Irena Rachunkov, Svatoslav Staněk

General Existence Principle for Singular BVPs and Its Application

Abstract:
We present a general existence principle which can be used for a large class of singular boundary value problems of the form $$u^{(n)}(t)=f\big(t,u(t),\dots,u^{(n-1)}(t)\big), \quad u\in \s,$$
where $f$ satisfies the local Carath\'{e}odory conditions on $[0,T] \times \D$, a set $\D\subset \R^n$ is not closed, $f$ has singularities in its phase variables on the boundary $\partial \D$ of $\D$, and $\s$ is a closed subset in $C^{n-1}([0,T])$. The proof is based on the regularization and sequential techniques. An application of the general existence principle to singular conjugate $(p,n-p)$ BVPs is also given.

Keywords:
General existence principle, singular BVP, conjugate BVP, regularization, Vitali's convergence theorem.

MSC 2000: 34B16, 34B18