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
$$u^{(n)}(t)=g(t,u(t),u^{\prime }(t)\cdots u^{(n-1)}(t)),\text{a.e.}t\in (0,\mathrm{\infty })$$
subject to the boundary conditions
$$\begin{array}{rcl}{u}^{(n-1)}(0)& =& \frac{(n-1)!}{{\xi }^{n-1}}u(\xi ),u^{(i)}(0)=0,i=1,2,\dots ,n-2,\\ u^{(n-1)}(\mathrm{\infty })& =& {\int }_0^{\xi }{u}^{(n-1)}(s)dA(s)\end{array}$$
where $$\xi >0,g:[0,\mathrm{\infty })\times {\mathrm{\Re }}^n⟶\mathrm{\Re }$$ is a Caratheodory's function,

$A:[0,\xi ]⟶[0,1)$ 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.