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On fractional powers of generators of fractional resolvent families. (English) Zbl 1203.47021
The authors show that if $-A$ generates a bounded $\alpha$-times resolvent family for some $\alpha \in \left(0,2\right]$, then $-{A}^{\beta }$ generates an analytic $\gamma$-times resolvent family for $\beta \in \left(0,\frac{2\pi -\pi \gamma }{2\pi -\pi \alpha }\right)$ and $\gamma \in \left(0,2\right)$. They also derive a generalized subordination principle. In particular, if $-A$ generates a bounded $\alpha$-times resolvent family for some $\alpha \in \left(1,2\right]$, then $-{A}^{1/\alpha }$ generates an analytic ${C}_{0}$-semigroup. Such relations are applied to study the solutions of Cauchy problems of fractional order and first order. These results allow to the authors to give a unitary perspective to the variegate phenomena related to fractional operators and to establish a connection between solutions of fractional Cauchy problems and Cauchy problems of first order.
##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 35K90 Abstract parabolic equations 47D60 $C$-semigroups, regularized semigroups
##### References:
 [1] Allouba, H.; Zheng, W.: Brownian-time processes: the PDE connection and the half-derivative generator, Ann. probab. 29, No. 2, 1780-1795 (2001) · Zbl 1018.60066 · doi:10.1214/aop/1015345772 [2] Arendt, W.; Batty, C.; Hieber, M.; Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems, Monogr. math. 96 (2001) · Zbl 0978.34001 [3] E.G. Bajlekova, Fractional evolution equations in Banach spaces, PhD thesis, Department of Mathematics, Eindhoven University of Technology, 2001. · Zbl 0989.34002 [4] Balakrishnan, A. V.: Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10, 419-437 (1960) · Zbl 0103.33502 [5] Baeumer, B.; Meerschaert, M. M.: Stochastic solutions for fractional Cauchy problems, Fract. calc. Appl. anal. 4, 481-500 (2001) · Zbl 1057.35102 [6] Baeumer, B.; Meerschaert, M. M.; Nane, E.: Brownian subordinators and fractional Cauchy problems, Trans. amer. Math. soc. 361, 3915-3930 (2009) · Zbl 1186.60079 · doi:10.1090/S0002-9947-09-04678-9 [7] Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M.: The fractional-order governing equation of Lévy motion, Water resources res. 36, No. 6, 1413-1424 (2000) [8] Chen, C.; Li, M.: On fractional resolvent operator functions, Semigroup forum 80, 121-142 (2010) · Zbl 1185.47040 · doi:10.1007/s00233-009-9184-7 [9] Cioranescu, I.; Keyantuo, V.: On operator cosine functions in UMD spaces, Semigroup forum 63, 429-440 (2001) · Zbl 1191.47056 · doi:10.1007/s002330010086 [10] Da Prato, G.; Iannelli, M.: Linear integro-differential equations in Banach space, Rend. sem. Mat. univ. Padova 62, 207-219 (1980) · Zbl 0451.45014 · doi:numdam:RSMUP_1980__62__207_0 [11] Deblassie, R. D.: Iterated Brownian motion in an open set, Ann. appl. Probab. 14, No. 3, 1529-1558 (2004) · Zbl 1051.60082 · doi:10.1214/105051604000000404 [12] Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order, Fractals and fractional calculus in continuum mechanics, 223-276 (1997) [13] Gorenflo, R.; Luchko, Y.; Mainardi, F.: Analytical properties and applications of the wright function, Fract. calc. Appl. anal. 2, No. 4, 383-414 (1999) · Zbl 1027.33006 [14] Haase, M.: The functional calculus for sectorial operators, Oper. theory adv. Appl. 169 (2006) [15] Huang, Y. Z.; Zheng, Q.: Regularization for ill-posed Cauchy problems associated with generators of analytic semigroups, J. differential equations 203, 38-54 (2004) · Zbl 1068.34055 · doi:10.1016/j.jde.2004.03.035 [16] Karczewska, A.; Lizama, C.: Stochastic Volterra equations driven by cylindrical Wiener process, J. evol. Equ. 7, 373-386 (2007) · Zbl 1120.60062 · doi:10.1007/s00028-007-0302-2 [17] Kato, T.: Note on fractional powers of linear operators, Proc. Japan acad. 36, 94-96 (1960) · Zbl 0097.31802 · doi:10.3792/pja/1195524082 [18] Keyantuo, V.: On analytic semigroups and cosine operator functions in Banach spaces, Studia math. 129, No. 2, 137-156 (1998) · Zbl 0910.47035 [19] Keyantuo, V.; Lizama, C.: On a connection between powers of operators and fractional Cauchy problems [20] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland math. Stud. 204 (2006) [21] Komatsu, H.: Fractional powers of operators, Pacific J. Math. 19, 285-346 (1966) · Zbl 0154.16104 [22] Li, F. B.; Li, M.: On maximal regularity and semivariation of $\alpha$-times resolvent families [23] Li, M.; Zheng, Q.: On spectral inclusions and approximations of $\alpha$-times resolvent families, Semigroup forum 69, 356-368 (2004) · Zbl 1096.47516 · doi:10.1007/s00233-004-0128-y [24] Lizama, C.: Regularized solutions for abstract Volterra equations, J. math. Anal. appl. 243, 278-292 (2000) · Zbl 0952.45005 · doi:10.1006/jmaa.1999.6668 [25] Lizama, C.: On approximation and representation of k-regularized resolvent families, Integral equations operator theory 41, 223-229 (2001) · Zbl 1011.45006 · doi:10.1007/BF01295306 [26] Martínez, C.; Sanz, M.: The theory of fractional powers of operators, North-holland math. Stud. 187 (2001) [27] Meerschaert, M. M.; Scheffler, H. P.: Limit distributions for sum of independent random vectors: heavy tails in theory and practice, Wiley ser. Probab. stat. (2001) [28] Prüss, J.: Evolutionary integral equations and applications, (1993) [29] Podlubny, I.: Fractional differential equations, Math. sci. Eng. 198 (1999) · Zbl 0924.34008 [30] Scalas, E.; Gorenflo, R.; Mainardi, F.: Fractional calculus and continuous-time finance, Phys. A 284, 376-384 (2000) [31] Schneider, W. R.; Wyss, W.: Fractional diffusion and wave equations, J. math. Phys. 30, 134-144 (1989) · Zbl 0692.45004 · doi:10.1063/1.528578 [32] Yosida, K.: Fractional powers of infinitesimal generators and the analyticity of the semigroups generated by them, Proc. Japan acad. 36, 86-89 (1960) · Zbl 0097.31801 · doi:10.3792/pja/1195524080 [33] Zaslavsky, G. M.: Fractional kinetic equations for Hamiltonian chaos, chaotic advection, tracer dynamics and turbulent dispersion, Phys. D 76, 110-122 (1994) · Zbl 1194.37163 · doi:10.1016/0167-2789(94)90254-2