Periodic solutions for some partial functional differentialequations

Abstract

We study the existence of a periodic solution for some partial functional differential equations. We assume that the linear part is nondensely defined and satisfies the Hille-Yosida condition. In the nonhomogeneous linear case, we prove the existence of a periodic solution under the existence of a bounded solution. In the nonlinear case, using a fixed-point theorem concerning set-valued maps, we establish the existence of a periodic solution.

Periodic solutions for some nonautonomous semilinear boundary evolution equations

“…Consider the 1-periodic functions 1], then the boundary evolution equation (5.7) takes the following abstract form…”

“…In [19], Massera explains the relationship between the boundedness of solutions and the existence of periodic solutions. The literature is rich for researches about the existence of periodic solutions, see e.g., [1,8,9,11,12,14,17,18,23]. In many of those studies, the most important feature is to show that the Poincaré map…”

“…In [13], using Horn's fixed point theorem, the existence of periodic solutions for functional differential equations with finite delay was established. In [1], the authors proved the existence of a periodic solution for partial functional differential equations.…”

Massera Type Theorems for Abstract Functional Differential Equations

“…The Massera Theorem has been extended to various kinds of di¤erential equations. We refer the reader to [4,15,16,27,3,11] and the references therein for more information in this direction. The method employed in these works is to prove the existence of periodic solutions via the existence of fixed points of the associated period maps.…”

Linear Impulsive Periodic System with Time-Varying Generating Operators on Banach Space

“…The periodic solution theory of dynamic equations has been developed over the last decades. We refer the readers to [1][2][3][4][5][6][7][8][9][10][11] for infinite dimensional cases, to [12][13][14][15] for finite dimensional cases. Especially, there are many results of periodic solutions (such as existence, the relationship between bounded solutions and periodic solutions, stability, and robustness) for non-autonomous impulsive periodic system on finite dimensional spaces (see [12,14,15]).…”