Periodic problems for ODEs via multivalued Poincaré operators

Lech Górniewicz

Archivum Mathematicum (1998)

  • Volume: 034, Issue: 1, page 93-104
  • ISSN: 0044-8753

Abstract

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We shall consider periodic problems for ordinary differential equations of the form x ' ( t ) = f ( t , x ( t ) ) , x ( 0 ) = x ( a ) , where f : [ 0 , a ] × R n R n satisfies suitable assumptions. To study the above problem we shall follow an approach based on the topological degree theory. Roughly speaking, if on some ball of R n , the topological degree of, associated to (), multivalued Poincaré operator P turns out to be different from zero, then problem () has solutions. Next by using the multivalued version of the classical Liapunov-Krasnoselskǐ guiding potential method we calculate the topological degree of the Poincaré operator  P . To do it we associate with f a guiding potential V which is assumed to be locally Lipschitzean (instead of C 1 ) and hence, by using Clarke generalized gradient calculus we are able to prove existence results for (), of the classical type, obtained earlier under the assumption that V is C 1 . Note that using a technique of the same type (adopting to the random case) we are able to obtain all of above mentioned results for the following random periodic problem: x ' ( ξ , t ) = f ( ξ , t , x ( ξ , t ) ) , x ( ξ , 0 ) = x ( ξ , a ) , where f : Ω × [ 0 , a ] × R n R n is a random operator satisfying suitable assumptions. This paper stands a simplification of earlier works of F. S. De Blasi, G. Pianigiani and L. Górniewicz (see: [gor7], [gor8]), where the case of differential inclusions is considered.

How to cite

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Górniewicz, Lech. "Periodic problems for ODEs via multivalued Poincaré operators." Archivum Mathematicum 034.1 (1998): 93-104. <http://eudml.org/doc/248204>.

@article{Górniewicz1998,
abstract = {We shall consider periodic problems for ordinary differential equations of the form \[ \{\left\lbrace \begin\{array\}\{ll\} x^\{\prime \}(t)= f(t,x(t)),\\ x(0) = x(a), \end\{array\}\right.\} \] where $ f:[0,a] \times R^n \rightarrow R^n$ satisfies suitable assumptions. To study the above problem we shall follow an approach based on the topological degree theory. Roughly speaking, if on some ball of $R^n$, the topological degree of, associated to (), multivalued Poincaré operator $P$ turns out to be different from zero, then problem () has solutions. Next by using the multivalued version of the classical Liapunov-Krasnoselskǐ guiding potential method we calculate the topological degree of the Poincaré operator $P$. To do it we associate with $f$ a guiding potential $V$ which is assumed to be locally Lipschitzean (instead of $C^1$) and hence, by using Clarke generalized gradient calculus we are able to prove existence results for (), of the classical type, obtained earlier under the assumption that $V$ is $C^1$. Note that using a technique of the same type (adopting to the random case) we are able to obtain all of above mentioned results for the following random periodic problem: \[ \{\left\lbrace \begin\{array\}\{ll\} x^\{\prime \}(\xi , t) = f(\xi , t, x(\xi ,t)),\\ x(\xi ,0) = x(\xi , a), \end\{array\}\right.\} \] where $f:\Omega \times [0,a]\times R^n\rightarrow R^n$ is a random operator satisfying suitable assumptions. This paper stands a simplification of earlier works of F. S. De Blasi, G. Pianigiani and L. Górniewicz (see: [gor7], [gor8]), where the case of differential inclusions is considered.},
author = {Górniewicz, Lech},
journal = {Archivum Mathematicum},
keywords = {Periodic processes; topological degree; Poincaré translation operator; periodic processes; topological degree; Poincaré translation operator},
language = {eng},
number = {1},
pages = {93-104},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Periodic problems for ODEs via multivalued Poincaré operators},
url = {http://eudml.org/doc/248204},
volume = {034},
year = {1998},
}

TY - JOUR
AU - Górniewicz, Lech
TI - Periodic problems for ODEs via multivalued Poincaré operators
JO - Archivum Mathematicum
PY - 1998
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 034
IS - 1
SP - 93
EP - 104
AB - We shall consider periodic problems for ordinary differential equations of the form \[ {\left\lbrace \begin{array}{ll} x^{\prime }(t)= f(t,x(t)),\\ x(0) = x(a), \end{array}\right.} \] where $ f:[0,a] \times R^n \rightarrow R^n$ satisfies suitable assumptions. To study the above problem we shall follow an approach based on the topological degree theory. Roughly speaking, if on some ball of $R^n$, the topological degree of, associated to (), multivalued Poincaré operator $P$ turns out to be different from zero, then problem () has solutions. Next by using the multivalued version of the classical Liapunov-Krasnoselskǐ guiding potential method we calculate the topological degree of the Poincaré operator $P$. To do it we associate with $f$ a guiding potential $V$ which is assumed to be locally Lipschitzean (instead of $C^1$) and hence, by using Clarke generalized gradient calculus we are able to prove existence results for (), of the classical type, obtained earlier under the assumption that $V$ is $C^1$. Note that using a technique of the same type (adopting to the random case) we are able to obtain all of above mentioned results for the following random periodic problem: \[ {\left\lbrace \begin{array}{ll} x^{\prime }(\xi , t) = f(\xi , t, x(\xi ,t)),\\ x(\xi ,0) = x(\xi , a), \end{array}\right.} \] where $f:\Omega \times [0,a]\times R^n\rightarrow R^n$ is a random operator satisfying suitable assumptions. This paper stands a simplification of earlier works of F. S. De Blasi, G. Pianigiani and L. Górniewicz (see: [gor7], [gor8]), where the case of differential inclusions is considered.
LA - eng
KW - Periodic processes; topological degree; Poincaré translation operator; periodic processes; topological degree; Poincaré translation operator
UR - http://eudml.org/doc/248204
ER -

References

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