Stability in nonlinear evolution problems by means of fixed point theorems

Jaromír J. Koliha; Ivan Straškraba

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 1, page 37-59
  • ISSN: 0010-2628

Abstract

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The stabilization of solutions to an abstract differential equation is investigated. The initial value problem is considered in the form of an integral equation. The equation is solved by means of the Banach contraction mapping theorem or the Schauder fixed point theorem in the space of functions decreasing to zero at an appropriate rate. Stable manifolds for singular perturbation problems are compared with each other. A possible application is illustrated on an initial-boundary-value problem for a parabolic equation in several space variables.

How to cite

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Koliha, Jaromír J., and Straškraba, Ivan. "Stability in nonlinear evolution problems by means of fixed point theorems." Commentationes Mathematicae Universitatis Carolinae 38.1 (1997): 37-59. <http://eudml.org/doc/248067>.

@article{Koliha1997,
abstract = {The stabilization of solutions to an abstract differential equation is investigated. The initial value problem is considered in the form of an integral equation. The equation is solved by means of the Banach contraction mapping theorem or the Schauder fixed point theorem in the space of functions decreasing to zero at an appropriate rate. Stable manifolds for singular perturbation problems are compared with each other. A possible application is illustrated on an initial-boundary-value problem for a parabolic equation in several space variables.},
author = {Koliha, Jaromír J., Straškraba, Ivan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {evolution equations; stabilization of solutions; parabolic problem; evolution equations; stabilization of solutions; parabolic problems},
language = {eng},
number = {1},
pages = {37-59},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Stability in nonlinear evolution problems by means of fixed point theorems},
url = {http://eudml.org/doc/248067},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Koliha, Jaromír J.
AU - Straškraba, Ivan
TI - Stability in nonlinear evolution problems by means of fixed point theorems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 1
SP - 37
EP - 59
AB - The stabilization of solutions to an abstract differential equation is investigated. The initial value problem is considered in the form of an integral equation. The equation is solved by means of the Banach contraction mapping theorem or the Schauder fixed point theorem in the space of functions decreasing to zero at an appropriate rate. Stable manifolds for singular perturbation problems are compared with each other. A possible application is illustrated on an initial-boundary-value problem for a parabolic equation in several space variables.
LA - eng
KW - evolution equations; stabilization of solutions; parabolic problem; evolution equations; stabilization of solutions; parabolic problems
UR - http://eudml.org/doc/248067
ER -

References

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  2. Crandall M., Liggett T., Generation of semigroups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1977), 265-298. (1977) MR0287357
  3. Hale J.K., Theory of Functional Differential Equations, Springer, New York, 1977. Zbl1092.34500MR0508721
  4. Hale J.K., Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs No. 25, Amer. Math. Soc., Providence, 1988. Zbl0642.58013MR0941371
  5. Iwamiya T., Takahashi T., Oharu T., Characterization of nonlinearly perturbed semigroups, in: Functional Analysis and Related Topics, Proceedings, Kyoto 1991, H. Komatsu (ed.), Lecture Notes in Math. No. 1540, Springer, New York, 1993. Zbl0819.47081
  6. Komatsu H., Fractional powers of generators, II Interpolation spaces, Pacific J. Math. 21 (1967), 89-111. (1967) MR0206716
  7. Krasnosel'skij M.A., Zabrejko P.P., Pustyl'nik E.I., Sobolevskij P.E., Integral Operators in the Spaces of Integrable Functions (in Russian), Nauka, Moscow, 1966. 
  8. Krein M., Dalecki J., Stability of the Solutions of Differential Equations in Banach Spaces, Amer. Math. Soc., Providence, 1974. MR0352639
  9. Lunardi A., Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. Zbl0816.35001MR1329547
  10. Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. Zbl0516.47023MR0710486
  11. Rauch J., Stability of motion for semilinear equations, in: Boundary Value Problems for Linear Evolution Partial Differential Equations, Proceedings of the NATO Advanced Study Institute held in Liège, Belgium, September 6-17, 1976, H. G. Garnir (ed.), NATO Advanced Study Institute Series C, Mathematical and Physical Sciences, vol. 29, Reidel Publishing Co., Dordrecht, 1977. Zbl0347.35015MR0492677
  12. Tanabe H., Equations of Evolution, Pitman, London, 1979. Zbl0417.35003MR0533824

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