Inverses of generators of integrated fractional resolvent operator functions
-
Miao Li
,
Javier Pastor
Abstract
This paper is devoted to the inverse generator problem in the setting of generators of integrated resolvent operator functions. It is shown that if the operator A is the generator of a tempered β-times integrated α-resolvent operator function ((α, β)-ROF) and it is injective, then the inverse operator A−1 is the generator of a tempered (α, γ)-ROF for all γ > β + 1/2, by means of an explicit representation of the integrated resolvent operator function based in Bessel functions of first kind. Analytic resolvent operator functions are also considered, showing that A−1 is in addition the generator of a tempered (δ, 0)-ROF for all δ < α. Moreover, the optimal decay rate of (α, β)-ROFs as t → ∞ is given. These result are applied to fractional Cauchy problem unsolved in the fractional derivative.
Acknowledgements
This work was done while the second author was visiting the Department of Mathematics of Sichuan University, supported by Universitat de València grant UV-INV_EPDI17-545952. He would like to thank the functional analysis group, and very specially Professor Miao Li, for their warm hospitality and support. The first author was supported by the NSFC-RFBR Programme of China (No. 11611530677). The third author was supported by the Russian Foundation for Basic Research, projects 15-01-00026_a, 16-01-00039_a and 17-51-53008.
References
[1] L. Abadías, P.J. Miana, A subordination principle on Wright functions and regularized resolvent families. J. Funct. Spaces2015 (2015), Art. ID 158145, 9 pp.10.1155/2015/158145Suche in Google Scholar
[2] W. Arendt, Resolvent positive operators. Proc. London Math. Soc. 54 (1987), 321–349.10.1112/plms/s3-54.2.321Suche in Google Scholar
[3] W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Math. 96, Birkhäuser Verlag, Basel (2001).10.1007/978-3-0348-5075-9Suche in Google Scholar
[4] W. Arendt, H. Kellermann, Integrated solutions of Volterra integrodifferential equations and applications. In: Volterra Integro-differential Equations in Banach Spaces and Applications, Proc. Conf., Trento/Italy 1987, Pitman Res. Notes Math. Ser. 190 (1989), 21–51.Suche in Google Scholar
[5] W. Arendt, F. Neubrander, U. Schlotterbeck, Interpolation of semigroups and integrated semigroups. Semigroup Forum45 (1992), 26–37.10.1007/BF03025746Suche in Google Scholar
[6] E.G. Bajlekova, Fractional Evolution Equations in Banach Spaces, PhD Thesis, Eindhoven University of Technology, Eindhoven (2001).Suche in Google Scholar
[7] G.I. Barenblatt, Iu.P. Zheltov, I.N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata). J. Appl. Math. Mech. 24, No 5 (1960), 1286–1303.10.1016/0021-8928(60)90107-6Suche in Google Scholar
[8] Y. Chang, A. Pereira, R. Ponce, Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators. Fract. Calc. Appl. Anal. 20, No 4 (2017), 963–987; 10.1515/fca-2017-0050; https://www.degruyter.com/view/j/fca.2017.20.issue-4/issue-files/fca.2017.20.issue-4.xml.Suche in Google Scholar
[9] C. Chen, M. Li, On fractional resolvent operator functions. Semigroup Forum80 (2010), 121–142.10.1007/s00233-009-9184-7Suche in Google Scholar
[10] R. deLaubenfels, Inverses of generators. Proc. Amer. Math. Soc. 104 (1988), 443–448.10.1090/S0002-9939-1988-0962810-6Suche in Google Scholar
[11] R. deLaubenfels, Inverses of generators of nonanalytic semigroups. Studia Math. 191 (2009), 11–38.10.4064/sm191-1-2Suche in Google Scholar
[12] S. Fackler, A short counterexample to the inverse generator problem on non-Hilbertian reflexive Lp-spaces. Arch. Math. 106, No 4 (2016), 383–389.10.1007/s00013-015-0863-1Suche in Google Scholar
