Periodic solutions for partial neutral non densely differential equations

Abstract

This work investigates the existence of periodic solutions for the following partial neutral nonautonomous functional differential equation (1) $\begin{array}{rlr}& \frac{\mathrm{d}}{\mathrm{d}t}\mathcal{D}{u}_t=(A+B(t\left)\right)\mathcal{D}{u}_t+F(t,u_t),& t\ge 0,\\ & u_0=\mathrm{\Phi }\in \mathcal{C}=C\left(\right[-r,0],X),\end{array}$(1) where the linear operator A is not necessarily densely defined and satisfies the Hille–Yosida condition, $B\left(t\right)$, $t\in {\mathbb{R}}_{+}$, is a family of bounded linear operators from $\overline{D\left(A\right)}$ into X and the nonlinear delayed part F satisfies some locally Lipschitz conditions. More precisely, we study the Massera problem for the existence of a τ-periodic solution of (1). Then, we prove for $\tau =1$, in the dichotomic case, the existence, uniqueness and conditional stability of the periodic solution. Finally, our results are illustrated by an application.

KEYWORDS:

• Nonautonomous partial neutral functional differential equation
• Hille–Yosida condition
• evolution family
• periodic solution
• Massera problem
• dichotomy

MATHEMATICS SUBJECT CLASSIFICATIONS (2010):

• 35R10
• 35B10
• 47D06
• 34G20

Disclosure statement

No potential conflict of interest was reported by the authors.