On the structure of the set solutions of a class of paratingent equation with delayed argument

Öz

In this paper we will study the main properties of the set solutions of the paratingent equation (type differential inclusion) with delayed argument of the form: (Ptx)(t) ⊂ F([x]t ) for t ≥ 0 with the initial condition: x(t) =z (t) for t ≤ 0. We will be interested  particularly in the topological properties of emission and zone of emission.

Anahtar Kelimeler

• Attainable set,convex,delayed argument,differential inclusion,emission,paratingent

Kaynakça

1. J. P. Aubin, A. Cellina; Differential inclusions, Springer-Verlag, 1984.
2. C. Berge; Espaces topologiques, fonctions set-valueds, Dunod, Paris, 1966.
3. L. Boudjenah; Existence of the solutions of the paratingent equation with delayed argument. Electron. J. Diff. Eqns., Vol. 2005, No.14, 1-8, 2005.
4. L Boudjenah; On the properties of the set solutions of a class of paratingent equation with delay. British Journal of Mathematics & Computer Science 4 (14): 1999-2003, 2014.
5. E. Campu; Equations diff´erentielles au paratingent `a retardement, dans les espaces de Banach. Th´eoreme d’existence des solutions. Rev. Roum. Math. Pures Appl. 20, 631-657 ,1975.
6. E, Campu ; Approximation des solutions des ´equations diff´erentielles au paratingent `a retardement, dans les espaces de Banach. Rev. Roum. Math. Pures Appl. 25, 509-518,1980.
7. C. Castaing, A. G. Ibrahim; Functional differential inclusions on closed sets in Banach spaces. Adv. Math. Eco. 2, 21-39, 2000.
8. K. Deimling; Multivalued differential equations, De Gruyter Ser. Nonlinear Anal. Appl. 1, Walter de Gruyter, Berlin, New York, 1992.

9. H.Frankowska, F.Rampazzo ; Filippov’s and Filippov-Wasewski theorems on closed domains. J. Diff. Equ, 161, 449-478, 2000.
10. G. Haddad; Monotone trajectories of differential inclusions and functional differential inclusions with memory, Israel J. Math. 39(1-2), 83-100. 1982.
11. G. Haddad; Monotone viable trajectories for functional differential inclusions with memory. J. Diff. Eq. 42, 1-24, 1981.
12. G. Haddad; Functional viability theorems for functional differential inclusions with memory. Ann. Inst. Henri Poincar´e I (3), 179-204, 1984.
13. G. Haddad, J. M. Lasry; Periodic solutions of functional differential inclusions and fixed points of -selectionable correspondences. J. Math. Anal. Appl. 96, 295-312, 1993.
14. M. Hukuhara; Sur l’application semi-continue dont la valeur est un compact convexe. RIMS-11, Res. Inst. Math. Sci. Kyoto. Univ. 941-945, 1963.
15. M. Kamenskii, V.Obukhovskii, P. Zecca; Condensing multivalued maps and semilinear differential inclusions in Banach spaces, De Gruyter Ser. Nonlinear Anal. Appl. 7, Walter de Gruyter, Berlin, New York, 2001.
16. M. Kisielewicz; Differential Inclusions and Optimal Control. Kluwer, Dordrecht, The Netherlands, 1991.
17. B. Kryzowa; Equation au paratingenta argument retard´e. Ann. Univ. Marie Curie-Sklodwska. Sectio A, 17, 7-18, 1965. B. Kryzowa; Sur les familles de solutions des ´equation au paratingenta argument retarde. Ann. Univ. Marie Curie-Sklodwska. Sectio A, 17, 19-24, 1965.
18. A. D. Myshkis; General theory of differential equations with delayed argument, Uspehi Matem. Nauk 4, no. 5, 1949, 99-141 (in Russian)
19. S. Raczynski; Differential inclusions in system simulation. Trans. Society for Computer Simulation, Vol. 13. No. 1, 47-54. 1996.
20. S. Raczynski; A market model: uncertainty and reachable sets. Int. J. Simul. Multisci. Des. Optim. 6, A2, 2015.
21. S. Raczynski; Creating galaxies on a PC. Simulation, Vol 74, No. 3, 161-166, 2000.
22. G. V. Smirnov; Introduction to the theory of differential inclusions, Amer. Math. Soc., Providence, R.I. 2002.
23. E.D Sontag; An infinite-time relaxation theorem for differential inclusions. in Proc. Amer.Math. Soc., V 131, N◦ 2, 487-499, 2001.
24. E.D Sontag ; A relaxation theorem for differential inclusions with applications to stability properties. in Mathematical Theory of Networks and Systems (D. Gilliam and J. Rosenthal, eds.), 12 pages, August 2002, Electronic Proceedings of MTNS-2002.
25. A. Syam; Contribution a letude des inclusion differentielles. Doctorat Thesis, Universite Montpellier II, 1993.
26. A. Turowicz; Sur les trajectoires et les quasitrajectoires des systemes de commande nonlineaires. Bull. Acad. Polon. Ser. Sci. Math., Astr., Phys, Vol 10, 529-531, 1963.
27. T. Wasewski; Sur une generalisation de la notion des solutions dune equation au contingent. Bull. Acad. Pol. Sci. Ser. Math. Astronom. Physi. 10 No. 1. Warszawa, 1962.
28. T. Wasewski; Sur les syst`emes de commande non lineaires dont le contredomaine n’est pas forcement convexe. Bull. Acad. Pol. Sci. Ser. Math. Astronom. Physi. 10 No. 1. Warswa. 1962.
29. T.Wazewski; Sur un systeme de commande dont les trajectoires coincident avec les quasitrajectoires d’un syst`eme de commande donn´e. Bull. Acad. Pol. Sci. Ser. Math., Astronom., Physi. 11, N◦ 3. Warszawa 1963.
30. W. Zygmunt; On a certain pratingent equation with a deviated argument. Ann. Univ. Marie. Curie-Sklodwska. Lublin. Polonia. 18, 14, 127-135, 1974.
31. W. Zygmunt; On some properties of a certain family of solutions of the paratingent equation. Ann. Univ.Marie Curie-Sklodowska, Lublin. Polonia. Sect. A -28, 136-141,1976.