143
Views
4
CrossRef citations to date
0
Altmetric
Original Articles

Abstract Volterra Integro-Differential Equations: Approximation and Convergence of Resolvent Operator Families

Pages 1579-1606 | Received 16 May 2013, Accepted 09 Mar 2014, Published online: 25 Aug 2014

REFERENCES

  • W. Arendt , C. J. K. Batty , M. Hieber , and F. Neubrander ( 2001 ). Vector-valued Laplace Transforms and Cauchy Problems . Monographs in Mathematics, Vol. 96. Birkhäuser Verlag, Basel .
  • E. Bazhlekova ( 2001 ). Fractional evolution equations in Banach spaces. Ph.D. dissertation, Eindhoven University of Technology.
  • S. Busenberg and B. H. Wu ( 1992 ). Convergence theorems for integrated semigroups . Differential Integral Equations 5 : 509 – 520 .
  • Y.-C. Chang and S.-Y. Shaw ( 1997 ). Optimal and non-optimal rates of approximation for integrated semigroups and cosine functions . J. Approx. Theory 90 : 200 – 223 .
  • K.-J. Engel and R. Nagel ( 2000 ). One–Parameter Semigroups for Linear Evolution Equations . Springer-Verlag , Berlin .
  • B. Hennig and F. Neubrander (1993). On representations, inversion and approximations of Laplace transform in Banach spaces. Appl. Analysis 49:151–170.
  • J. A. Goldstein ( 1974 ). On the convergence and approximations of cosine functions . Aequationes Math. 10 : 201 – 205 .
  • D. Guidetti , B. Karasrözen , and S. Piskarev ( 2004 ). Approximations of abstract differential equations . J. Math. Sci. (N. Y.) 122 : 3013 – 3054 .
  • T. Kato ( 1959 ). Remarks on pseudo-resolvents and infinitesimal generators of semi-groups . Proc. Japan Acad. 35 : 467 – 468 .
  • M. Kim ( 2004 ). Trotter-Kato theorems for convoluted resolvent families . Commun. Korean Math. Soc. 19 : 293 – 305 .
  • M. Kostić ( 2011 ). Generalized Semigroups and Cosine Functions. Mathematical Institute Belgrade.
  • M. Kostić ( 2009 ). (a, k)-regularized C-resolvent families: Regularity and local properties. Abstr. Appl. Anal., Article ID 858242.
  • M. Kostić ( 2012 ). Abstract Volterra equations in locally convex spaces . Science China Math. 55 : 1797 – 1825 .
  • M. Kostić , C.-G. Li , and M. Li ( 2012 ). On a class of abstract time-fractional equations on locally convex spaces . Abstr. Appl. Anal. , Article ID 131652.
  • M. Kostić ( 2014 ). Generalized well-posedness of hyperbolic Volterra equations of non-scalar type . Ann. Acad. Rom. Sci. Ser. Math. Appl. 6 : 21 – 49 .
  • M. Kostić ( 2013 ). (a, k)-regularized (C 1, C 2)-existence and uniqueness families . Bull. Cl. Sci. Math. Nat. Sci. Math. 38 : 11 – 28 .
  • M. Kostić (accepted). Abstract differential operators generating fractional resolvent families. Acta Math. Sin. (Engl. Ser.).
  • C.-C. Kuo ( 2009 ). On existence and approximation of solutions of abstract Cauchy problem . Taiwanese J. Math. 13 : 137 – 155 .
  • C.-C. Kuo ( 2010 ). On existence and approximation of solutions of second order abstract Cauchy problem . Taiwanese J. Math. 14 : 1093 – 1109 .
  • C.-G. Li , M. Kostić , M. Li , and S. Piskarev ( 2012 ). On a class of time-fractional differential equations . Fract. Calc. Appl. Anal. 15 : 639 – 668 .
  • M. Li and S. Piskarev ( 2010 ). On approximations of integrated semigroups . Taiwanese J. Math. 14 : 2137 – 2161 .
  • M. Li and Q. Zheng ( 2004 ). On spectral inclusions and approximations of α-times resolvent families . Semigroup Forum 69 : 356 – 368 .
  • M. Li and Q. Zheng ( 2009 ). On product formulas for C-semigroups . Semigroup Forum 78 : 536 – 546 .
  • Y.-C. Li and S.-Y. Shaw ( 1997 ). N-times integrated C-semigroups and the abstract Cauchy problem . Taiwanese J. Math. 1 : 75 – 102 .
  • X.-M. Li , X.-Q. Song , and Y.-Y. Zhao ( 2008 ). The Trotter-Kato approximation theorems of N − times integrated C cosine functions . Journal of Huaiyin Institute of Technology 01 (in chinese) .
  • C. Lizama ( 1994 ). On the convergence and approximation of integrated semigroups . J. Math. Anal. Appl. 181 : 89 – 103 .
  • C. Lizama ( 2001 ). On approximation and representation of K-regularized resolvent families . Integral Equations Operator Theory 41 : 223 – 229 .
  • C. Lizama and H. Prado ( 2003 ). Rates of approximations and ergodic limits of regularized operator families . J. Approx. Theory 122 : 42 – 61 .
  • C. Müller ( 2002 ). Approximation of local convoluted semigroups . J. Math. Anal. Appl. 269 : 401 – 420 .
  • S. Nicaise ( 1993 ). The Hille–Yosida and Trotter–Kato theorems for integrated semigroups . J. Math. Anal. Appl. 180 : 303 – 316 .
  • H. F. Trotter ( 1958 ). Approximation of operator-semigroups . Pacific J. Math. 8 : 887 – 919 .
  • J. Prüss ( 1993 ). Evolutionary Integral Equations and Applications . Birkhäuser-Verlag , Basel .
  • S.-Y. Shaw ( 2000 ). Ergodic theorems and approximation theorems with rates . Taiwanese J. Math. 4 : 365 – 383 .
  • T.-J. Xiao and J. Liang ( 1998 ). The Cauchy Problem for Higher-Order Abstract Differential Equations . Springer-Verlag , Berlin .
  • T.-J. Xiao and J. Liang ( 2000 ). Approximations of Laplace transforms and integrated semigroups . J. Funct. Anal. 172 : 202 – 220 .
  • T.-J. Xiao , R. Nagel , and J. Liang (2007). Approximation theorems for the propagators of higher order abstract Cauchy problems. Trans. Amer. Math. Soc. 360:1723–1739.
  • T.-J. Xiao and J. Liang ( 2001 ). Abstract degenerate Cauchy problems in locally convex spaces . J. Math. Anal. Appl. 259 : 398 – 412 .
  • Q. Zheng and Y.-S. Lei ( 1993 ). Exponentially bounded C-semigroup and integrated semigroup with nondensely defined generators I: Approximation . Acta Math. Sinica 13 : 251 – 260 .
  • Q. Zheng ( 1996 ). Integrated cosine functions . Int. J. Math. Math. Sci. 19 : 575 – 580 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.