A new theorem on exponential stability of periodic evolution families on Banach spaces

• Main Authors: Constantin Buse, Oprea Jitianu
• Format: Article
• Language: English
• Published: Texas State University 2003-02-01
• Series: Electronic Journal of Differential Equations
• Subjects: Almost periodic functions; exponential stability; periodic evolution families of operators; integral inequality
• Online Access: http://ejde.math.txstate.edu/Volumes/2003/14/abstr.html

We consider a mild solution $v_f(cdot, 0)$ of a well-posed inhomogeneous Cauchy problem $dot v(t)=A(t)v(t)+f(t)$, $v(0)=0$ on a complex Banach space $X$, where $A(cdot)$ is a 1-periodic operator-valued function. We prove that if $v_f(cdot, 0)$ belongs to $AP_0(mathbb{R}_+, X)$ for each $fin AP_0(mathbb{R}_+, X)$ then for each $xin X$ the solution of the well-posed Cauchy problem $dot u(t)=A(t)v(t)$, $u(0)=x$ is uniformly exponentially stable. The converse statement is also true. Details about the space $AP_0(mathbb{R}_+, X)$ are given in the section 1, below. Our approach is based on the spectral theory of evolution semigroups.