## Abstract and Applied Analysis

### Perturbation Theory for Abstract Volterra Equations

Marko Kostić

#### Abstract

We consider additive perturbation theorems for subgenerators of (a, k)-regularized C-resolvent families. A major part of our research is devoted to the study of perturbation properties of abstract time-fractional equations, primarily from their importance in modeling of various physical phenomena. We illustrate the results with several examples.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 307684, 26 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511790

Digital Object Identifier
doi:10.1155/2013/307684

Mathematical Reviews number (MathSciNet)
MR3035374

Zentralblatt MATH identifier
1273.47073

#### Citation

Kostić, Marko. Perturbation Theory for Abstract Volterra Equations. Abstr. Appl. Anal. 2013 (2013), Article ID 307684, 26 pages. doi:10.1155/2013/307684. https://projecteuclid.org/euclid.aaa/1393511790

#### References

• M. Kostić, Generalized Semigroups and Cosine Functions, Mathematical Institute Belgrade, 2011.
• W. Arendt and H. Kellermann, “Integrated solutions of Volterra integrodifferential equations and applications,” in Volterra Integrodifferential Equations in Banach Spaces and Applications, vol. 190 of Pitman Research Notes in Mathematics Series, pp. 21–51, Longman Science and Technology, Harlow, UK, 1989.
• C.-C. Kuo, “Perturbation theorems for local integrated semigroups,” Studia Mathematica, vol. 197, no. 1, pp. 13–26, 2010.
• A. Rhandi, “Positive perturbations of linear Volterra equations and sine functions of operators,” Journal of Integral Equations and Applications, vol. 4, no. 3, pp. 409–420, 1992.
• J. Zhang and Q. Zheng, “On $\alpha$-times integrated cosine functions,” Mathematica Japonica, vol. 50, no. 3, pp. 401–408, 1999.
• Q. Zheng, “Integrated cosine functions,” International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 3, pp. 575–580, 1996.
• C.-C. Kuo, “On perturbation of local integrated cosine functions,” Taiwanese Journal of Mathematics, vol. 16, no. 5, pp. 1613–1628, 2012.
• S. W. Wang, M. Y. Wang, and Y. Shen, “Perturbation theorems for local integrated semigroups and their applications,” Studia Mathematica, vol. 170, no. 2, pp. 121–146, 2005.
• C. Lizama and J. Sánchez, “On perturbation of $K$-regularized resolvent families,” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 217–227, 2003.
• J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, Switzerland, 1993.
• J.-C. Chang and S. Y. Shaw, “Perturbation theory of abstract Cauchy problems and Volterra equations,” in Proceedings of the 2nd World Congress of Nonlinear Analysts, vol. 30 of Nonlinear Analysis, part 6, pp. 3521–3528, Athens, Ga, USA, 1997.
• H. Oka, “Linear Volterra equations and integrated solution families,” Semigroup Forum, vol. 53, no. 3, pp. 278–297, 1996.
• S.-Y. Shaw, “Cosine operator functions and Cauchy problems,” Conferenze del Seminario di Matematica dell'Università di Bari, no. 287, pp. 1–75, 2002.
• M. Hieber, Integrated semigroups and dierential operators on Lp spaces [Ph.D. thesis], Tübingen, Germany, 1989.
• V. Keyantuo and M. Warma, “The wave equation in ${L}^{p}$-spaces,” Semigroup Forum, vol. 71, no. 1, pp. 73–92, 2005.
• H. Komatsu, “Operational calculus and semi-groups of operators,” in Functional Analysis and Related Topics (Kyoto), vol. 1540, pp. 213–234, Springer, Berlin, Germany, 1991.
• M. Kostić and S. Pilipović, “Convoluted $C$-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1224–1242, 2008.
• I. V. Melnikova and A. Filinkov, Abstract Cauchy Problems: Three Approaches, vol. 120 of Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2001.
• C. Kaiser and L. Weis, “Perturbation theorems for $\alpha$-times integrated semigroups,” Archiv der Mathematik, vol. 81, no. 2, pp. 215–228, 2003.
• A. Karczewska and C. Lizama, “Solutions to stochastic fractional oscillation equations,” Applied Mathematics Letters, vol. 23, no. 11, pp. 1361–1366, 2010.
