On a elastic beam fully equation with nonlinear boundary conditions

Abstract

We study the fourth-order nonlinear boundary value problem u^{iv}=f(t,u,u′,u′′,u′′′), 0<t<1, u(0)=A, u′(0)=B, g(u′′(0), u′′′(0))=0, h(u′′(1),u′′′(1))=0, with f:[0,1]×R⁴→R a continuous function veryfing a Nagumo-type condition, A,B∈R and g,h:R²→R are continuous functions with adequate monotonicities. For this model of the bending of an elastic beam, clamped at the left end-point, we obtained an existence and location result by lower and upper-solution method and degree theory. Similar results are presented for the same beam fully equation with different types of boundary conditions.