 1, 2) continuous functions such that gi(k, ·) is strictly decreasing for k ∈ [1, T],. There exists λ0 ≥ 0 such that for every λ ≥ λ0, problem (1.1) has at least one solution u ∈ C[0, T + 1] and u(k) > 0 for k ∈ [1, T]. From Schauder’s fixed point theorem (note that Ψz : [u, u] → [u, u]), there exists un ∈ C[0, T + 1] such that un(k) ≤ un(k) ≤ un(k) and Ψ(un)(k) = un(k) for k ∈ [1, T].
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