Local attractivity in nonautonomous semilinear evolution equations

Abstract

We study the local attractivity of mild solutions of equations in the formwhere A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at in nity.

Periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations in Banach spaces

“…For each φ ∈ PC([−r, 0], X), let u(·, φ) be the unique solution of (1.1) and define the operators F [1], F [2], F [3], and F by…”

“…Next, we will investigate the basic properties of the operators F [1], F [2], F [3], and F , respectively.…”

“…Thirdly, we pay attention to the operator F [3]. For every φ [1], φ [2] ∈ PC([−r, 0], X) and s ∈ [−r, 0], we have…”

“…T + s = t n , then the properties of the evolution family U (·, ·) does not imply the compactness of F [3] directly. Thus, to prove the operator F [3] is compact, we have to go a new way. From that I n is compact, it follows that for each s ∈ [t n − T, 0],…”

“…Consequently, we know that F [3] is also a compact operator. Moreover, we see that the nonlinear operator F is continuous from PC([−r, 0], X) to PC([−r, 0], X) and compact.…”