[13] V.E. Fedorov, R.R. Nazhimov, D.M. Gordievskikh, Initial value problem for a class of fractional order inhomogeneous equations in Banach spaces. AIP Conf. Proc. 1759, Art. # 020008 (2016); 10.1063/1.4959622.Suche in Google Scholar
[14] A. Gomilko, On the inverse of the generator of a bounded C0-semigroup. Funktsional. Anal. Prilozhen. 38, No 4 (2004), 6–12 (in Russian); English transl.: Funct. Anal. Appl. 38 (2004), 243–248.Suche in Google Scholar
[15] A. Gomilko, H. Zwart, Y. Tomilov, On the inverse of the generator of C0-semigroup. Mat. Sb. 198, No 8 (2007), 35–50 (in Russian); English transl.: Sb. Math. 198 (2007), 1095–1110.Suche in Google Scholar
[16] M. Haase, The Functional Calculus for Sectorial Operators. Birkhäuser Verlag, Basel (2006).10.1007/3-7643-7698-8Suche in Google Scholar
[17] M. Hieber, Integrated semigroups and differential operators in Lp spaces. Math. Ann. 291 (1991), 1–16.10.1007/BF01445187Suche in Google Scholar
[18] M. Hieber, Integrated semigroups and the Cauchy problem for systems in Lp spaces, J. Math. Anal. Appl. 162 (1991), 300–308.10.1016/0022-247X(91)90196-7Suche in Google Scholar
[19] H. Kellerman, M. Hieber, Integrated semigroups. J. Funct. Anal. 84 (1989), 160–180.10.1016/0022-1236(89)90116-XSuche in Google Scholar
[20] V. Keyantuo, C. Lizama, M. Warma, Existence, regularity and representation of solutions of time fractional diffusion equations. Adv. Differential Equations21, No 9/10 (2016), 837–886.10.57262/ade/1465912585Suche in Google Scholar
[21] A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Suche in Google Scholar
[22] A. Kochubei, Asymptotic properties of solutions of the fractional diffusion-wave equation. Fract. Calc. Appl. Anal. 17, No 3 (2014), 881–896; 10.2478/s13540-014-0203-3; https://www.degruyter.com/view/j/fca.2014.17.issue-3/issue-files/fca.2014.17.issue-3.xml.Suche in Google Scholar
[23] H. Komatsu, Fractional powers of operators. Pacific J. Math. 19 (1966), 285–346.10.2140/pjm.1966.19.285Suche in Google Scholar
[24] M. Li, C. Chen, F. Li, On fractional powers of generators of fractional resolvent families. J. Funct. Anal. 259 (2010), 2702–2726.10.1016/j.jfa.2010.07.007Suche in Google Scholar
[25] Z. Li, Y. Luchko, M. Yamamoto, Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations. Fract. Calc. Appl. Anal. 17, No 4 (2014), 1114–1136; 10.2478/s13540-014-0217-x; https://www.degruyter.com/view/j/fca.2014.17.issue-4/issue-files/fca.2014.17.issue-4.xml.Suche in Google Scholar
[26] Y.N. Li, H.R. Sun, Integrated fractional resolvent operator function and fractional abstract Cauchy problem. Abstr. Appl. Anal. 2014 (2014), Art. ID 430418, 9 pp.10.1155/2014/430418Suche in Google Scholar
[27] Y.C. Li, S.Y. Shaw, On generators of integrated C-semigroups and C-cosine functions. Semigroup Forum47 (1993), 29–35.10.1007/BF02573738Suche in Google Scholar
[28] M. Li, Q. Zheng, On spectral inclusions and approximations of α-times resolvent families. Semigroup Forum69 (2004), 356–368.10.1007/s00233-004-0128-ySuche in Google Scholar
[29] R. Liu, M. Li, J. Pastor, S.I. Piskarev, On the approximation of fractional resolution families. Translation of: Differ. Uravn. 50, No 7 (2014), 937–946; Differ. Equ. 50, No 7 (2014), 927–937.Suche in Google Scholar
[30] Y. Luchko, Asymptotics of zeros of the Wright function. J. for Anal. and its Appl. 19, No 1 (2000), 1–12.10.4171/ZAA/970Suche in Google Scholar
[31] O.I. Marichev, S.G. Samko, A.A. Kilbas, Fractional Integrals and Derivates. Theory and Applications. Gordon and Breach Science Publishers, Yverdon (1993).Suche in Google Scholar
[32] C. Martínez, M. Sanz, The Theory of Fractional Powers of Operators. North-Holland Math. Stud. 187, North-Holland, Amsterdam (2001).Suche in Google Scholar
