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 αix(2i)(0) −βix(2i+1)(0) = 0, γix(2i)(1)+ δix(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. © 2005 Published by Elsevier Inc.