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Banach Center Publications

1996 | 35 | 1 | 159-169
Tytuł artykułu

On the topological structure of the solution set for a semilinear ffunctional-differential inclusion in a Banach space

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper we show that the set of all mild solutions of the Cauchy problem for a functional-differential inclusion in a separable Banach space E of the form x'(t) ∈ A(t)x(t) + F(t,x_t) is an $R_δ$-set. Here {A(t)} is a family of linear operators and F is a Carathéodory type multifunction. We use the existence result proved by V. V. Obukhovskiĭ [22] and extend theorems on the structure of solutions sets obtained by N. S. Papageorgiou [23] and Ya. I. Umanskiĭ [32].
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
159-169
Opis fizyczny
Daty
wydano
1996
Twórcy
autor
• Istituto di Matematica Applicata, Facoltà di Architettura, Università di Firenze, Italy
autor
• Department of Physics and Mathematics, Voronezh Pedagogical University, Russia
autor
• Dipartimento di Sistemi e Informatica, Università di Firenze, via S. Marta 3, 50139 Firenze, Italy
Bibliografia
• [1] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser Verlag, Basel-Boston-Berlin, 1992.
• [2] G. Anichini, G. Conti and P. Zecca, Using solution sets for solving boundary value problems for ordinary differential equations, Nonlinear Anal., Theory, Meth. and Appl. 17 No. 5 (1991), 465-472.
• [3] G. Anichini and P. Zecca, Multivalued differential equations in Banach spaces, an application to control theory, J. Optim. Theory and Appl. 21 No. 4 (1977), 477-486.
• [4] Yu. G. Borisovich, B. D. Gelman, A. D. Myshkis and V. V. Obukhovskiĭ, Introduction to the Theory of Multivalued Maps, Voronezh Univ. Press, Voronezh, 1986 (in Russian).
• [5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lect. Notes in Math. 580, Springer, Berlin, 1977.
• [6] S. N. Chow and J. D. Schuur, Fundamental theory of contingent differential equations in Banach spaces, Trans. Amer. Math. Soc. 179 (1973), 133-144.
• [7] L. J. Davy, Properties of the solution set of a generalized differential equation, Bull. Australian Math. Soc. 6 (1972), 379-398.
• [8] F. De Blasi, Existence and stability of solutions for autonomous multivalued differential equations in Banach spaces, Rend. Acad. Naz. Lincei, Serie VII, 60 (1976), 767-774.
• [9] F. De Blasi and J. Myjak, On the solution sets for differential inclusions, Bull. Pol. Acad. Sci. 33 (1985), 17-23.
• [10] J. Diestel, Remarks on weak compactness in $L_1(μ_1,X),$ Glasgow Math. J. 18, No. 1 (1977), 87-91.
• [11] G. Dragoni, J. Macki, P. Nistri and P. Zecca, Solution sets of differential equations in abstract spaces, Pitman Res. Notes in Math., Longman, to appear.
• [12] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976.
• [13] C. Himmelberg and F. Van Vleck, A note on the solution sets for differential inclusions, Rocky Mountain J. Math. 12 (1982), 621-625.
• [14] D. M. Hyman, On decreasing sequences of compact absolute retracts, Fund. Math. 64 (1969), 91-97.
• [15] M. I. Kamenskiĭ, P. Nistri, V. V. Obukhovskiĭ and P. Zecca, Optimal feedback control for a semilinear evolution equation, J. Optim. Theory and Appl. 82 No. 3 (1994), 503-517.
• [16] M. Kisiliewicz, Multivalued differential equations in separable Banach spaces, J. Optim. Th. Appl. 37 (1982), 231-249.
• [17] S. G. Krein, Linear Differential Equations in Banach Spaces, Amer. Math. Soc., Providence, 1971.
• [18] J. M. Lasry and R. Robert, Acyclicité de l'ensemble des solutions de certaines équations fonctionnelles, C. R. Acad. Sci. Paris 282 (1976), 1283-1286.
• [19] R. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976.
• [20] A. M. Muhsinov, On differential inclusions in Banach spaces, Soviet Math. Dokl. 15 (1974), 1122-1125.
• [21] P. Nistri, V. V. Obukhovskiĭ and P. Zecca, On the solvability of systems of inclusions involving noncompact operators, Trans. Amer. Math. Soc. 342, No. 2 (1994), 543-562.
• [22] V. V. Obukhovskiĭ, Semilinear functional differential inclusions in a Banach space and controlled parabolic systems, Soviet J. Automat. Inform. Sci. 24, No. 3 (1991), 71-79 (1992).
• [23] N. S. Papageorgiou, On multivalued evolution equations and differential inclusions in Banach spaces, Comment. Math. Univ. Sancti Pauli 36, No. 1 (1987), 21-39.
• [24] N. S. Papageorgiou, On the solution set of differential inclusions in Banach space, Appl. Anal. 25 (1987), 319-329.
• [25] N. H. Paovel and J. Vrabie, On the solution set of differential inclusions with state constraints, Appl. Anal. 31 (1989), 279-289.
• [26] R. M. Sentis, Convergence de solutions d'équations différentielles multivoques, C.R. Acad. Sci. Paris, Série A, 278 (1974), 1623-1626.
• [27] A. A. Tolstonogov, On differential inclusions in Banach spaces and continuous selectors, Dokl. Akad. Nauk SSSR 244 (1979), 1088-1092.
• [28] A. A. Tolstonogov, On properties of solutions of differential inclusions in Banach spaces, Dokl. Akad. Nauk SSSR 248 (1979), 42-46.
• [29] A. A. Tolstonogov, On the structure of the solution set for differential inclusions in a Banach space, Math. Sbornik 46 (1983), 1-15.
• [30] A. A. Tolstonogov, Differential Inclusions in a Banach Space, Nauka, Novosibirsk, 1986 (in Russian).
• [31] A. A. Tolstonogov and Ya. I. Umanskiĭ, On solutions of evolution inclusions II, Sibirsk. Mat. Zh. 33, No. 4 (1992), 163-174 (in Russian).
• [32] Ya. I. Umanskiĭ, On a property of solutions set of differential inclusions in a Banach space, Differ. Uravneniya 28, No. 8 (1992), 1346-1351 (in Russian).
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Bibliografia
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