Solvability of some two-point boundary value problems of Dirichlet, Neumann or periodic type. (English) Zbl 0785.34025

O’Regan, Donal

Dyn. Syst. Appl. 2, No. 2, 163-182 (1993).

The paper deals with the existence of solutions for two point boundary value problems \({1 \over p} (py')'=qf(t,y,py')\), \(0<t<1\), of Dirichlet, Neumann or periodic type. The tools are the nonlinear alternative of Leray-Schauder and the “a priori bounds” technique. The novelty is that the conditions ensuring the a priori boundedness of the solutions are expressed with the aid of a number \(c\) located between two consecutive eigenvalues of the problem: \({1 \over p} (py')'+\lambda y=0\), \(y\) satisfies the corresponding Dirichlet, Neumann or periodic boundary condition.

Reviewer: R.Precup (Cluj-Napoca)

MSC:

34B15

Nonlinear boundary value problems for ordinary differential equations

Keywords:

a priori bounds; two point boundary value problems; Dirichlet, Neumann or periodic type; nonlinear alternative of Leray-Schauder