Two-point boundary value problems for 4th order ordinary differential equations

Abstract

The new optimal efficient sufficient conditions are established for solvability and uniqueness of a solution of the linear and nonlinear fourth order ordinary differential equations u^(4)(t) = p(t)u(t) + q(t) for t ∈ [a,b], u^(4)(t) = p(t)u(t) + f(t, u(t)) for t ∈ [a,b], under the following two-point boundary conditions u^(i)(a) = 0, u^(i)(b) = 0 (i = 0, 1), and u^(i)(a) = 0 (i = 0, 1, 2), u(b) = 0, where p ∈ L([a,b]; R) is a nonconstant sign function and f ∈ K([a,b] × R; R).