An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem

Abstract

In this article, we show the existence of an antisymmetric solution to the second order boundary value problem x + f (x(t)) = 0, t ∈ (0, n) satisfying antiperiodic boundary conditions x(0) + x(n) = 0, x (0) + x (n) = 0 using an Avery et. al. fixed point theorem which itself is an extension of the traditional Leggett-Williams fixed point theorem. The antisymmetric solution satisfies x(t) = −x(n − t) for t ∈ [0, n] and is nonnegative, nonincreasing, and concave for t ∈ [0, n/2]. To conclude, we present an example.