Lower and Upper Solutions for System of Differential Equations Involving Homeomorphism and Nonlinear Boundary Conditions

Abstract

We study the existence of solution to the system of differential equations (φ(u)) = f(t, u, u) with nonlinear boundary conditionsg(u(0), u, u)=0, h(u(1), u, u)=0, where f : [0, 1]×Rn ×Rn → Rn, g, h : Rn ×C([0, 1], Rn)×C([0, 1], Rn) →Rn are continuous, φ : ni=1(−ai, ai) → Rn, 0 < ai ≤ +∞, φ(s) =(φ1(s1),...,φn(sn)) and φi : (−ai, ai) → R is a one dimensional regular or singular homeomorphism. Our proofs are based on the concept of the lower and upper solutions