Applications of the spectral radius to some integral equations

Mirosława Zima

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 4, page 695-703
  • ISSN: 0010-2628

Abstract

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In the paper [13] we proved a fixed point theorem for an operator 𝒜 , which satisfies a generalized Lipschitz condition with respect to a linear bounded operator A , that is: m ( 𝒜 x - 𝒜 y ) ≺ A m ( 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 .

How to cite

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Zima, Mirosława. "Applications of the spectral radius to some integral equations." Commentationes Mathematicae Universitatis Carolinae 36.4 (1995): 695-703. <http://eudml.org/doc/247754>.

@article{Zima1995,
abstract = {In the paper [13] we proved a fixed point theorem for an operator $\mathcal \{A\}$, which satisfies a generalized Lipschitz condition with respect to a linear bounded operator $A$, that is: \[ m(\mathcal \{A\} x-\mathcal \{A\} y)\prec Am(x-y). \] The purpose of this paper is to show that the results obtained in [13], [14] can be extended to a nonlinear operator $A$.},
author = {Zima, Mirosława},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {fixed point theorem; spectral radius; integral-functional equation; spectral radius; integral-functional equation; fixed point theorem; generalized Lipschitz condition},
language = {eng},
number = {4},
pages = {695-703},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Applications of the spectral radius to some integral equations},
url = {http://eudml.org/doc/247754},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Zima, Mirosława
TI - Applications of the spectral radius to some integral equations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 4
SP - 695
EP - 703
AB - In the paper [13] we proved a fixed point theorem for an operator $\mathcal {A}$, which satisfies a generalized Lipschitz condition with respect to a linear bounded operator $A$, that is: \[ m(\mathcal {A} x-\mathcal {A} y)\prec Am(x-y). \] The purpose of this paper is to show that the results obtained in [13], [14] can be extended to a nonlinear operator $A$.
LA - eng
KW - fixed point theorem; spectral radius; integral-functional equation; spectral radius; integral-functional equation; fixed point theorem; generalized Lipschitz condition
UR - http://eudml.org/doc/247754
ER -

References

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  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. (1991) MR1124954
  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. (1993) Zbl0772.34047MR1200990
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  7. Riesz F., Sz.-Nagy B., Functional analysis, Ungar, New York, 1955. Zbl0732.47001MR0071727
  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. (1960) MR0126109
  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. (1990) MR1095667
  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. (1967) MR0208390
  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. (1987) MR0911361
  12. Zeidler E., Nonlinear functional analysis and its applications I, Springer Verlag, New York Heidelberg Berlin, 1993. Zbl0583.47050MR0816732
  13. Zima M., A certain fixed point theorem and its applications to integral-functional equations, Bull. Austral. Math. Soc. 46 (1992), 179-186. (1992) Zbl0761.34048MR1183775
  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. (1993) Zbl0795.34069MR1248046

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