S. Staněk

Nonlocal Singular Boundary Value Problems for Even-Order Differential Equations

abstract:
Differential equations of the type $x^{(2n)}=f(t,x,\dots,x^{(2n-1)})$ are considered. Here a positive function $f$ satisfies local Carathéodory conditions on a subset of $[0,T] \times \mathbb{R}^{2n}$ and $f$ may be singular at the value 0 of all its phase variables. The paper presents conditions guaranteeing the existence of a solution of the above differential equation satisfying nonlocal boundary conditions whose special case are the $(2p,2n-2p)$ right focal boundary conditions $x^{(j)}(0)=0$ for $0\leq j\leq 2p-1$ and $x^{(j)}(T)=0$ for $2p\leq j\leq 2n-1$, where $p\in \mathbb{N}$, $1\leq p\leq n-1$.

Mathematics Subject Classification: 34B16, 34B15

Key words and phrases: Singular boundary value problem, even-order differential equation, nonlocal boundary conditions, focal boundary conditions, existence