Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

Abstract

The well-known Favard's theorem states that the linear differential equation(1)with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equationsthere are bounded solutions which are nonseparated, sometimes almost periodic solutions do not exist (R. Johnson, R. Ortega and M. Tarallo, V. Zhikov and B. Levitan).In this paper we prove that linear differential equation (1) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution, and the bounded solutions of the homogeneous equationare homoclinic to zero (i.e. lim |t|→+∞ |ϕ(t)| = 0 for all bounded solutions ϕ of (3)). If the coefficients of (1) are Bohr almost periodic and all bounded solutions of all limiting equations (2) are homoclinic to zero, then Eq. (1) admits a unique almost automorphic solution.The analogue of this result for difference equations is also given.We study the problem of existence of Bohr/Levitan almost periodic solutions of Eq. (1) in the framework of general non-autonomous dynamical systems (cocycles).

Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations

“…The theory of almost periodic functions and systems initiated by Bohr was largely developed in the last century (see [5,14]), and many mathematicians consider almost periodic problems that exist in the change and development in nature (see [8,9]). To find a class of functions which are more general than almost periodic functions, the notion of pseudo-almost periodic functions was introduced by Zhang in [35,36].…”

Almost periodicity and almost automorphy for some evolution equations using Favard’s theory in uniformly convex Banach spaces

“…Recently, Almost periodic and almost automorphic solutions for Eq. (2) via Favard's approach has been largely investigated in the literature [9,10,20,26,29,32]. More results about almost periodic differential equations in finite dimensional spaces can be found in [17].…”

Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales

“…Many papers have been published on the theory of dynamic equations on time scales [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. In addition, the existence of almost periodic, asymptotically almost periodic, pseudo-almost periodic solutions is among the most attractive topics in the qualitative theory of differential equations and difference equations due to their applications, especially in biology, economics and physics [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. Recently, in [14,35], the almost periodic functions and the uniformly almost periodic functions on time scales were presented and investigated, as applications, the existence of almost periodic solutions to a class of functional differential equations and neural networks were studied effectively (see [13,14,35]).…”