Publication Date

2018-08-10

Availability

Open access

Embargo Period

2018-08-10

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2018-08-03

First Committee Member

Shigui Ruan

Second Committee Member

Robert Stephen Cantrell

Third Committee Member

Chris Cosner

Fourth Committee Member

Don DeAngelis

Abstract

We study the existence of periodic solutions to the abstract semi linear evolution equation du/dt=A(t)u(t)+F(t,u(t)), t \geq 0 in a Banach space X, where A(t) is a T-periodic linear operator on X (not necessarily densely defined) satisfying the hyperbolic conditions, and F is continuous and T-periodic in t. The idea is to combine Poincare map technique with fixed point theorems to derive various conditions on the operator A(t) and the map F(t, u) to ensure that the abstract evolution equation has periodic solutions. Three cases are considered: (i) If A(t)=A is time-independent and is a Hille-Yoshida operator, conditions on F are given to guarantee the existence of mild periodic solutions; (ii) If A(t) is time-dependent and satisfies the hyperbolic condition, sufficient conditions on A(t) and F are presented to ensure the existence of mild periodic solutions; (iii) If A(t)=A is time-independent, is a Hille-Yoshida operator and generates a compact semigroup, the existence of mild periodic solutions is also discussed. As applications, the main results are applied to establish the existence of periodic solutions in a delayed periodic red-blood cell model; age-structured models with periodic harvesting, diffusive logistic equations with periodic coefficients, and periodic diffusive Nicholson' blowflies equation with delay.

Keywords

Non-densely defined Cauchy problem; Periodic solutions

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