A Dirichelet-Neumann m-point BVP with a p-Laplacian-like operator

Abstract

Let phi, theta be odd increasing homeomorphisms from R onto R satisfying phi(0) = theta(0) = 0, and let f : [a, b] x R x R -> R be a function satisfying Caratheodory's conditions. Let alpha(i) is an element of R, xi(i) is an element of (a, b), i = 1, ..., m-2, a < xi(1) < xi(2) <... < xi(m -2) < b be given. We are interested in the problem of existence of solutions for the m-point boundary value problem: [GRAPHICS]

in the resonance and non-resonance cases. We say that this problem is at resonance if the associated problem

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has a non-trivial solutions. This is the case if and only if Sigma(i=1)(m-1) alpha(i) = 1. Our results use topological degree methods. Interestingly enough in the non-resonance case, i.e., when Sigma(i=1)(m-2) alpha(i) not equal 1 the sign of degree for the relevant operator depends on whether Sigma(i=1)(m-2) alpha(i) > 1 or Sigma(i=1)(m-2) alpha(i) < 1.