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 form x' (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, det A≠0, v( (j)(t)= ρj1(t,x(ρj1 (⋯x(ρjmj (t,x(t)))⋯))), j=1,k̄, and functions F:ℝ+×(ℝn)k+l+1→ℝ n, gj:ℝ+→ℝ+, j=1,l̄, ρji:R+× ℝ n→ℝ+, j=1,k̄, i=1, mj̄, 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 lim t→+∞|x(t+1)-Ax(t)|=0.