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Three-point boundary value problems with solutions that change sign. (English) Zbl 1055.34023
The authors study the existence of nonzero solutions of the second-order differential equation \[ u^{''}(t)+g(t)f(u(t))=0, \;\;0<t<1,\eqno{(1)} \] under one of the boundary conditions \[ u^{'}(0)=0,\;\alpha u(\eta)=u(1),\;\;0<\eta<1,\eqno{(2)} \] \[ u(0)=0,\;\alpha u(\eta)=u(1),\;\;0<\eta<1.\eqno{(3)} \] By using the theory of fixed-point index, results on the existence of at least one or of multiple nonzero solutions for BVP (1), (2), and BVP (1), (3), are obtained.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
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