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Solvability of Neumann boundary-value problems with Caratheodory nonlinearities. (English) Zbl 1095.34516
From the introduction: The authors give an existence result for the Neumann boundary value problem associated to a second-order equation of the form u '' +f(t,u,u ' )=0, where f satisfies the Carathéodory conditions. It is assumed that there exists M 0 >0 such that [ 0 1 f(t,M 0 ,0)dt][ 0 1 f(t,-M 0 ,0)dt]<0, together with a Nagumo-type condition. The authors provide a priori bounds on the C 1 -norm of the (possible) solutions to a parameter-dependent problem associated to the given one; existence is then achieved by using degree arguments (more precisely, the “topological transversality theorem”).
MSC:
34B15Nonlinear boundary value problems for ODE
47H11Degree theory (nonlinear operators)
47J05Equations involving nonlinear operators (general)