Green’s function for a third-order generalized right focal problem. (English) Zbl 1045.34008

The first part is concerned with the third-order three-point generalized right focal boundary value problem \(x'''(t)=0,\;t_1\leq t\leq t_3,\;x(t_1)=x'(t_2)=0,\;\gamma x(t_3)+\delta x''(t_3)=0,\) with \(\gamma\geq0\) and \(\delta>0\). Green’s function for this homogeneous problem is determined. Then sufficient conditions in terms of coefficients and boundary points are given, which guarantee its positivity. The second part of this paper is concerned with proving the existence of positive solutions of the boundary value problem (BVP) \(x'''(t)=f(t,x(t)),\;x(t_1)=x'(t_2)=0,\;\gamma x(t_3)+\delta x''(t_3)=0,\) (*), where \(f\) is nonnegative for \(x\geq0\). More precisely, Krasnoselskii’s fixed-point theorem is used to find sufficient conditions for the existence of a positive solution of BVP (*) whose norm lies between given numbers. Further, the fixed-theorem of Legget and Williams is used to find conditions for the existence of at least three positive solutions of BVP (*).
Reviewer: Pavel Rehak (Brno)


34B27 Green’s functions for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


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