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Positive solutions of some nonlocal boundary value problems. (English) Zbl 1072.34014
For two 4-point BVP $$u''(t)+g(t)f(u(t))=0\quad \text{a.e. on }[0,1],$$ $$u'(0)=0,\quad u(1)=\alpha_1u(\eta_1)+\alpha_2u(\eta_2),$$ or $$u(0)=0,\quad u(1)=\alpha_1u(\eta_1)+\alpha_2u(\eta_2),$$ the authors determine a region in the $(\alpha_1,\,\alpha_2)$-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.

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
Full Text: DOI EuDML