Extremal solutions to fourth-order functional boundary value problems including multipoint conditions

This paper concerns the fully fourth-order nonlinear functional equation equation(E)−ddt(ϕ∘u‴)(t)=f(t,u″(t),u‴(t),u,u′,u″),for a.a. t∈I=[a,b], with the functional boundary conditions (BC) B1(u(b),u,u′,u″)=0=B2(u′(b),u,u′,u″),B1(u(b),u,u′,u″)=0=B2(u′(b),u,u′,u″),B3(u″(a),u″(b),u‴(a),u‴(b),u,u′,u″)=0=L2(u″(a),u″(b)),B3(u″(a),u″(b),u‴(a),u‴(b),u,u′,u″)=0=L2(u″(a),u″(b)), where ϕ:R⟶Rϕ:R⟶R is an increasing homeomorphism, f:I×R2×(C(I))3⟶R,Bi,i=1,2,3, and L2L2 are suitable functions. The existence of extremal solutions for problem (E)-(BC) is proved by defining a convenient partial ordering. Some sufficient conditions to obtain lower and upper solutions are given.