| Title:
|
Resolvents, integral equations, limit sets (English) |
| Author:
|
Burton, T. A. |
| Author:
|
Dwiggins, D. P. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
135 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
337-354 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we study a linear integral equation $x(t)=a(t)-\int ^t_0 C(t,s) x(s) {\rm d} s$, its resolvent equation $R(t,s)=C(t,s)-\int ^t_s C(t,u)R(u,s) {\rm d} u$, the variation of parameters formula $x(t)=a(t)-\int ^t_0 R(t,s)a(s) {\rm d} s$, and a perturbed equation. The kernel, $C(t,s)$, satisfies classical smoothness and sign conditions assumed in many real-world problems. We study the effects of perturbations of $C$ and also the limit sets of the resolvent. These results lead us to the study of nonlinear perturbations. (English) |
| Keyword:
|
integral equation |
| Keyword:
|
resolvent |
| MSC:
|
34D20 |
| idZBL:
|
Zbl 1224.45001 |
| idMR:
|
MR2681008 |
| DOI:
|
10.21136/MB.2010.140824 |
| . |
| Date available:
|
2010-11-24T08:22:18Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140824 |
| . |
| Reference:
|
[1] Burton, T. A.: Liapunov Functionals for Integral Equations.Trafford, Victoria, B. C., Canada (2008) (www.trafford.com/08-1365). |
| Reference:
|
[2] Burton, T. A.: Averaged neural networks.Neural Networks 6 (1993), 667-680. 10.1016/S0893-6080(05)80111-X |
| Reference:
|
[3] Ergen, W. K.: Kinetics of the circulating-fuel nuclear reactor.J. Appl. Phys. 25 (1954), 702-711. Zbl 0055.23003, 10.1063/1.1721720 |
| Reference:
|
[4] Islam, M. N., Neugebauer, J. T.: Qualitative properties of nonlinear Volterra integral equations.Electron. J. Qual. Theory Differ. Equ. 12 (2008), 1-16. Zbl 1178.45009, MR 2385416, 10.14232/ejqtde.2008.1.12 |
| Reference:
|
[5] Levin, J. J.: The asymptotic behavior of the solution of a Volterra equation.Proc. Amer. Math. Soc. 14 (1963), 534-541. Zbl 0115.32403, MR 0152852, 10.1090/S0002-9939-1963-0152852-8 |
| Reference:
|
[6] Miller, Richard K.: Nonlinear Volterra Integral Equations.Benjamin, New York (1971). Zbl 0448.45004, MR 0511193 |
| Reference:
|
[7] Reynolds, David W.: On linear singular Volterra integral equations of the second kind.J. Math. Anal. Appl. 103 (1984), 230-262. Zbl 0557.45003, MR 0757637, 10.1016/0022-247X(84)90171-9 |
| Reference:
|
[8] Strauss, A.: On a perturbed Volterra equation.J. Math. Anal. Appl. 30 (1970), 564-575. MR 0261291, 10.1016/0022-247X(70)90141-1 |
| Reference:
|
[9] Volterra, V.: Sur la théorie mathématique des phénomès héréditaires.J. Math. Pur. Appl. 7 (1928), 249-298. |
| . |
