Some Invariant Manifolds for Functional Difference Equations with Infinite Delay

A note on discrete maximal regularity for functional difference equations with infinite delay

“…Questions concerning boundedness, convergence and asymptotic behavior of perturbations (1.1) has been studied by Cuevas [6], Cuevas and Pinto [8][9][10], Cuevas and Vidal [11], Cuevas and Del Campo [7]. Recently, a very interesting article has been published by Matsunaga and Murakami [16] concerning to the existence of the local stable manifolds, together with the local unstable manifolds and the local center-unstable manifolds for nonlinear autonomous functional difference equations in phase spaces. The results in [16] are based on a representation formula for solutions of nonhomogeneous linear functional difference equations.…”

“…Recently, a very interesting article has been published by Matsunaga and Murakami [16] concerning to the existence of the local stable manifolds, together with the local unstable manifolds and the local center-unstable manifolds for nonlinear autonomous functional difference equations in phase spaces. The results in [16] are based on a representation formula for solutions of nonhomogeneous linear functional difference equations. As application of the general results given in [16] have been obtained some results on stabilities and instabilities for the zero solution of equation…”

“…The results in [16] are based on a representation formula for solutions of nonhomogeneous linear functional difference equations. As application of the general results given in [16] have been obtained some results on stabilities and instabilities for the zero solution of equation…”

“…For these reasons from then the theory of functional difference equations with infinite delay has drawn the attention of several authors (see [6][7][8][9][10][11][12][16][17][18]). The fact that the state space for functional difference equations is infinite dimensional require the development of methods and techniques from functional analysis.…”

Asymptotic Expansions for Higher-Order Scalar Difference Equations

“…The simple short proof presented below is based on the inversion formula for the z-transform and the residue theorem. Finally, we mention the recent remarkable work of Matsunaga and Murakami [7] which is relevant to our study. In this paper, the authors described the structure of the solutions of nonlinear functional difference equations in a neighborhood of an equilibrium.…”