Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

Abstract

The paper presents existence principles for the nonlocal boundary-value problem $$ (\phi (u^(p-1)))' = g(t, u,...,u^{(p-1)}), \alpha_k(u)=0, 1 \leq k \leq p-1$$ where $p\geq2,\quad \phi: {\mathbb R}\rightarrow{\mathbb R}$ is an increasing and odd homeomorphism, $g$ is a Caratheodory function which is either regular or has singularities in its space variables and $\alpha_k: C^{p-1}[0,T]\rightarrow{\mathbb R}$ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems $(-1)^n(\phi(u^{(2n-1)}))' = f (t,u,...,u^{(2n-1)}),\quad u^{(2k)}(0) = 0,\quad$ $a_ku^{(2k)}(T) + b_k u^{(2k+1)}(T)=0,\quad 0\leq k\leq n-1$ is given.