Spectral Theory for Compact and Self-Adjoint Fractional Resolvent Families
Kun-Yi Zhang
Department of Mathematics, Sichuan University, Chengdu, Sichuan, China
Contribution: Writing - original draft, Writing - review & editing
Search for more papers by this authorCorresponding Author
Miao Li
Department of Mathematics, Sichuan University, Chengdu, Sichuan, China
Correspondence:
Miao Li ([email protected])
Contribution: Writing - original draft, Writing - review & editing
Search for more papers by this authorKun-Yi Zhang
Department of Mathematics, Sichuan University, Chengdu, Sichuan, China
Contribution: Writing - original draft, Writing - review & editing
Search for more papers by this authorCorresponding Author
Miao Li
Department of Mathematics, Sichuan University, Chengdu, Sichuan, China
Correspondence:
Miao Li ([email protected])
Contribution: Writing - original draft, Writing - review & editing
Search for more papers by this authorABSTRACT
This paper concerns with self-adjoint fractional resolvent families. We show that a self-adjoint operator generating a fractional resolvent family if and only if is bounded above. And a spectral decomposition form for compact and self-adjoint fractional resolvent families is provided. We apply such decomposition to study the ergodic limits for the solutions of some inhomogeneous fractional differential equations.
Conflicts of Interest
The authors declare no conflicts of interest.
References
- 1E. Bajlekova, (2001). “ Fractional Evolution Equations in Banach Spaces,” Ph.D. Thesis, Eindhoven University of Technology.
- 2J. Prüss, Evolutionary Integral Equations and Applications (Modern Birkhäuser Classics, 1993).
10.1007/978-3-0348-8570-6 Google Scholar
- 3W. Arendt, C. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96 (Birkhäuser, 2001).
10.1007/978-3-0348-5075-9 Google Scholar
- 4A. Pazy, Semigroups of Linear Operators and Aplications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44 (Springer, 1983).
- 5C. Travis and G. Webb, “Compactness, Regularity, and Uniform Continuity Properties of Strongly Continuous Cosine Families,” Houston Journal of Mathematics 3 (1977): 555–567.
- 6A. V. Antoniouk, A. N. Kochubei, and S. I. Piskarev, “On the Compactness and the Uniform Continuity of a Resolvent Family for a Fractional Differential Equation,” Reports of National Academy of Sciences of Ukraine- N 6 (2014): 7–12.
- 7Z. Fan, “Characterization of Compactness for Resolvents and Its Applications,” Applied Mathematics and Computation 232 (2014): 60–67.
- 8C. Lizama, A. Pereira, and R. Ponce, “On the Compactness of Fractional Resolvent Operator Functions,” Semigroup Forum 93 (2016): 363–374.
- 9D. Applebaum, Semigroups of Linear Operators: With Applications to Analysis Probability and Physics (Cambridge University Press, 2019).
10.1017/9781108672641 Google Scholar
- 10B. D. Rouhani, G. R. Goldstein, and J. A. Goldstein, “Ergodic Limits for Inhomogeneous Evolution Equations,” Electronic Journal of Qualitative Theory 77 (2020): 1–7.
- 11L. Chen, Z. Fan, and F. Wang, “Ergodicity and Stability for -Regularized Resolvent Operator Families,” Annals of Functional Analysis 12, no. 6 (2021): 14.
- 12C. Y. Li and M. Li, “Asymptotic Stability of Fractional Resolvent Families,” Journal of Evolution Equations 21 (2021): 2523–2545.
- 13C. Lizama, “A Mean Ergodic Theorem for Resolvent Operators,” Semigroup Forum 47 (1993): 227–230.
- 14Y. C. Li and S. Y. Shaw, “Mean Ergodicity and Mean Stability of Regularized Solution Families,” Mediterranean Journal of Mathematics 1 (2004): 175–193.
10.1007/s00009-004-0010-x Google Scholar
- 15R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications (Springer Monographs in Mathematics, Springer, 2014).
10.1007/978-3-662-43930-2 Google Scholar
- 16C. Y. Li, “Fractional Resolvent Family Generated by Normal Operators,” AIMS Mathematics 8, no. 10 (2023): 23815–23832.
- 17J. W. Hanneken, D. M. Vaught, and B. N. N. Achar, Enumeration of the Real Zeros of the Mittag-Leffler function , eds. J. Sabatier, O. P. Agrawal, and J. A. T. Machado (Advances in Fractional Calculus, Springer, 2007).
- 18M. Reed and B. Simon, Methods of Modern Mathematical Physics, Functional Analysis (revised and enlarged edition), vol. 1 (Academic Press, 1980).
- 19C. Lizama, “ Abstract Linear Fractional Evolution Equations,” in Fractional Differential Equations, vol. 2, eds. A. Kochubei and Y. Luchko (De Gruyter, 2019), 465–498.
10.1515/9783110571660-021 Google Scholar
- 20M. Li and Q. Zheng, “On Spectral Inclusions and Approximations of -Times Resolvent Families,” Semigroup Forum 69 (2004): 356–368.
- 21K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, GTM, vol. 194 (Springer-Verlag, 2000).
- 22M. E. Taylor, Partial Differential Equations II. Applied Mathematical Sciences, vol. 116 (Springer, 2011).
10.1007/978-1-4419-7052-7 Google Scholar
- 23P. D. Lax, Functional Analysis (Wiley-Interscience, 2014).
- 24F. Li and M. Li, “On Maximal Regularity and Semivariation of
-Times Resolvent Families,” Advanced Pure Mathematics 03 (2013): 680–684.
10.4236/apm.2013.38091 Google Scholar
