Laplace Transform Inversions And Perturbations Of The Operator Families

In this paper, We study the inversions of the Laplace transform for the C-regularized resolventfamiliy and k-regularized resolvent families in UMD spaces and Banach spaces.The important dif-ference of the C-regularized resolvent families and k-regularized resolvent families in this spacesand them defined in Banach spaces is that the former holds for all x∈X while the later holds onlyfor x∈D(A). In addition , the paper also studies the additive and multiplicative perturbation ofk-convoluted solutions operator .Second,the multiplicative perturbations of k-convoluted C-cosine functions and k-convolutedC-semigroups are studied . We will prove that if (C?1)AC,θ) generates an exponentially boundedconvoluted C-cosine functions or convoluted C-semigroups,then(BA,θ), (AB,θ)or(A(I +B),θ),((I + B)A,θ)also generates an exponential bounded k-convoluted C-cosine functions andk-convoluted C-semigroups.Third ,We study the multiplicative perturbations of k-convoluted solution operator families .Let k∈C(R+)and B be a linear bounded operator families. We prove that if(A,μ)generates an ex-ponential bounded k-convoluted solution operator families under the some condition,then (BA,μ), (AB,μ),or (A(I + B),μ) ,((I + B)A,μ)also generate an exponential bounded k-convoluted so-lution operator families. In addition , the additive perturbation of k-convoluted solution operatorfamilies is proved, i.e. if(A,μ)generates an exponential bounded k-convoluted solution opera-tor families under the some condition , then (A + B,μ) also generate an exponential boundedk-convoluted solution operator families.