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Positive solutions of some nonlocal boundary value problems. (English) Zbl 1072.34014

For two 4-point BVP

u '' (t)+g(t)f(u(t))=0a.e.on[0,1],
u ' (0)=0,u(1)=α 1 u(η 1 )+α 2 u(η 2 ),

or

u(0)=0,u(1)=α 1 u(η 1 )+α 2 u(η 2 ),

the authors determine a region in the (α 1 ,α 2 )-plane which ensures the existence of positive solutions. Further, they conclude that one can obtain the existence of positive solutions for an m-point boundary value problem under the weaker assumption that all parameters occurring in the boundary conditions are not required to be positive. Hence, their results allow more general behavior on f than being either sub- or superlinear.

MSC:
34B10Nonlocal and multipoint boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47H10Fixed point theorems for nonlinear operators on topological linear spaces
34B15Nonlinear boundary value problems for ODE