Existence, uniqueness and regularity of solutions to a class of third-kind Volterra integral equations. (English) Zbl 1329.45001
Summary: 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.
MSC:
| 45A05 | Linear integral equations |
| 45D05 | Volterra integral equations |
| 45E99 | Singular integral equations |
Keywords:
Volterra integral equation of the third kind; cordial Volterra integral equation; existence and regularity of solutionsReferences:
| [1] | G.R. Bart and R.L. Warnock, Linear integral equations of the third kind , SIAM J. Math. Anal. 4 (1973), 609-622. · Zbl 0265.45001 · doi:10.1137/0504053 |
| [2] | H. Brunner, Collocation methods for Volterra integral and related functional equations , Cambridge University Press, Cambridge, 2004. · Zbl 1059.65122 · doi:10.1017/CBO9780511543234 |
| [3] | H.G. Bžihatlov, A certain integral equation of the third kind , Izv. Akad. Nauk. 2 (1970), 18-23. |
| [4] | A.R. Chvoles, On Fredholm’s integral equations of the third kind , Bull. Acad. Sci. Georgian 2 (1941), 389-395. |
| [5] | G.C. Evans, Volterra’s integral equation of the second kind with discontinuous kernel , Trans. Amer. Math. Soc. 11 (1910), 393-413. · JFM 41.0386.02 |
| [6] | —-, Volterra’s integral equation of the second kind with discontinuous kernel II, Trans. Amer. Math. Soc. 11 (1911), 429-472. · JFM 42.0381.01 |
| [7] | T. Fényes, On the operational solution of a convolution type integral equation of the third kind , Stud. Sci. Math. Hungar. 12 (1977), 65-75. |
| [8] | 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. · Zbl 0603.45001 |
| [9] | —-, Approximate solution of integral equations of the third kind , Izv. Vyssh. Uchebn. Zaved. Mat. 82 (1986), 49-52. · Zbl 0615.65133 |
| [10] | 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. · Zbl 1258.45003 · doi:10.3103/S1066369X12080038 |
| [11] | P. Grandits, A regularity theorem for a Volterra integral equation of the third kind , J. Integral Equations Appl. 20 (2008), 507-526. · Zbl 1161.45002 · doi:10.1216/JIE-2008-20-4-507 |
| [12] | D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen , Teubner, Leipzig, 1912. |
| [13] | 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. · Zbl 1157.45305 · doi:10.1134/S1064562407040035 |
| [14] | 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. |
| [15] | 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. · Zbl 0907.65136 · doi:10.1515/jiip.1997.5.6.565 |
| [16] | É. Picard, Sur les équations intégrales de troisième espèce , Ann. Sci. École Norm. 28 (1911), 459-472. · JFM 42.0371.04 |
| [17] | 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. · Zbl 0419.45001 |
| [18] | 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. · Zbl 0052.10703 · doi:10.2969/jmsj/00520145 |
| [19] | E. Schock, Integral equations of the third kind , Stud. Math. 81 (1985), 1-11. · Zbl 0536.45002 |
| [20] | G. Vainikko, Cordial Volterra integral equations 1, Numer. Funct. Anal. Optim. 30 (2009), 1145-1172. · Zbl 1195.45004 · doi:10.1080/01630560903393188 |
| [21] | —-, Cordial Volterra integral equations 2, Numer. Funct. Anal. Optim. 31 (2010), 191-219. · Zbl 1194.65152 · doi:10.1080/01630561003666234 |
| [22] | 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. · Zbl 1118.45005 · doi:10.1016/j.jmaa.2006.09.042 |
| [23] | Z.W. Yang, Second-kind linear Volterra integral equations with noncompact operators , Numer. Funct. Anal. Optim. 36 (2015), 104-131. · Zbl 1333.47041 · doi:10.1080/01630563.2014.951769 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
