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Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions. (English) Zbl 1132.34022
The Sturm-Liouville problem with integral boundary value conditions $$ -(au')' + bu = g(t)f(t,u), \quad t \in (0,1), $$ $$ (\cos \gamma_0) u(0) - (\sin \gamma_0) u'(0) = \int_0^1 u(\tau) \,d\alpha(\tau), $$ $$ (\cos \gamma_1) u(1) + (\sin \gamma_1) u'(1) = \int_0^1 u(\tau) \,d\beta(\tau), $$ is considered, where $\int_0^1 u(\tau)\,d\alpha(\tau)$ and $\int_0^1 u(\tau) \,d\beta(\tau)$ denote Riemann-Stieltjes integrals. Existence results of a nontrivial solution are established. The proofs are based on the Leray-Schauder degree theory.

34B24Sturm-Liouville theory
34B15Nonlinear boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
47H11Degree theory (nonlinear operators)
Full Text: DOI
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