On third order coupled systems with full nonlinearities

Abstract

This work studies the solvability of the nonlinear third order coupled system composed by the differential equations −u′′′ (t) = f (t, v(t), v′(t), v′′(t)) −v′′′ (t) = h (t, u(t), u′(t), u′′(t)) , with f, h : [0, 1] × R3 → R L1-Carathéodory functions and the two-point boundary conditions u (0) = u′ (0) = u′ (1) = 0 v (0) = v′ (0) = v′ (1) = 0. An adequate truncature together with Nagumo-type conditions allow the dependence of the nonlinearities on the second derivatives. By lower and upper solutions method we obtain strips where the unknown functions and their derivatives must lie, which provides some qualitative data on the solutions.