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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