Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems - Staněk, S. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
The paper presents existence principles for the nonlocal boundary-value problem $$ (φ (u^(p-1)))' = g(t, u,...,u^{(p-1)}), α_k(u)=0, 1 ≤ k ≤ p-1$$where $p≥2,\quad φ: {\mathbb R}→{\mathbb R}$is an increasing and odd homeomorphism, $g$ is a Caratheodory function which is either regular or has singularities in its space variables and $α_k: C^{p-1}[0,T]→{\mathbb R}$ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems $(-1)^n(φ(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≤ k≤ n-1$ is given.