On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions

We prove the existence of extremal solutions for the third order discontinuous functional nonlinear problem − φ u (t ) = f t,u,u (t ),u (t ) for a.e. t ∈ [a, b], u(a) =A (A∈ R), L1 u, u ,u (a),u (b),u (a) = 0, L2 u, u (a),u (b),u (b) = 0, by using a fixed point theorem after having established some existence results for some auxiliary second order nonlinear problems. We observe that, together with the discontinuities allowed on the spacial variable u, with adequate modifications of technical type, analogous results can be obtained when the equation has a second member not necessarily continuous in u .  2003 Elsevier Inc. All rights reserved.