Existence and uniqueness of solutions of a fourth-order boundary value problem with non-homogeneous boundary conditions

Let m >= 2 and a, b, c > 0. We consider the existence and uniqueness of solutions for the fourth order iterative boundary value problem, x((4))(t) = - f (t, x(t), x([2])(t),..., x([m])( t)), -a <= t <= a where x([2])(t) =x x(t)) and for j = 3,..., m, x([j])(t) = x x([j-1])(t)), with solutions satisfying one of the following sets of conjugate boundary conditions: x(- a) = -a, x '(-a) = b, x ''(- a) = c, x(a) = a, x(- a) = -a, x(a) = a, x '( a) = b, x ''(a) = c. The main tool used is the Schauder fixed point theorem.