• C. Lizama and H. Prado, “Fractional relaxation equations on Banach spaces,” Applied Mathematics Letters, vol. 23, no. 2, pp. 137–142, 2010.
• W. von Wahl, “Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen,” Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische, vol. 11, pp. 231–258, 1972.
• F.-B. Li, M. Li, and Q. Zheng, “Fractional evolution equations governed by coercive differential operators,” Abstract and Applied Analysis, vol. 2009, Article ID 438690, 14 pages, 2009.
• M. Kostić, “Abstract differential operators generating fractional regularized resolvent families,” submitted to Acta Mathematica Sinica.
• W. Arendt and C. J. K. Batty, “Rank-1 perturbations of cosine functions and semigroups,” Journal of Functional Analysis, vol. 238, no. 1, pp. 340–352, 2006.
• W. Desch, G. Schappacher, and W. Schappacher, “Relatively bounded rank one perturbations of non-analytic semigroups can generate large point spectrum,” Semigroup Forum, vol. 75, no. 2, pp. 470–476, 2007.
• T.-J. Xiao, J. Liang, and J. van Casteren, “Time dependent Desch-Schappacher type perturbations of Volterra integral equations,” Integral Equations and Operator Theory, vol. 44, no. 4, pp. 494–506, 2002.
• E. Bazhlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], Eindhoven University of Technology, Eindhoven, the Netherland, 2001.
• Y. Lin, “Time-dependent perturbation theory for abstract evolution equations of second order,” Studia Mathematica, vol. 130, no. 3, pp. 263–274, 1998.
• H. Serizawa and M. Watanabe, “Time-dependent perturbation for cosine families in Banach spaces,” Houston Journal of Mathematics, vol. 12, no. 4, pp. 579–586, 1986.
• M. Kostić, “Time-dependent perturbations of abstract Volterra equations,” Bulletin: Classe des Sciences Mathmatiques et Natturalles–-Sciences Natturalles, vol. 143, no. 36, pp. 89–104, 2011.
• M. Kostić, “$(a,k)$-regularized $C$-resolvent families: regularity and local properties,” Abstract and Applied Analysis, vol. 2009, Article ID 858242, 27 pages, 2009.
• M. Kostić, “Abstract Volterra equations in locally convex spaces,” Science China Mathematics, vol. 55, no. 9, pp. 1797–1825, 2012.
• M. Li, Q. Zheng, and J. Zhang, “Regularized resolvent families,” Taiwanese Journal of Mathematics, vol. 11, no. 1, pp. 117–133, 2007.
• C. Lizama, “Regularized solutions for abstract Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 243, no. 2, pp. 278–292, 2000.
• F. Neubrander, “Integrated semigroups and their applications to the abstract Cauchy problem,” Pacific Journal of Mathematics, vol. 135, no. 1, pp. 111–155, 1988.
• D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 11, pp. 3692–3705, 2008.
• M. Kostić, “Abstract time-fractional equations: existence and growth of solutions,” Fractional Calculus and Applied Analysis, vol. 14, no. 2, pp. 301–316, 2011.
• W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace transforms and Cauchy problems, Birkhäuser, Basel, Switzerland, 2001.
• C. J. K. Batty, “Differentiability of perturbed semigroups and delay semigroups,” in Banach Center Publications, vol. 75, pp. 39–53, Polish Academy of Science, Warsaw, Poland, 2007.
• M. Kostić, “Differential and analytical properties of semigroups of operators,” Integral Equations and Operator Theory, vol. 67, no. 4, pp. 499–557, 2010.
• H. J. Haubold, A. M. Mathai, and R. K. Saxena, “Mittag-Leffler functions and their applications,” Journal of Applied Mathematics, vol. 2011, Article ID 298628, 51 pages, 2011.
• F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 283–299, 2000.
• R. Wong and Y.-Q. Zhao, “Exponential asymptotics of the Mittag-Leffler function,” Constructive Approximation, vol. 18, no. 3, pp. 355–385, 2002.
• M. Kostić, “Distribution cosine functions,” Taiwanese Journal of Mathematics, vol. 10, no. 3, pp. 739–775, 2006.
• R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5 of Evolution Problems I, Springer, Berlin, Germany, 2000.
