Functional Differential Equations with Infinite Delay

Bounded and almost periodic solutions and evolution semigroups associated with nonautonomous functional differential equations

“…We then demonstrate an application of the obtained results to study the boundedness and almost periodicity of solutions to functional differential equations. As a result we get a sufficient condition for the existence of bounded and almost periodic solutions which is an extension to fully nonlinear functional differential equations of previous results by other authors (see [10,12,16,20,21] for related results and methods, [6,25] for more information on ordinary differential equations with almost periodic coefficients, and [8] for various conditions for almost periodicity of solutions of equations with infinite delay). In this paper, we put an emphasis on an application of evolution semigroup method to the study of boundedness and almost periodicity of solutions to nonlinear nonautonomous functional differential equations via nonlinear semigroup techniques.…”

“…In this paper, we put an emphasis on an application of evolution semigroup method to the study of boundedness and almost periodicity of solutions to nonlinear nonautonomous functional differential equations via nonlinear semigroup techniques. For other various applications of nonlinear semigroup theory to nonlinear functional differential equations, we refer the reader to [5,8,9,22,24] and the references therein for some fundamental notions and results which may have direct relations with the present paper. We now give a brief outline of our constructions.…”

Some results in stochastic functional integro-differential equations with infinite delays

“…We utilise the axiomatic definition of the phase space B introduced by Hale and Kato (1978) and Hinto, Murakami and Naito (1991). The axioms of the space B are established for F 0 -measurable functions from (−∞, 0] into X, endowed with a norm · B is given as the following axioms: (A1) If x : (−∞, T ] → X, T > 0 is such that x 0 ∈ B then for every t ∈ [0, T ], the following conditions hold:…”

Existence of Solutions for Nonlinear Evolution Equations With Infinite Delay

“…A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [7]. For a detailed discussion on the topic, we refer to the book by Hino et al [9]. In the last decades, the theory of functional differential equations of various classes with delay has attracted widespread attention.…”