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 boundary value problems of the form
$$u^{(n)}(t) = g(t, u(t), u'(t)\cdots u^{(n-1)}(t)), \;\mbox{a.e.}\; t\in(0,\infty)$$
subject to the boundary conditions
\begin{eqnarray*}
u^{(n-1)}(0) &=& \frac{(n-1)!}{\xi^{n-1}}u(\xi), u^{(i)}(0) = 0,\; i=1,2,\dots, n-2,\\
u^{(n-1)}(\infty) &=& \int^\xi_0 u^{(n-1)}(s)dA(s)
\end{eqnarray*}
where $$\xi > 0, g:[0,\infty) \times \Re^n \longrightarrow \Re$$ is a Caratheodory's function,

$A:[0,\xi] \longrightarrow [0,1)$ is a non-decreasing function with $A(0)=0,A(\xi)=1$. The differential operator is a Fredholm map of index zero and non-invertible. We shall employ coincidence degree arguments and construct suitable operators to establish existence of solutions for the above higher order nonlocal boundary value problems at resonance.