Periodic Solution for some Class of Linear Partial Differential Equation with infinite Delay using Semi-Fredholm perturbations

Abstract

In this work, we study the existence of periodic solutions for a class of linear partial functional differential equations with infinite delay. Inspiring by an existing study, by applying the perturbation theory of semi-Fredholm operators, we introduce a suitable a priori estimate on the norm of the operator L to establish the periodicity of solutions in the case where the linear part is nondensely defined and satisfies the Hille-Yosida condition and without considering the exponential stability condition on the semigroup generated by the part of this operator on the closure of it’s domain. Moreover, in the special case where the linear part generates a strongly continuous semigroup and perturbed by a compact linear operator, we give some sufficient conditions to derive periodic solution from bounded ones. Finally, our theoretical results are illustrated by applications in both densely and nondensely cases.

Keywords

• hille-yosida condition
• semi-fredholm operators
• semigroup
• chow and hale’s fixed point theorem
• poincaré map
• periodicity
• 34k14
• 34k30
• 35b10
• 35b40