A Global Description of the Positive Solutions of Sublinear Second-Order Discrete Boundary Value Problems

Abstract

Let T∈ℕ be an integer with T>1, 𝕋:={1,…,T}, 𝕋^:={0,1,…,T+1}. We consider boundary value problems of nonlinear second-order difference equations of the form Δ2u(t−1)+λa(t)f(u(t))=0, t∈𝕋, u(0)=u(T+1)=0, where a:𝕋→ℝ+, f∈C([0,∞),[0,∞)) and, f(s)>0 for s>0, and f0=f∞=0, f0=lim⁡s→0+f(s)/s, f∞=lim⁡s→+∞f(s)/s. We investigate the global structure of positive solutions by using the Rabinowitz's global bifurcation theorem.