Abstract
ABSTRACTIn this paper we focus on the abstract Cauchy problems of the time-dependent evolution equation ẋ = A(t)x(t). If the operator A(t) is time-independent, we can use the C0-semigroups theory to solve the abstract Cauchy problem. In this case A is a infinitesimal generator of the C0-semigroups. However, if A(t) is time-dependent, we can not apply directly the C0-semigroups theory to solve the problem. In this situation we can use the quasi semigroups theory as development of the two parameters semigroups. This semigroups is induced by bounded evolution operators U(t, s) that satisfy some assumptions. In this paper we determine the fundamental properties of the quasi semigroups included its generator related to the time-dependent evolution equation.
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