S-asymptotically ω-periodic Functions and Sequences and Applications to Evolution Equations

Abstract

The contribution of this dissertation is threefold: First, we investigate further properties of S-asymptotically $\omega$-periodic functions, a concept that was introduced in 2008. Second, we introduce the new concept of S-asymptotically $\omega$-periodic sequences. Finally we use the results to study S-asymptotically $\omega$-periodic solutions to some evolution equations.

The first equation is the general abstract semilinear integrodifferential equation $u'(t)=Au(t)+\int_{0}^{t}B(t-\xi)u(\xi)d\xi+f(t,u(t)),\;\;t\geq 0$ with nonlocal conditions $u(0)=u_0 + g(u)$ in a general Banach space $(X, \| \cdot \|)$. The second is the semilinear fractional differential equation $D_t^\alpha u(t)=Au(t)+D_t^{\alpha-1} f(t,u(t)),\;\; t\geq 0$ with non-local conditions $u(0)=u_0 + g(u)$ where $D_t^\alpha(\cdot)$ ($1< \alpha < 2,$) is the Riemann-Liouville derivative, $A: D(A) \subset X \to X$ is a linear densely defined operator of sectorial type on a complex Banach space $X$, $f:\R^+\times X\to X$ is S-asymptotically $\omega$-periodic with respect to the first variable. The results obtained are new even in the context of asymptotically $\omega$-periodic functions. An example to fractional relaxation-oscillation equations is given.

Also, in this work, we prove some properties of S-asymptotically $\omega$-periodic sequences, including discrete convolutions, some infinite discrete sums and a composition result. These results are then utilized to find S-asymptotically $\omega$-periodic solutions of first order difference equations, linear and then semi-linear. For the linear case, an application to an equation of the distribution of heat through a thin bar is presented.