• O. El-Mennaoui and V. Keyantuo, “Trace theorems for holomorphic semigroups and the second order Cauchy problem,” Proceedings of the American Mathematical Society, vol. 124, no. 5, pp. 1445–1458, 1996.
• V. Keyantuo, “Integrated semigroups and related partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 212, no. 1, pp. 135–153, 1997.
• V. Keyantuo and M. Warma, “The wave equation with Wentzell-Robin boundary conditions on ${L}^{p}$-spaces,” Journal of Differential Equations, vol. 229, no. 2, pp. 680–697, 2006.
• Q. Zheng, “Coercive differential operators and fractionally integrated cosine functions,” Taiwanese Journal of Mathematics, vol. 6, no. 1, pp. 59–65, 2002.
• T.-J. Xiao and J. Liang, “Abstract degenerate Cauchy problems in locally convex spaces,” Journal of Mathematical Analysis and Applications, vol. 259, no. 2, pp. 398–412, 2001.
• Q. Zheng, “Matrices of operators and regularized cosine functions,” Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 68–75, 2006.
• P. C. Kunstmann, “Stationary dense operators and generation of non-dense distribution semigroups,” Journal of Operator Theory, vol. 37, no. 1, pp. 111–120, 1997.
• H. Komatsu, “Ultradistributions. I. Structure theorems and a characterization,” Journal of the Faculty of Science. University of Tokyo, vol. 20, pp. 25–105, 1973.
• J. L. Walsh, “On the location of the roots of certain types of polynomials,” Transactions of the American Mathematical Society, vol. 24, no. 3, pp. 163–180, 1922.
• D. Kovačević, On integral transforms and convolution equations on the spaces of tempered ultradistributions [Ph.D. thesis], University of Novi Sad, Novi Sad, Serbia, 1992.
• P. C. Kunstmann, “Banach space valued ultradistributions and applications to abstract Cauchy problems,” preprint.
• T.-J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, Springer, Berlin, Germany, 1998.
• R. Meise and D. Vogt, Introduction to Functional Analysis, vol. 2 of Translated from the German by M. S. Ramanujan and Revised by the Authors, Oxford Graduate Texts in Mathematics, Clarendon Press, New York, NY, USA, 1997.
• C. Chen and M. Li, “On fractional resolvent operator functions,” Semigroup Forum, vol. 80, no. 1, pp. 121–142, 2010.
• C. Kaiser and L. Weis, “A perturbation theorem for operator semigroups in Hilbert spaces,” Semigroup Forum, vol. 67, no. 1, pp. 63–75, 2003.
• C. J. K. Batty, “On a perturbation theorem of Kaiser and Weis,” Semigroup Forum, vol. 70, no. 3, pp. 471–474, 2005.
• J. M. A. M. van Neerven, “Some recent results on adjoint semigroups,” CWI Quarterly, vol. 6, no. 2, pp. 139–153, 1993.
• K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, Berlin, Germany, 2000.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, Germany, 1983.
• A. N. Carvalho, T. Dlotko, and M. J. D. Nascimento, “Non-autonomous semilinear evolution equations with almost sectorial operators,” Journal of Evolution Equations, vol. 8, no. 4, pp. 631–659, 2008.
• F. Periago and B. Straub, “A functional calculus for almost sectorial operators and applications to abstract evolution equations,” Journal of Evolution Equations, vol. 2, no. 1, pp. 41–68, 2002.
• J. Chazarain, “Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes,” Journal of Functional Analysis, vol. 7, pp. 386–446, 1971.
• I. Ciorănescu, “Abstract Beurling spaces of class $({M}_{p})$ and ultradistribution semi-groups,” Bulletin des Science Mathematiques, vol. 102, no. 2, pp. 167–192, 1978.
• A. Borichev and Y. Tomilov, “Optimal polynomial decay of functions and operator semigroups,” Mathematische Annalen, vol. 347, no. 2, pp. 455–478, 2010.
• S. Ōuchi, “Hyperfunction solutions of the abstract Cauchy problem,” Proceedings of the Japan Academy, vol. 47, pp. 541–544, 1971.
• P. S. Iley, “Perturbations of differentiable semigroups,” Journal of Evolution Equations, vol. 7, no. 4, pp. 765–781, 2007.
• M. Kostić, “On a class of (a, k)-regularized C-resolvent families,” Electronic Journal of Qualitative Theory of Differential Equations, no. 94, pp. 1–27, 2012.