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**1 - 3**of**3**### Research Article Fractional Evolution Equations Governed by Coercive Differential Operators

"... This paper is concerned with evolution equations of fractional order Dαut Aut;u0 u0, u ′0 0, where A is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than π and 1 < α < 2. We show that such equations are well posed in the sense t ..."

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This paper is concerned with evolution equations of fractional order Dαut Aut;u0 u0, u ′0 0, where A is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than π and 1 < α < 2. We show that such equations are well posed in the sense that there always exists an α-times resolvent family for the operator A. Copyright q 2009 Fu-Bo Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.

### SYSTEMS OF ABSTRACT TIME-FRACTIONAL EQUATIONS

"... Abstract. We analyze systems of abstract time-fractional equations in cer-tain classes of sequentially complete locally convex spaces. We also consider arbitrary matrices of operators as generators of fractional regularized resol-vent families, improving in such a way the results known for semigroup ..."

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Abstract. We analyze systems of abstract time-fractional equations in cer-tain classes of sequentially complete locally convex spaces. We also consider arbitrary matrices of operators as generators of fractional regularized resol-vent families, improving in such a way the results known for semigroups of operators. 1. Introduction and

### REGULARIZED FUNCTIONAL CALCULI, SEMIGROUPS, AND COSINE FUNCTIONS FOR PSEUDODIFFERENTIAL OPERATORS

"... Abstract. Let iAj(1 ≤ j ≤ n) be generators of commuting bounded strongly continuous groups, A ≡ (A1, A2,..., An). We show that, when f has sufficiently many polynomially bounded derivatives, then there exist k, r> 0 such that f(A) has a (1+ |A|2)−r-regularized BCk(f(Rn)) functional calcu-lus. Thi ..."

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Abstract. Let iAj(1 ≤ j ≤ n) be generators of commuting bounded strongly continuous groups, A ≡ (A1, A2,..., An). We show that, when f has sufficiently many polynomially bounded derivatives, then there exist k, r> 0 such that f(A) has a (1+ |A|2)−r-regularized BCk(f(Rn)) functional calcu-lus. This immediately produces regularized semigroups and cosine functions with an explicit representation; in particular, when f(Rn) ⊆ R, then, for appropriate k, r, t → (1 − it)−ke−itf(A)(1 + |A|2)−r is a Fourier-Stieltjes transform, and when f(Rn) ⊆ [0,∞), then t → (1+t)−ke−tf(A)(1+|A|2)−r is a Laplace-Stieltjes transform. With A ≡ i(D1,..., Dn), f(A) is a pseudo-differential operator on Lp(Rn)(1 ≤ p <∞) or BUC(Rn). In finite dimensions, the Jordan canonical form for matrices guarantees that, although a linear operator may not be diagonalizable, which is equiv-alent to having a BC(C) functional calculus, it will be generalized scalar, that is, have a BCk(C) functional calculus, for some k; specifically, k may