[33] O. El-Mennaoui, Traces de Semi-groupes Holomorphes Singuliers á L’origine et Comportement Asymptotique. Thése, Besançon (1992).Suche in Google Scholar
[34] O. El-Mennaoni, V. Keyantuo, Trace theorems for holomorphic semigroups and the second order Cauchy problems. Proc. Amer. Math. Soc. 124, No 5 (1996), 1445–1458.10.1090/S0002-9939-96-03133-4Suche in Google Scholar
[35] J.M.A.M. van Neerven, B. Straub, On the existence and growth of mild solutions of the abstract Cauchy problem for operators with polynomially bounded resolvent. Houston J. Math. 24 (1998), 137–171.Suche in Google Scholar
[36] S. Piskarev, H. Zwart, Crank-Nicolson scheme for abstract linear systems. Numer. Funct. Anal. Optim. 28 (2007), 717–736.10.1080/01630560701380981Suche in Google Scholar
[37] J. Prüss, Evolutionary Integral Equations and Applications. Birkhäuser Verlag, Basel (1993).10.1007/978-3-0348-8570-6Suche in Google Scholar
[38] G. N. Watson, Theory of Bessel Functions. 2nd Ed., Cambridge Univ. Press, Cambridge (1962).Suche in Google Scholar
[39] G.B. Whitham, Linear and Nonlinear Waves. John Wiley & Sons, New York (1999).10.1002/9781118032954Suche in Google Scholar
[40] T.J. Xiao, J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations. Springer-Verlag, Berlin (1998).10.1007/978-3-540-49479-9Suche in Google Scholar
[41] H. Zwart, Is A−1 an infinitesimal generator? In: Perspectives in Operator Theory, Banach Center Publ. 75, Inst. Math., Polish Acad. Sci. (2007), 303–313.10.4064/bc75-0-18Suche in Google Scholar
[42] H. Zwart, Growth estimates for (A−1t) on a Hilbert space. Semigroup Forum74 (2007), 487–494.10.1007/s00233-006-0679-1Suche in Google Scholar
© 2018 Diogenes Co., Sofia
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books
- Research Paper
- Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions
- Finite-time attractivity for semilinear tempered fractional wave equations
- Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas
- Extrapolating for attaining high precision solutions for fractional partial differential equations
- Time optimal controls for fractional differential systems with Riemann-Liouville derivatives
- Inverses of generators of integrated fractional resolvent operator functions
- A variational approach for boundary value problems for impulsive fractional differential equations
- Infinitely many solutions to boundary value problem for fractional differential equations
- A semi-analytic method for fractional-order ordinary differential equations: Testing results
- Blow-up and global existence of solutions for a time fractional diffusion equation
- A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity
- Short Paper
- Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books
- Research Paper
- Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions
- Finite-time attractivity for semilinear tempered fractional wave equations
- Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas
- Extrapolating for attaining high precision solutions for fractional partial differential equations
- Time optimal controls for fractional differential systems with Riemann-Liouville derivatives
- Inverses of generators of integrated fractional resolvent operator functions
- A variational approach for boundary value problems for impulsive fractional differential equations
- Infinitely many solutions to boundary value problem for fractional differential equations
- A semi-analytic method for fractional-order ordinary differential equations: Testing results
- Blow-up and global existence of solutions for a time fractional diffusion equation
- A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity
- Short Paper
- Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions
