Non ordered lower and upper solutions to fourth order functional BVP

In this paper, given a L1-Carath éodory function, it is considered the functional fourth order equation u^(iv) (x) = f(x; u; u'; u'' (x) ; u''' (x)) together with the nonlinear functional boundary conditions L_0(u; u'; u''; u (a)) = 0 = L_1(u; u'; u''; u' (a)) L_2(u; u'; u''; u'' (a) ; u''' (a)) = 0 = L_3(u; u'; u''; u'' (b) ; u''' (b)): Here L_i, i = 0; 1; 2; 3, are continuous functions satisfying some adequate monotonicity assumptions. It will be proved an existence and location result in presence of non ordered lower and upper solutions and without monotone assumptions on the right hand side of the equation