×

zbMATH — the first resource for mathematics

A nonlocal singular boundary value problem for second-order differential equations. (English) Zbl 1048.34041
Summary: We discuss the singular differential equation \((g(x'))'=f(t,x,x')\) together with the nonlocal boundary conditions \(\max\{x(t): t \in [0,T]\}=A\), \(x(0)=x(T)\). Here, \(g \in C^0({\mathbb R})\) is an increasing and odd function, whereas the negative \(f\) satisfying the local Carathéodory conditions on \([0,T] \times {\mathbb R} \times ({\mathbb R} \setminus\{0\})\) may be singular at the value \(0\) of its second phase variable and \(A \in {\mathbb R}\). An existence result for the above boundary value problem is proved by regularization and sequential techniques. The proofs use the Leray-Schauder degree principle and the Vitali convergence theorem.

MSC:
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite