A necessary and sufficient condition for the existence of symmetric positive solutions of higher-order boundary value problems

where n≥  is an integer, f : (, )× (,∞)→ [,∞) is continuous, αi, βi are nonnegative constants, α i + αiβi > , i = , . . . ,n. f (t,u) may be singular at u = , t =  (and/or t = ). If a function u : [, ]→ R is continuous and satisfies u(t) = u( – t) for t ∈ [, ], then we say that u(t) is symmetric on [, ]. By a symmetric positive solution of BVP (.) we mean a symmetric function u ∈ Cn[, ] such that u(t) >  for t ∈ (, ) and u(t) satisfies (.). In recent years, many authors have studied BVP (.), they only considered that f is nondecreasing or nonincreasing in u, or the boundary condition depends only on derivatives of even orders; see [–] and references cited therein. To the best of the author’s knowledge, there is no such results involving (.). In this note, we intend to fill in such gaps in the literature. The organization of this paper is as follows. After this introduction, in Section , we state the assumptions and some preliminary lemmas. By applying the monotone iterative technique, we discuss the existence of symmetric positive solutions for (.) and obtain the main results in Section .