Cordial Volterra integral equations. II. (English) Zbl 1194.65152
Summary: We study the mapping properties and spectra of the (cordial) Volterra integral operators of the form \((V_{\varphi,a},u)(t)=\int^t_0 t^{-1}\varphi(t^{-1}\varphi(t^{-1}s)a(t,s)u(s)\,ds\), \(0\leq t\leq T\), where \(\varphi\in L^1(0, 1)\), \(a\in C^m\) \((0\leq s\leq t\leq T)\), \(m \geq 0\). Also polynomial collocation methods for the solving the (cordial) Volterra integral equation \(\mu u = V_{\varphi, a}u + f\) is examined. Here \(\mu\) is real or complex parameter outside the spectrum of \(V_{\varphi, a}\) as an operator in the space \(C[0,T]\).
For part I, cf. Numer. Funct. Anal. Optim. 30, No. 9–10, 1145–1172 (2009; Zbl 1195.45004).
For part I, cf. Numer. Funct. Anal. Optim. 30, No. 9–10, 1145–1172 (2009; Zbl 1195.45004).
MSC:
| 65R20 | Numerical methods for integral equations |
| 45A05 | Linear integral equations |
| 45C05 | Eigenvalue problems for integral equations |
| 45D05 | Volterra integral equations |
| 45P05 | Integral operators |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
non-compact operators; polynomial collocation; spectrum; (cordial) Volterra integral operatorsCitations:
Zbl 1195.45004References:
| [1] | DOI: 10.1017/CBO9780511543234 · Zbl 1059.65122 · doi:10.1017/CBO9780511543234 |
| [2] | Brunner H., The Numerical Solution of Volterra Equations 3 (1986) · Zbl 0634.65143 |
| [3] | DOI: 10.1090/S0025-5718-99-01073-X · Zbl 0941.65136 · doi:10.1090/S0025-5718-99-01073-X |
| [4] | Daugavet I.K., Introduction to the Theory of Approximation of Functions (1977) · Zbl 0414.41001 |
| [5] | Dunford N., Linear Operators, Part 1: General Theory (1958) |
| [6] | DOI: 10.1002/mana.19901461703 · Zbl 0709.65117 · doi:10.1002/mana.19901461703 |
| [7] | DOI: 10.1016/j.apnum.2007.08.004 · Zbl 1159.65105 · doi:10.1016/j.apnum.2007.08.004 |
| [8] | DOI: 10.1007/s00607-004-0088-9 · Zbl 1063.65147 · doi:10.1007/s00607-004-0088-9 |
| [9] | Proessdorf S., Numerical Analysis for Integral and Related Equations (1991) |
| [10] | Saranen J., Periodic Integral and Pseudodifferential Equations with Numerical Approximation (2002) · Zbl 0991.65125 |
| [11] | Vainikko E., American Inst. Physics Conf. Proc. 936 pp 570– (2007) |
| [12] | DOI: 10.1080/01630560903393188 · Zbl 1195.45004 · doi:10.1080/01630560903393188 |
| [13] | Vainikko G., Numer. Funct. Anal. Optim. |
| [14] | Vainikko G., Proc. Eston. Acad. Sci., Phys. Math. 53 pp 124– (2004) |
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.
