General existence principles for nonlocal boundary value problems withφ-Laplacian and their applications

Abstract

The paper presents general existence principles which can be used for a large class of nonlocal boundary value problems of the form(φ(x′))′=f1(t,x,x′)+f2(t,x,x′)F1x+f3(t,x,x′)F2x,α(x)=0,β(x)=0, wherefjsatisfy local Carathéodory conditions on some[0,T]×𝒟j⊂ℝ2,fjare either regular or have singularities in their phase variables(j=1,2,3),Fi:C1[0,T]→C0[0,T](i=1,2), andα,β:C1[0,T]→ℝare continuous. The proofs are based on the Leray-Schauder degree theory and use regularization and sequential techniques. Applications of general existence principles to singular BVPs are given.