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In this paper, we consider the existence of positive solutions for a second-order discrete boundary value problem Δ(g(k−1)Δu(k−1))+w(k)f(k,u(k))=0 subject to the boundary conditions: au(0)−bg(0)Δu(0)=∑i=1n−1h(i)u(i), au(n)+bg(n−1)Δu(n−1)=∑i=1n−1h(i)u(i), where a,b>0, Δu(k)=u(k+1)−u(k) for k∈{0,1,…,n−1}, g(k)>0 is symmetric on {0,1,…,n−1}, w(k) is symmetric on {0,1,…,n}, f:{0,1,…,n}×[0,+∞) is continuous, f(k,u)=f(n−k,u) for all (k,u)∈{0,1,…,n}×[0,+∞), and h(i) is nonnegative and symmetric on {0,1,…,n}. By the fixed point theorem and the Hölder inequality, we study the existence of symmetric positive solutions for the above difference equation with sum form boundary conditions.

The existence of symmetric positive solutions for a seconder-order difference equation with sum form boundary conditions