# 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

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topTumajer, 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- J. L. Massera J. J. Schäffler, Linear differential equations and function spaces, Academic press, New York and London, 1966. (1966) MR0212324
- 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
- M. Greguš M. Švec V. Šeda, Ordinary differential equations, Praha, 1985. (In Slovak.) (1985)
- 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
- 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)
- B. Rzepecki, 10.1017/S0004972700002161, Bull. Austral. Soc. 30 no. 3 (1984), 449-456. (1984) Zbl0561.34042MR0766802DOI10.1017/S0004972700002161
- 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

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