## 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, Nemytskiĭ, Uryson, etc.)

### Keywords:

fixed-point index; cone; nonzero solution
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### References:

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