Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay

Abstract

In this work, we give a variation of constants formula for partial functional differential equations with infinite delay. We assume that the linear part is not necessarily densely defined and its resolvent operator satisfies the Hille-Yosida condition. We establish a reduction of the problem to a finitedimensional space which allows us to prove the existence of almost periodic solutions.

Approximate Controllability for a Semilinear Evolution System with Infinite Delay

“…A variety of mathematical models for physical, chemical, or biological processes are most appreciatively formed as PFDEs with infinite delay, for example the reaction-diffusion logistic equation with infinite delay and equations of heat conduction in materials with fading memory. In the past decades, existence, uniqueness, periodicity, regularity, and stability of the solutions for this kind of evolution equations have been studied by many authors, see [1,2,5,10,11,23,24] among others, and see [12,25] for more practical background of this kind of equations.…”

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“…Owing to the infinite dimensionality of the DDEs, inclusion of time delays into the equation of motion makes analysis more difficult and challenging. The theory of the equations with time-delays is an active topic of research and some recent developments can be found in, for example, Hu & Wang (2002), Adimy et al (2006) and Laurent et al (2006).…”