Geodesic
Resolvents, integral equations, limit sets
Mathematica Bohemica, Tome 135 (2010) no. 4, pp. 337-354

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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.
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.
DOI : 10.21136/MB.2010.140824
Classification : 34D20
Keywords: integral equation; resolvent 
Burton, T. A.; Dwiggins, D. P. Resolvents, integral equations, limit sets. Mathematica Bohemica, Tome 135 (2010) no. 4, pp. 337-354. doi: 10.21136/MB.2010.140824
@article{10_21136_MB_2010_140824,
     author = {Burton, T. A. and Dwiggins, D. P.},
     title = {Resolvents, integral equations, limit sets},
     journal = {Mathematica Bohemica},
     pages = {337--354},
     year = {2010},
     volume = {135},
     number = {4},
     doi = {10.21136/MB.2010.140824},
     mrnumber = {2681008},
     zbl = {1224.45001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2010.140824/}
}
TY  - JOUR
AU  - Burton, T. A.
AU  - Dwiggins, D. P.
TI  - Resolvents, integral equations, limit sets
JO  - Mathematica Bohemica
PY  - 2010
SP  - 337
EP  - 354
VL  - 135
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2010.140824/
DO  - 10.21136/MB.2010.140824
LA  - en
ID  - 10_21136_MB_2010_140824
ER  - 
%0 Journal Article
%A Burton, T. A.
%A Dwiggins, D. P.
%T Resolvents, integral equations, limit sets
%J Mathematica Bohemica
%D 2010
%P 337-354
%V 135
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2010.140824/
%R 10.21136/MB.2010.140824
%G en
%F 10_21136_MB_2010_140824

[1] Burton, T. A.: Liapunov Functionals for Integral Equations. Trafford, Victoria, B. C., Canada (2008) (www.trafford.com/08-1365).

[2] Burton, T. A.: Averaged neural networks. Neural Networks 6 (1993), 667-680. | DOI

[3] Ergen, W. K.: Kinetics of the circulating-fuel nuclear reactor. J. Appl. Phys. 25 (1954), 702-711. | DOI | Zbl

[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 | MR | Zbl

[5] Levin, J. J.: The asymptotic behavior of the solution of a Volterra equation. Proc. Amer. Math. Soc. 14 (1963), 534-541. | DOI | MR | Zbl

[6] Miller, Richard K.: Nonlinear Volterra Integral Equations. Benjamin, New York (1971). | MR | Zbl

[7] Reynolds, David W.: On linear singular Volterra integral equations of the second kind. J. Math. Anal. Appl. 103 (1984), 230-262. | DOI | MR | Zbl

[8] Strauss, A.: On a perturbed Volterra equation. J. Math. Anal. Appl. 30 (1970), 564-575. | DOI | MR

[9] Volterra, V.: Sur la théorie mathématique des phénomès héréditaires. J. Math. Pur. Appl. 7 (1928), 249-298.

Cité par Sources :