The reduction square root and perturbation for a class of strongly continuous operator families. (English) Zbl 1251.47042
Summary: We introduce the concept of \(\alpha \) times \(C\)-second resolvent families and present the relationship between \(\alpha \) times \(C\)-resolvent families and \(\alpha \) times \(C\)-second resolvent families. Moreover, the perturbation and square root for \(\alpha \) times \(C\)-resolvent families are considered in this paper which generalize the counterparts of \(C\)-cosine operator functions.
MSC:
| 47D60 | \(C\)-semigroups, regularized semigroups |
| 26A33 | Fractional derivatives and integrals |
| 45N05 | Abstract integral equations, integral equations in abstract spaces |
Keywords:
fractional calculus; \(\alpha \) times \(C\)-resolvent families; \(\alpha \) times \(C\)-second resolvent families; Laplace transforms; fractional evolution equationsReferences:
| [1] | Prüss, J.: Evolutionary Integral Equations and Applications, Birkhäuser, Basel-Boston-Berlin, 1993 · Zbl 0784.45006 |
| [2] | Fujita, Y.: Integrodifferential equations which interpolate the heat equation and the wave equation. Osaka J. Math., 27, 309–321 (1990) · Zbl 0790.45009 |
| [3] | Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons and Fractals, 7, 1461–1477 (1996) · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5 |
| [4] | Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983 · Zbl 0516.47023 |
| [5] | Arendt, W., Batty, C., Hieber M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel-Boston-Berlin, 2001 · Zbl 0978.34001 |
| [6] | Takenaka, T., Okazawa, N.: A Phillips-Miyadera type perturbation theorem for cosine functions of operators. Tohoku Math. J., 30, 107–115 (1978) · Zbl 0435.47045 · doi:10.2748/tmj/1178230101 |
| [7] | deLaubenfels, R.: Existence Families, Functional Calculus and Evolution Equations, Lect. Note Math., 1570, Spinger-Verlag, Berlin, 1994 · Zbl 0811.47034 |
| [8] | Takenaka, T., Miyadera, I.: Exponentially bounded C-semigroups and integrated semigroups. Tokyo J. Math., 12, 99–115 (1989) · Zbl 0702.47028 · doi:10.3836/tjm/1270133551 |
| [9] | Takenaka, T.: On the exponentially bounded C-cosine family. Gokujutsu Kenkyu (Acacemic Studies) Math., 37, 37–44 (1988) · Zbl 0657.60061 |
| [10] | Zhen, Q., Lei, Y. S.: Exponentially bounded C-cosine functions of operators. J. Sys. Sci. Math. Scis., 16, 242–252 (1996) · Zbl 0897.47039 |
| [11] | Li, M., Zheng, Q.: Stability of C-regularized semigroups. Acta Mathematica Sinica, English Series, 20(5), 821–828 (2004) · Zbl 1082.47037 · doi:10.1007/s10114-004-0349-8 |
| [12] | Bajlekova, E.: Fractional Evolution Equations in Banach Spaces (Ph. D. Thesis), Eindhoven University of Technology, 2001 · Zbl 0989.34002 |
| [13] | Li, M., Zheng, Q.: On spectral inclusions and approximations of {\(\alpha\)}-times resolvent families. Semigroup Forum, 69, 356–368 (2004) · Zbl 1096.47516 |
| [14] | Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Analysis, 69, 3692–3705 (2008) · Zbl 1166.34033 · doi:10.1016/j.na.2007.10.004 |
| [15] | Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. J. R. Astr. Soc., 13, 529–539 (1967) |
| [16] | Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo. Cimento, 1, 161–198 (1971) · doi:10.1007/BF02820620 |
| [17] | Erdélyi, A. (Ed.): Tables of Integral Transforms, Vol. 1, McGraw-Hill, Inc., New York-Toronto-London, 1954 · Zbl 0055.36401 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
