EXISTENCE OF SOLUTIONS TO BOUNDARY VALUE PROBLEMS AT FULL RESONANCE

The focus of this paper is the study of nonlinear differential equations of the form x(i()t) = a(i()t)x(i)(t) + f(i)(epsilon,t,x1(t), ... , x(n)(t)), i = 1,2 ... ,n, subject to two-point boundary conditions b(i)x(i)(0) d(i)x(i)(1) = 0, i = 1,2, ... ,n. We formulate sufficient conditions for the existence of solutions based on the dimension of the solution space of the corresponding linear, homogeneous equation and the properties of the nonlinear term when epsilon = 0. We focus on the case when the solution space of the corresponding linear, homogeneous equation is n-dimensional; that is, when the system is at full resonance. The argument we use relies on the Lyapunov-Schmidt procedure and the Schauder fixed point theorem.