Existence theory for functional p-Laplacian equations with variable exponents

Abstract

In this paper we consider the solvability of equations of the form − d dt ϕ(t,u,u(t),u′(t))=f(t,u,u(t),u′(t)) for a.e. t∈I=[a,b] subject to a general type of functional-boundary conditions which cover Dirichlet and periodic boundary data as particular cases. Our approach is that of upper and lower solutions together with growth restrictions of Nagumo's type. An example is provided where a p-Laplacian with variable p is shown to have a solution between given upper and lower solutions.