Periodic problems for ODEs via multivalued Poincaré operators

Lech Górniewicz

Archivum Mathematicum (1998)

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

Abstract

top
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

top

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

top
  1. Andres J., Periodic derivatives of solutions to nonlinear differential equations, Czech. Math. J. 40, 3 (1990), 353–360 (1990) MR1065015
  2. Andres J., Derivo-periodic boundary value problems for nonautonomous ordinary differential equations, Riv. Mat. Pura Appl., 13 (1993), 62–90 (1993) Zbl0792.34016MR1231317
  3. Andres J., On the multivalued Poincaré operator, Topological Methods in Nonlinear Analysis (to appear) MR1646627
  4. Andres J., Górniewicz L., Lewicka M., Partially dissipative periodic processes, Topology in Nonlinear Analysis (K. Geba and L. Górniewicz, eds.) 35 (1996), Banach Center Publications, Warszawa, 109–118 (1996) MR1448430
  5. Andres J., Gabor G., Górniewicz L., Boundary value problems on infinite intervals, Trans. Amer. Math. Soc. (to appear) MR1603870
  6. Bader R., Kryszewski W., Fixed point index for compositions of set-valued maps with proximally ∞ -connected values on arbitrary A N R ’s, Set Valued Analysis 2, 3 (1994), 459–480 (1994) Zbl0846.55001MR1304049
  7. De Blasi F. S., Górniewicz L., Pianigiani G., Topological degree and periodic solutions of differential inclusions, Nonlinear Analysis TMA (to appear) Zbl0936.34009MR1661342
  8. De Blasi F. S., Górniewicz L., Pianigiani G., Random topological degree and differential inclusions, reprint 
  9. Clarke F., Optimization and nonsmooth analysis, John Wiley, New York, 1983 (1983) Zbl0582.49001MR0709590
  10. Górniewicz L., Topological approach to differential inclusions, Topological Methods in Differential Equations and Inclusions (M. Frigon and A. Granas, eds.), NATO ASI Series C 472, Kluwer Academic Publ., 1995, 129–190 (1995) MR1368672
  11. Górniewicz L., Granas A., Kryszewski W., On the homotopy method in the fixed point index theory of multivalued mappings of compact A N R ’s, J. Math. Anal. Appl. 161 (1991), 457–473 (1991) MR1132121
  12. Górniewicz L., Plaskacz S., Periodic solutions of differential inclusions in R n , Boll. Un. Mat. Ital. 7 (7-A 1993), 409–420 (1993) MR1249117
  13. Krasnoselskiǐ M. A., Zabreiko P., Geometric methods of nonlinear analysis, Springer-Verlag, Berlin, 1975 (1975) MR0500310
  14. Mawhin J., Topological degree methods in nonlinear boundary value problems, Regional Conference in Mathematics, vol. 40, American Mathematical Society, Providence, R. I., 1979 (1979) Zbl0414.34025MR0525202
  15. Plaskacz S., Periodic solutions of differential inclusions on compact subsets of R n , J. Math. Anal. Appl. 148 (1990), 202–212 (1990) MR1052055
  16. Yorke J. A., Extending Liapunov’s second method to non Lipschitz Liapunov functions, Bull. Amer. Math. Soc. 2 (1968), 322–325 (1968) Zbl0179.13401MR0224925
  17. Yorke J. A., Differential inequalities and non Lipschitz scalar functions, Math. Systems Theory 4 (1970), 140–151 (1970) Zbl0231.34047MR0268476
  18. Yoshizawa T., Stability theory by Liapunov’s second method, Math. Soc. Japan, Tokyo, 1996 (1996) MR0208086

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.