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

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations