zbMATH — the first resource for mathematics

Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. (English) Zbl 0763.34009
The paper deals with the second order three-point nonlinear boundary value problem (1) \(u''=f(x,u(x),u'(x))-e(x)\), \(0<x<1\), \(u(0)=0\), \(u(\eta)=u(1)\), where \(f\) is a Carathéodory function and \(e\) is a Lebesgue integrable function. Provided \(f\) has at most linear growth in its phase variables, the author establishes conditions for the existence of solutions to (1) and for the uniqueness of problem (1). The proofs are based on the topological degree theory and the Leray-Schauder continuation theorem. A priori estimates are obtained by Wirtinger-type inequalities.

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
[1] Kiguradze, I.T; Lomtatidze, A.G, In certain boundary value problems for second-order linear ordinary differential equations with singularities, J. math. anal. appl., 101, 325-347, (1984) · Zbl 0559.34012
[2] Mawhin, J, Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.