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

Samuel Iyase, Abiodun Opanuga

Abstract

This paper investigates the solvability of a class of higher order nonlocal boundary
value problems of the form
u
(n)
(t) = g(t, u(t), u0
(t)· · · u
(n−1)(t)), a.e. t ∈ (0, ∞)

subject to the boundary conditions

u
(n−1)(0) = (n − 1)!
ξ
n−1
u(ξ), u(i)
(0) = 0, i = 1, 2, . . . , n − 2,

u
(n−1)(∞) = Z ξ
0
u
(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 operator
is a Fredholm map of index zero and non-invertible. We shall employ coicidence degree arguments
and construct suitable operators to establish existence of solutions for the above higher order
nonlocal boundary value problems at resonance.

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