On the structure of the solution set of abstract inclusions with infinite delay in a Banach space
Keywords
Abstract inclusion, functional differential inclusion, infinite delay, measures of noncompactness and condensing mappings, $R_\delta$-setAbstract
In this paper we study the topological structure of the solution set of abstract inclusions, not necessarily linear, with infinite delay on a Banach space defined axiomatically. By using the techniques of the theory of condensing maps and multivalued analysis tools, we prove that the solution set is a compact $R_\delta$-set. Our approach makes possible to give a unified scheme in the investigation of the structure of the solution set of certain classes of differential inclusions with infinite delay.References
R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, B.N. Rodkina and B.N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhäuser, 1992.
N. Aronszajn, Le correspondant topologique de l’unicité dans la théorie des équations différentielles, Ann Math. 43 (1942), 730–738.
J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980.
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Bucharest–Noordhoff, Leyden, 1976.
D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108 (1998), 109–138.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science & Business Media, 2010.
T. Cardinali and P. Rubbioni, On the existence of mild solutions of semilinear evolution differential inclusions, J. Math. Anal. Appl. 308 (2005), 620–635.
M. Cichoń and I. Kubiaczyk, Some remarks on the structure of the solution set for differential inclusions in Banach spaces, J. Math. Anal. Appl. 233 (1999), 597–606.
B.D. Coleman and D.R. Owen, On the initial value problem for a class of functionaldifferential equations, Arch. Ration. Mech. Anal. 55 (1974), 275–299.
G. Conti, V.V. Obukhovskiı̆ and P. Zecca, On the topological structure of the solution set for a semilinear functional-differential inclusion in a Banach space, Banach Center Publications 35 (1996), 159–169.
J.F. Couchouron and M. Kamenskiı̆, A unified topological point of view for integrodifferential inclusions, Lecture Notes in Nonlinear Anal. 2 (1998), 123–137.
F.S. De Blasi and J. Myjak, On the solution sets for differential inclusions, Bull. Polon. Acad. Sci. 33 (1985), 17–23.
K. Deimling, Multivalued Differential Equations, de Gruyter, Berlin, 1992.
K. Deimling and M.R. Mohana Rao, On solution sets of multivalued differential equations, Appl. Anal. 30 (1988), 129–135.
J. Diestel, W.M. Ruess and W. Schachermayer, On weak compactness in L1 (µ, X), Proc. Amer. Math. Soc. 118 (1993), 447–453.
S. Djebali, L. Górniewicz and A. Ouahab, Solution sets for differential equations and inclusions, Walter de Gruyter, 2012.
C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, On some properties of semilinear functional differential inclusions in abstract spaces, J. Concr. Appl. Math. 4 (2006), no. 2, 183–214.
C. Gori, V.V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay, Nonlinear Anal. 51 (2002), 765–782.
L. Górniewicz, Topological fixed point theory of multivalued mappings, Springer, 2006 (second edition).
L. Górniewicz, Topological Structure of Solution Sets: Current Results, Arch. Math. (Brno) 36 (2000), 343–382.
J.K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978), 11–41.
C.J. Himmelberg and F.S. Van Vleck, A note on the solution sets of differential inclusions, Rocky Mountain J. Math. 12 (1982), 621–625.
Y. Hino, S. Murakami and T. Naito, Functional-differential equations with infinite delay, Springer–Verlag, 1991.
Y. Hino, T. Naito, N.V. Minh and J.S. Shin, Almost periodic solutions of differential equations in Banach spaces, Taylor & Francis, 2002.
M.I. Kamenskii, V.V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter & Co., 2001.
F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations. 37 (1980), 141–183.
N.S. Papageorgiou, On the solution set of differential inclusions in Banach space, Appl. Anal. 25 (1987), 319–329.
K. Schumacher, Existence and continuous dependence for functional-differential equations with unbounded delay, Arch. Rational Mech. Anal. 67 (1978), 315–335.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
