# zbMATH — the first resource for mathematics

On fractional resolvent operator functions. (English) Zbl 1185.47040
Summary: In this paper, we introduce three kinds of resolvent families defined by purely algebraic equations, which extend the classical semigroup property and cosine functional equation. We give their basic properties and analyticity criteria. Moreover, the relations between integrated resolvent families and resolvent families are discussed as well.

##### MSC:
 47D03 Groups and semigroups of linear operators
Full Text:
##### References:
 [1] Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. Ser. A, Theory Methods Appl. 69, 3692–3705 (2008) · Zbl 1166.34033 · doi:10.1016/j.na.2007.10.004 [2] Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96, Birkhäuser, Basel (2001) · Zbl 0978.34001 [3] Bajlekova, E.: Fractional evolution equations in Banach spaces. PhD Thesis, Eindhoven University of Technology (2001) · Zbl 0989.34002 [4] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. J. R. Astron. Soc. 13, 529–539 (1967) [5] Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento 1, 161–198 (1971) · doi:10.1007/BF02820620 [6] Cuesta, E.: Asymptotic behavior of the solutions of fractional integro-differential equations and some time discretizations. Discrete Contin. Dyn. Syst. (Suppl.) 277–285 (2007) · Zbl 1163.45306 [7] Da Prato, G., Iannelli, M.: Linear integrodifferential equations in Banach space. Rend. Sem. Mat. Univ. Padova 62, 207–219 (1980) · Zbl 0451.45014 [8] deLaubenfels, R.: Existence Families, Functional Calculi and Evolution Equations. Lecture Notes in Mathematics, vol. 1570, Springer, Berlin (1994) · Zbl 0811.47034 [9] Kostić, M.: On analytic integrated semigroups. Novi Sad J. Math. 35(1), 127–135 (2005) · Zbl 1268.47051 [10] 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 [11] Lizama, C.: On approximation and representation of k-regularized resolvent families. Integral Equ. Oper. Theory 41, 223–229 (2001) · Zbl 1011.45006 · doi:10.1007/BF01295306 [12] Lizama, C., Prado, H.: Rates of approximation and ergodic limits of regularized operator families. J. Approx. Theory 122, 42–61 (2003) · Zbl 1032.47024 · doi:10.1016/S0021-9045(03)00040-6 [13] Lizama, C., Sánchez, J.: On perturbation of k-regularized resolvent families. Taiwan. J. Math. 7, 217–227 (2003) · Zbl 1051.45009 [14] Li, M., Zheng, Q.: On spectral inclusions and approximations of $$\alpha$$-times resolvent families. Semigroup Forum 69, 356–368 (2004) · Zbl 1096.47516 [15] Li, M., Zheng, Q., Zhang, J.Z.: Regularized resolvent families. Taiwan. J. Math. 11, 117–133 (2007) · Zbl 1157.45006 [16] Oka, H.: Linear Volterra equations and integrated solution families. Semigroup Forum 53, 278–297 (1996) · Zbl 0862.45017 · doi:10.1007/BF02574144 [17] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) · Zbl 0516.47023 [18] Prüss, J.: Evolutionary Integral Equations and Applications. Birkhäuser, Basel (1993) · Zbl 0784.45006 [19] Shaw, S.Y., Chen, J.C.: Asymptotic behavior of (a,k)-regularized resolvent families at zero. Taiwan. J. Math. 10, 531–542 (2006) · Zbl 1106.45004 [20] Tanabe, H.: Equations of Evolution. Pitman, London (1979) · Zbl 0417.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.