Journal of Integral Equations and Applications

Existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations

Sonia Seyed Allaei, Zhan-wen Yang, and Hermann Brunner

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Abstract

We analyze the existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations, including equations with weakly singular kernels. Of particular interest are those integral equations that can be transformed into cordial Volterra integral equations whose underlying integral operator may be non-compact.

Article information

Source
J. Integral Equations Applications, Volume 27, Number 3 (2015), 325-342.

Dates
First available in Project Euclid: 17 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1450388938

Digital Object Identifier
doi:10.1216/JIE-2015-27-3-325

Mathematical Reviews number (MathSciNet)
MR3435803

Zentralblatt MATH identifier
1329.45001

Subjects
Primary: 45A05: Linear integral equations 45D05: Volterra integral equations [See also 34A12] 45E99: None of the above, but in this section

Keywords
Volterra integral equation of the third kind cordial Volterra integral equation existence and regularity of solutions

Citation

Allaei, Sonia Seyed; Yang, Zhan-wen; Brunner, Hermann. Existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations. J. Integral Equations Applications 27 (2015), no. 3, 325--342. doi:10.1216/JIE-2015-27-3-325. https://projecteuclid.org/euclid.jiea/1450388938


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References

  • G.R. Bart and R.L. Warnock, Linear integral equations of the third kind, SIAM J. Math. Anal. 4 (1973), 609–622.
  • H. Brunner, Collocation methods for Volterra integral and related functional equations, Cambridge University Press, Cambridge, 2004.
  • H.G. B\u zihatlov, A certain integral equation of the third kind, Izv. Akad. Nauk. 2 (1970), 18–23.
  • A.R. Chvoles, On Fredholm's integral equations of the third kind, Bull. Acad. Sci. Georgian 2 (1941), 389–395.
  • G.C. Evans, Volterra's integral equation of the second kind with discontinuous kernel, Trans. Amer. Math. Soc. 11 (1910), 393–413.
  • ––––, Volterra's integral equation of the second kind with discontinuous kernel II, Trans. Amer. Math. Soc. 11 (1911), 429–472.
  • T. Fényes, On the operational solution of a convolution type integral equation of the third kind, Stud. Sci. Math. Hungar. 12 (1977), 65–75.
  • N.S. Gabbasov, On the theory of Fredholm integral equations of the third kind in a space of generalized functions, Izv. Vyssh. Uchebn. Zaved. Mat. 85 (1986), 68–70.
  • ––––, Approximate solution of integral equations of the third kind, Izv. Vyssh. Uchebn. Zaved. Mat. 82 (1986), 49–52.
  • N.S. Gabbasov and S.A. Solov'eva, Special versions of the collocation method for a class of integral equations of the third kind, Russ. Math. 56 (2012), 22–27.
  • P. Grandits, A regularity theorem for a Volterra integral equation of the third kind, J. Integral Equations Appl. 20 (2008), 507–526.
  • D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, Leipzig, 1912.
  • M.I. Imanaliev and A. Asanov, Regularization and uniqueness of solutions of systems of nonlinear Volterra integral equations of the third kind, Dokl. Math. 76 (2007), 490–493.
  • N.N. Juhanonov, General theorems on the solvability of a certain convolution type singular integral equation of the third kind on the half-axis, Izv. Akad. Nauk. 3 (1973), 22–28.
  • S.V. Pereverzev and S.A. Prössdorf, Discretization of Volterra integral equations of the third kind with weakly singular kernels, J. Inv. Ill-Posed Prob. 5 (1997), 565–577.
  • É. Picard, Sur les équations intégrales de troisième espèce, Ann. Sci. École Norm. 28 (1911), 459–472.
  • V.S. Rogo\uzin and S.N. Raslambekov, The Noether theory for integral equations of the third kind in spaces of continuous and generalized functions, Sov. Math. 23 (1979), 48–53.
  • T. Sato, Sur l'équation intégrale $xu(x)=f(x)+\int_0^xK(x,t,u(t))\,dt$, J. Math. Soc. Japan 5 (1953), 145–153.
  • E. Schock, Integral equations of the third kind, Stud. Math. 81 (1985), 1–11.
  • G. Vainikko, Cordial Volterra integral equations 1, Numer. Funct. Anal. Optim. 30 (2009), 1145–1172.
  • ––––, Cordial Volterra integral equations 2, Numer. Funct. Anal. Optim. 31 (2010), 191–219.
  • L.V. Wolfersdorf, On the theory of convolution equations of the third kind, J. Math. Anal. Appl. 331 (2007), 1314–1336; 342 (2008), 838–863.
  • Z.W. Yang, Second-kind linear Volterra integral equations with noncompact operators, Numer. Funct. Anal. Optim. 36 (2015), 104–131.