| Title:
|
Applications of the spectral radius to some integral equations (English) |
| Author:
|
Zima, Mirosława |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
36 |
| Issue:
|
4 |
| Year:
|
1995 |
| Pages:
|
695-703 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the paper [13] we proved a fixed point theorem for an operator $\Cal A$, which satisfies a generalized Lipschitz condition with respect to a linear bounded operator $A$, that is: $$ m(\Cal A x-\Cal A y)\prec Am(x-y). $$ The purpose of this paper is to show that the results obtained in [13], [14] can be extended to a nonlinear operator $A$. (English) |
| Keyword:
|
fixed point theorem |
| Keyword:
|
spectral radius |
| Keyword:
|
integral-functional equation |
| MSC:
|
34K10 |
| MSC:
|
45G10 |
| MSC:
|
47G10 |
| MSC:
|
47H07 |
| MSC:
|
47H10 |
| MSC:
|
47J10 |
| idZBL:
|
Zbl 0845.47047 |
| idMR:
|
MR1378690 |
| . |
| Date available:
|
2009-01-08T18:21:00Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118796 |
| . |
| Reference:
|
[1] Bainov D.D., Mishev D.P.: Oscillation theory for neutral differential equations with delay.Adam Hilger, Bristol Philadelphia New York, 1991. Zbl 0747.34037, MR 1147908 |
| Reference:
|
[2] Förster K.-H., Nagy B.: On the local spectral radius of a nonnegative element with respect to an irreducible operator.Acta Sci. Math. 55 (1991), 155-166. MR 1124954 |
| Reference:
|
[3] Hristova S.G., Bainov D.D.: Monotone-iterative techniques of V. Lakshmikantham for a boundary value problem for systems of impulsive differential equations with ``supremum''.J. Math. Anal. Appl. 172 (1993), 339-352. Zbl 0772.34047, MR 1200990 |
| Reference:
|
[4] Krasnoselski M.A. et al.: Näherungsverfahren zur Lösung von Operatorgleichungen.Akademie Verlag, Berlin, 1973. Zbl 0269.65001 |
| Reference:
|
[5] Kwapisz M.: On the existence and uniqueness of solutions of a certain integral-functional equation.Ann. Polon. Math. 31 (1975), 23-41. MR 0380329 |
| Reference:
|
[6] Myshkis A.D.: On some problems of the theory of differential equations with deviating argument (in Russian).Uspehi Mat. Nauk 32 (1977), 173-202. MR 0492443 |
| Reference:
|
[7] Riesz F., Sz.-Nagy B.: Functional analysis.Ungar, New York, 1955. Zbl 0732.47001, MR 0071727 |
| Reference:
|
[8] Waẓewski T.: Sur un procédé de prouver la convergence des approximations successives sans utilisation des séries de comparaison.Bull. Acad. Polon. Sci. 1 (1960), 45-52. MR 0126109 |
| Reference:
|
[9] Zabrejko P.P.: The contraction mapping principle in $K$-metric and locally convex spaces (in Russian).Dokl. Akad. Nauk BSSR 34 (1990), 1065-1068. MR 1095667 |
| Reference:
|
[10] Zabrejko P.P., Krasnoselski M.A., Stecenko V.Ya.: On estimations of the spectral radius of the linear positive operators (in Russian).Mat. Zametki 1 (1967), 461-470. MR 0208390 |
| Reference:
|
[11] Zabrejko P.P., Makarevich T.A.: On some generalization of the Banach-Caccioppoli principle to operators in pseudometric spaces (in Russian).Diff. Uravn. 23 (1987), 1497-1504. MR 0911361 |
| Reference:
|
[12] Zeidler E.: Nonlinear functional analysis and its applications I.Springer Verlag, New York Heidelberg Berlin, 1993. Zbl 0583.47050, MR 0816732 |
| Reference:
|
[13] Zima M.: A certain fixed point theorem and its applications to integral-functional equations.Bull. Austral. Math. Soc. 46 (1992), 179-186. Zbl 0761.34048, MR 1183775 |
| Reference:
|
[14] Zima M.: A theorem on the spectral radius of the sum of two operators and its application.Bull. Austral. Math. Soc. 48 (1993), 427-434. Zbl 0795.34069, MR 1248046 |
| . |
