Higher Order Nonlocal Boundary Value Problems at Resonance on the Half-line

Abstract

This paper investigates the solvability of a class of higher order nonlocal boundaryvalue problems of the formu(n)(t) = g(t, u(t), u0(t)· · · u(n−1)(t)), a.e. t ∈ (0, ∞)subject to the boundary conditionsu(n−1)(0) = (n − 1)!ξn−1u(ξ), u(i)(0) = 0, i = 1, 2, . . . , n − 2,u(n−1)(∞) = Z ξ0u(n−1)(s)dA(s)where ξ > 0, g : [0, ∞) × <n −→ < is a Caratheodory’s function,A : [0, ξ] −→ [0, 1) is a non-decreasing function with A(0) = 0, A(ξ) = 1. The differential operatoris a Fredholm map of index zero and non-invertible. We shall employ coicidence degree argumentsand construct suitable operators to establish existence of solutions for the above higher ordernonlocal boundary value problems at resonance.