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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.

MSC:

34B24 Sturm-Liouville theory
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
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