Strong solutions for some nonlinear partial functional differential equations with infinite delay

Abstract

In this work, we use the Kato approximation to prove the existence of strong solutions for partial functional differential equations with infinite delay. We assume that the undelayed part is $m$-accretive in Banach space and the delayed part is Lipschitz continuous. The phase space is axiomatically defined. Firstly, we show the existence of the mild solution in the sense of Evans. Secondly, when the Banach space has the Radon-Nikodym property, we prove the existence of strong solutions. Some applications are given for parabolic and hyperbolic equations with delay. The results of this work are extensions of the Kato-approximation results of Kartsatos and Parrot [8,9].

Keywords

• Partial functional differential equations
• infinite delay
• m-accretive operator
• Kato approximation
• mild solution in the sense of Evans
• strong solution
• Radon-Nikodym property