## Topological Methods in Nonlinear Analysis

### On the structure of the solution set of abstract inclusions with infinite delay in a Banach space

Lahcene Guedda

#### Abstract

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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 567-595.

Dates
First available in Project Euclid: 21 December 2016

https://projecteuclid.org/euclid.tmna/1482289230

Digital Object Identifier
doi:10.12775/TMNA.2016.060

Mathematical Reviews number (MathSciNet)
MR3642774

Zentralblatt MATH identifier
1365.34133

#### Citation

Guedda, Lahcene. On the structure of the solution set of abstract inclusions with infinite delay in a Banach space. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 567--595. doi:10.12775/TMNA.2016.060. https://projecteuclid.org/euclid.tmna/1482289230

#### References

• R.R. Akhmerov, M.I. Kamenskĭ, A.S. Potapov, B.N. Rodkina and B.N. Sadovskĭ, 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 functional-differential equations, Arch. Ration. Mech. Anal. 55 (1974), 275–299.
• G. Conti, V.V. Obukhovskiĭ 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. Kamenskiĭ, A unified topological point of view for integro-differential 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 $L^1(\mu, X)$, 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. Obukhovskiĭ, 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. Obukhovskiĭ, 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).
• ––––, 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. Kamenskiĭ, V.V. Obukhovskiĭ 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.