Previous |  Up |  Next


integral equation; resolvent
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.
[1] Burton, T. A.: Liapunov Functionals for Integral Equations. Trafford, Victoria, B. C., Canada (2008) (
[2] Burton, T. A.: Averaged neural networks. Neural Networks 6 (1993), 667-680. DOI 10.1016/S0893-6080(05)80111-X
[3] Ergen, W. K.: Kinetics of the circulating-fuel nuclear reactor. J. Appl. Phys. 25 (1954), 702-711. DOI 10.1063/1.1721720 | Zbl 0055.23003
[4] Islam, M. N., Neugebauer, J. T.: Qualitative properties of nonlinear Volterra integral equations. Electron. J. Qual. Theory Differ. Equ. 12 (2008), 1-16. DOI 10.14232/ejqtde.2008.1.12 | MR 2385416 | Zbl 1178.45009
[5] Levin, J. J.: The asymptotic behavior of the solution of a Volterra equation. Proc. Amer. Math. Soc. 14 (1963), 534-541. DOI 10.1090/S0002-9939-1963-0152852-8 | MR 0152852 | Zbl 0115.32403
[6] Miller, Richard K.: Nonlinear Volterra Integral Equations. Benjamin, New York (1971). MR 0511193 | Zbl 0448.45004
[7] Reynolds, David W.: On linear singular Volterra integral equations of the second kind. J. Math. Anal. Appl. 103 (1984), 230-262. DOI 10.1016/0022-247X(84)90171-9 | MR 0757637 | Zbl 0557.45003
[8] Strauss, A.: On a perturbed Volterra equation. J. Math. Anal. Appl. 30 (1970), 564-575. DOI 10.1016/0022-247X(70)90141-1 | MR 0261291
[9] Volterra, V.: Sur la théorie mathématique des phénomès héréditaires. J. Math. Pur. Appl. 7 (1928), 249-298.
Partner of
EuDML logo