Stability of C₀-quasi semigroups in Banach spaces

We concern on the non-autonomous abstract Cauchy problems $\dot{x}=A(t)x(t)$ on Banach spaces X. If A(t) is the infinitesimal generator of a Co-quasi semigroup R(t, s) on X and x₀ 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.