Existence of solutions bounded together with their first derivatives of some systems of nonlinear functional differential equations

Abstract

We give sufficient conditions for the system of nonlinear functional differential equations of the formx′(t+1)=Ax′(t)+F(t,x(t),x(v(1)(t)),…,x(v(k)(t)),x′(g1(t)),…,x′(gl(t))),where A is a matrix, detA≠0, v(j)(t)=φj1(t,x(φj2(…x(φjmj(t,x(t)))…))), j=1,k¯, and functions F:R+×(Rn)k+l+1→Rn, gj:R+→R+, j=1,l¯, φji:R+×Rn→R+, j=1,k¯, i=1,mj¯, are continuous, for the unique existence of continuously differentiable solutions x(t) which are bounded together with their first derivatives on R+=[0,∞) and satisfy the condition limt→+∞|x(t+1)-Ax(t)|=0.