# The fixed point theorem and the boundedness of solutions of differential equations in the Banach space

Mathematica Bohemica (1993)

• Volume: 118, Issue: 1, page 1-9
• ISSN: 0862-7959

top

## Abstract

top
The properties of solutions of the nonlinear differential equation ${x}^{\text{'}}=A\left(s\right)x+f\left(s,x\right)$ in a Banach space and of the special case of the homogeneous linear differential equation ${x}^{\text{'}}=A\left(s\right)x$ are studied. Theorems and conditions guaranteeing boundedness of the solution of the nonlinear equation are given on the assumption that the solutions of the linear homogeneous equation have certain properties.

## How to cite

top

Tumajer, František. "The fixed point theorem and the boundedness of solutions of differential equations in the Banach space." Mathematica Bohemica 118.1 (1993): 1-9. <http://eudml.org/doc/29170>.

@article{Tumajer1993,
abstract = {The properties of solutions of the nonlinear differential equation $x^\{\prime \}=A(s)x+f(s,x)$ in a Banach space and of the special case of the homogeneous linear differential equation $x^\{\prime \}=A(s)x$ are studied. Theorems and conditions guaranteeing boundedness of the solution of the nonlinear equation are given on the assumption that the solutions of the linear homogeneous equation have certain properties.},
author = {Tumajer, František},
journal = {Mathematica Bohemica},
keywords = {differential equation; Banach space; existence; uniqueness; boundedness; bounded solution; derivative of the norm of a linear mapping; fixed point; differential equation; Banach space; existence; uniqueness; boundedness},
language = {eng},
number = {1},
pages = {1-9},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The fixed point theorem and the boundedness of solutions of differential equations in the Banach space},
url = {http://eudml.org/doc/29170},
volume = {118},
year = {1993},
}

TY - JOUR
AU - Tumajer, František
TI - The fixed point theorem and the boundedness of solutions of differential equations in the Banach space
JO - Mathematica Bohemica
PY - 1993
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 118
IS - 1
SP - 1
EP - 9
AB - The properties of solutions of the nonlinear differential equation $x^{\prime }=A(s)x+f(s,x)$ in a Banach space and of the special case of the homogeneous linear differential equation $x^{\prime }=A(s)x$ are studied. Theorems and conditions guaranteeing boundedness of the solution of the nonlinear equation are given on the assumption that the solutions of the linear homogeneous equation have certain properties.
LA - eng
KW - differential equation; Banach space; existence; uniqueness; boundedness; bounded solution; derivative of the norm of a linear mapping; fixed point; differential equation; Banach space; existence; uniqueness; boundedness
UR - http://eudml.org/doc/29170
ER -

## References

top
1. J. L. Massera J. J. Schäffler, Linear differential equations and function spaces, Academic press, New York and London, 1966. (1966) MR0212324
2. F. Tumajer, The derivative of the norm of the linear mapping and its application to differential equations, Aplikace matematiky 57 (1992), 193-200. (1992) MR1157455
3. M. Greguš M. Švec V. Šeda, Ordinary differential equations, Praha, 1985. (In Slovak.) (1985)
4. S. G. Krein M. I. Khazan, Differential equations in a Banach space, Mathem. analysis Vol. 21, Itogi Nauki i Tekhniky, Akad. Nauk SSSR, Vsesojuz. Inst. Nauki i Tekh, Informatsii, Moscow, 1983, pp. 130-264. (1983) MR0736523
5. V. V. Vasil'ev S. G. Krejn S. I. Piskarev, Pologruppy operatorov, kosinus operator-funkcii i linejnye differencial'nye uravnenija, Itogi Nauki i Tekhniki, Matematiceskij analiz T. 28, Moskva, 1990, pp. 87-203. (1990)
6. B. Rzepecki, 10.1017/S0004972700002161, Bull. Austral. Soc. 30 no. 3 (1984), 449-456. (1984) Zbl0561.34042MR0766802DOI10.1017/S0004972700002161
7. B. Rzepecki, An existence theorem for bounded solutions of differential equations in Banach spaces, Rend. Sem. Mat. Univ. Padova 13 (1985), 89-94. (1985) Zbl0586.34052MR0799899

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.