Stability of C0-quasi semigroups in Banach spaces

Abstract

We concern on the non-autonomous abstract Cauchy problems on Banach spaces X. If A(t) is the infinitesimal generator of a Co-quasi semigroup R(t, s) on X and x0 ϵ D, domain of A(t), then the solution of the equation has uniquely representation x(t) = R(0,t)x0. This representation shows that the stability of the quasi semigroup R(t, s) influences the stability of the solution. In this paper, we investigate the stabilities of C0-quasi semigroups following the existing theory of stabilities of C0-semigroups T(t) and bounded evolution operators U(t, s). We devote the uniform, exponential, and strong stability of C0-quasi semigroups in Banach spaces. The results are applicable for a large class of the time-dependent differential equations with unbounded coefficients in Banach spaces.