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

*Correspondence: [email protected] 2College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, P.R. China Full list of author information is available at the end of the article Abstract In this paper, we consider the existence of positive solutions for a second-order discrete boundary value problem (g(k – 1) u(k – 1)) +w(k)f (k,u(k)) = 0 subject to the boundary conditions: au(0) – bg(0) u(0) = ∑n–1 i=1 h(i)u(i), au(n) + bg(n – 1) u(n – 1) = ∑n–1 i=1 h(i)u(i), where a,b > 0, u(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.