Solutions of two-point boundary value problems for even-order differential equations

The existence of solutions of the two-point boundary value problems consisting of the even-order differential equations x((2n)) (t) = f (t, x(t), x'(t),..., x((2n-2)) (t)) + r(t), 0 < t < 1, and the boundary value conditions alpha(i)x((2i)) (0) - beta(i)x((2i+1)) (0) = 0, gamma(i)x((2i)) (1) + delta(i)x((2i+1)) (1) = 0, i = 0, 1,..., n-1, is studied. Sufficient conditions for the existence of at least one solution of above BVPs are established. It is interesting that the nonlinearity f in the equation depends on all lower derivatives, especially, odd order derivatives, and the growth conditions imposed on f are allowed to be super-linear (the degrees of phases variables are allowed to be greater than 1 if it is a polynomial). The results are different from known ones since we do not apply the Green's functions of the corresponding problem and the method to obtain a priori bounds of solutions is different from known ones. Examples that cannot be solved by known results are given to illustrate our theorems. (c) 2005 Published by Elsevier Inc.