Besicovitch almost periodic solutions for a class of second order differential equations involving reflection of the argument

Abstract

In this paper we investigate the existence and uniqueness of bounded, periodic and almost periodic solutions for second order differential equations involving reflection of the argument.The relationship between frequency modules of forced term and the solution of the equation is considered.

On almost periodicity of solutions of second-order differential equations involving reflection of the argument

“…They proved that t → x(t) is almost periodic by assuming the existence of bounded solutions. Further results were extended and improved by several authors; see, for instance, the papers by Hai [18], O'Regan [22], Piao [23,24], Piao and Sun [25], and Zima [30]. In particular, Piao [23,24] investigated the existence and uniqueness of periodic, almost periodic, and pseudo almost periodic solutions of the equations…”

Response Solutions for a Singularly Perturbed System Involving Reflection of the Argument

“…He proved the existence of solutions for the boundary value problems in a simple and straightforward manner. The existence and uniqueness of periodic, almost periodic, pseudo almost periodic, Besicovitch almost periodic, and pseudo almost automorphic solutions of this kind of equations were investigated in [14][15][16][17][18][19]. Cabada et al [20][21][22] studied the first-order equation with two-point boundary conditions and the nth-order differential equations involving reflection, con-stant coefficients, and initial conditions, adding a new element to the previous studies: the existence of Green's function.…”

Nonlinear perturbed integral equations related to nonlocal boundary value problems

“…On the other hand, BVPs where nonlocal terms occur in the differential equation have been studied by a number of authors. For example, the case of equations with reflection of the argument has been investigated by Andrade and Ma [3], Cabada and co-authors [6], Piao [45,46], Piao and Xin [47], Wiener and Aftabizadeh [61], the case of equations with deviated arguments has be en studied by Jankowski [34][35][36], Figueroa and Pouso [14] and Szatanik [51,52] and the case of equations that involve the average of the solution has been considered by Andrade and Ma [3], Chipot and Rodrigues [8] and Infante [27].…”