On the solvability of systems of inclusions involving noncompact operators
Authors:
P. Nistri, V. V. Obukhovskiĭ and P. Zecca
Journal:
Trans. Amer. Math. Soc. 342 (1994), 543562
MSC:
Primary 47H15; Secondary 34G20, 34K30, 47H04, 47N20, 47N70, 49J45
MathSciNet review:
1232189
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Abstract: We consider the solvability of a system of setvalued maps in two different cases. In the first one, the map is supposed to be closed graph with convex values and condensing in the second variable and is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, singlevalued map), satisfying a condensivity condition in the first variable. In the second case is as before with compact, not necessarily convex, values and is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.
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 [1]
 Yu. G. Borisovich, B. D. Gel'man, A. D. Myskis, and V. V. Obukhovskiĭ, Topological methods in fixedpoint theory of multivalued mappings, Uspekhi Mat. Nauk 35 (1980), 59126; English transl. in Russian Math. Surveys 35 (1980), 65143. MR 565568 (81e:55004)
 [2]
 , Introduction to the theory of multivalued mappings, Voronezh Univ. Press, Voronezh, (1986); English transl. in J. Soviet Math. 39 (1987), 27722811. MR 928179 (89f:49014)
 [3]
 , Multivalued analysis and operator inclusions, Ser. Sov. Probl. Mat. Noveishie Dostizheniya, vol. 29, Itogi Nauki i Tekhniki, 1986, pp. 151211. (Russian) MR 892745 (88k:58014)
 [4]
 G. Conti, P. Nistri, and P. Zecca, Systems of setvalued equations in Banach spaces, Delay Differential Equations and Dynamical Systems (Proc. Claremont 1990), (S. Busenberg and M. Martelli, eds.), Lecture Notes in Math., vol. 1475, SpringerVerlag, 1991, pp. 98109. MR 1132022 (93e:34087)
 [5]
 , Non convex setvalued systems in Banach spaces, Funkcial. Ekvac. (to appear).
 [6]
 G. Conti and J. Pejsachowicz, Fixed points theorems for multivalued weighted maps, Ann. Mat. Pura Appl. 126 (1980), 319341. MR 612366 (82f:55002)
 [7]
 Do Hong Tan, On continuity of fixed points of multivalued collectively condensing mappings, Indian J. Pure Appl. Math. 15 (1984), 631632. MR 750213 (86b:54049)
 [8]
 L. Gorniewicz, Homological methods in fixed point theory of multivalued mappings, Dissertationes Math. 129 (1976), 171. MR 0394637 (52:15438)
 [9]
 A. Lasota and Z. Opial, An approximation theorem for multivalued mappings, Podstawy Sterowania 1 (1971), 7175. MR 0305336 (46:4466)
 [10]
 I. Massabò, P. Nistri, and J. Pejsachowicz, On the solvability of nonlinear equations in Banach spaces, Fixed Point Theory, (Proc. Sherbrooke, Quebec 1980), (E. Fadell and G. Fournier, eds.), Lecture Notes in Math., vol. 886, SpringerVerlag, 1980, pp. 270289. MR 643012 (83a:47068)
 [11]
 V. V. Obukhovskiĭ, Some fixedpoint principles for multivalued condensing operators, Voronezh. Gos. Univ. Trudy Mat. Fak. 4 (1970), 7079. (Russian)
 [12]
 , On the topological degree for a class of non compact multivalued mappings, Funktsional Anal. 23 (1984), 8293. (Russian)
 [13]
 , On semilinear functional differential inclusions in a Banach space and control systems of a parabolic type, Avtomatika 3 (1991), 7381. (Russian)
 [14]
 V. V. Obukhovskiĭ and E. V. Gorokhov, On the definition of the rotation of a class of compactly restrictible multivalued vector fields, Voronezh. Gos. Univ. Trudy Mat. Fak. 12 (1974), 4554. (Russian)
 [15]
 N. S. Papageorgiou, On multivalued evolution equations and differential inclusions in Banach spaces, Comment. Math. Univ. St. Paul. 36 (1987), 2139. MR 892378 (89f:34020)
 [16]
 B. N. Sadovskiĭ, Limitcompact and condensing operators, Uspekhi Mat. Nauk 27 (1972), no. 1, 81146. (Russian) MR 0428132 (55:1161)
 [17]
 E. U. Tarafdar and H. B. Thompson, On the solvability of nonlinear noncompact operator equations, J. Austral. Math. Soc. Ser. A 43 (1987), 103126. MR 886808 (88g:47123)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412321890
PII:
S 00029947(1994)12321890
Keywords:
Condensing multivalued map,
degree theory,
BorsukUlam condition,
control problem,
periodic solution
Article copyright:
© Copyright 1994
American Mathematical Society
