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.


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, Nemytskiĭ, Uryson, etc.)
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