Extremal solutions to fourth order discontinuous functional boundary value problems

摘要

In this paper, given f : [a, b] × C ([a, b]) × R~2 → R a L~1-Caratheodory function, it is considered the functional fourth order equation u~((iv)) (x)=f(x, u, u"(x), u'''(x)), a.a. x∈[a, b], together with the nonlinear functional boundary conditions L_0(u, u", u(a))=0, L_1(u, u", u(b))=0, L_2(u, u", u"(a), u'''(a))=0, L_3(u, u"(b), u'''(b))=0. Here L_i, i = 0, 1, 2, 3, satisfy some adequate monotonicity assumptions and are not necessarily continuous functions. It will be proved an existence and location result in presence of non ordered lower and upper solutions.