LECTURE NOTES

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prof, dr hab. Jerzy Motyl. Centrum Badan Nieliniowych im. JuJiusza Schaudera. Um'wersytet Mikolaja Kopernika ul. Chopina 12/18, 87-100 Torun. Redakcja: tel.
LECTURE NOTES IN NONLINEAR ANALYSIS

Existence and Structure of Solution Sets for Impulsive Différent!al Inclusions: a Survey

Smail Djebali Lech Gôrniewicz Abdelghani Ouahab

Juliusz Schauder University Centre for Nonlinear Studies Nicolaus Copernicus University ISSN 2082-4335

Juliusz Schauder University Centre for Nonlinear Studies Nicolaus Copernicus University

Existence and Structure of Solution Sets for Impulsive Differential Inclusions: a Survey

Smaïl Djebali Ecole Normale Suppérieure, Kouba, Algeria

Lech Gorniewicz Juliusz Schauder University Centre for Nonlinear Studies Nicolaus Copernicus University of Torurï, Polaiid Kazimierz Wielki University, Bydgoszcz, Poland

Abdelghani Ouahab Sidi-Bel-Abbés University, Algeria

Torun, 2012

ISSN 2082-4335

Recenzenci: prof, dr hab. Andrzej Fryszkowski prof, dr hab. Jerzy Motyl

Centrum Badan Nieliniowych im. JuJiusza Schaudera Um'wersytet Mikolaja Kopernika ul. Chopina 12/18, 87-100 Torun Redakcja: tel. +48 (56) 611 34 28, faks: +48 (56) 622 89 79 e-mail: [email protected] http://www.cbn.ncu.pl Wydanie pierwsze. Naklad 200 egz.

Dedicated to Professor Francesco S. De Blasi on thé occasion of his 70th birthday

ABSTRACT

In this survey paper. we présent some existence results of mild solutions and study thé topological structure of solution sets for first-order impulsive semilinear differential inclusions with initial value and periodic boundary conditions. Under various assumptions on thé nonlinear terra, we présent several existence results for thé Cauchy problem. We appeal to thé topological fixed point theory as well as to some results and properties from multi-valued analysis, functional analysis, and measure of noncompactness. Further to thé compactness of thé solution sets, we prove some géométrie properties about thé structure of thé solution sets such as AR, R$, contractibility, and acyclicity, corresponding to Aronszajn-Browder-Gupta type results. This is achieved by using some éléments from algebraic topology and homology. Regarding thé periodic boundary conditions, thé problem is formulated as a fixed point problem either for an intégral operator or for a Poincaré translation operator. In particular, one existence resuit relies on a new nonlinear alternative for compact u.s.c. maps defined in infinite-dimensional Banach spaces. Then, we investigate thé topological structure of thé solution sets. A continuons version of Filippov's theorem is provided and thé continuous dependence of solutions on parameters in thé both convex and thé nonconvex cases are proved. More generally, a class of impulsive functional differential inclusions is considered and thé same theory is developed. Finally, we consider thé question of existence and thé structure of solution sets for firstorder impulsive differential inclusions in Fréchet space settings and thé initial and terminal problems are considered. The results on thé géométrie structure of thé solution sets are obtained via thé method of thé inverse System limit of some Banach spaces. 2010 Mathematics Subject Classifications. 34A37, 34A60, 34K30, 34K45, 47D60, 47H09, 47H10. Key words and phrases. Impulsive functional differential inclusions, mild solution, periodic problem, infinity interval, Filippov's theorem, fixed point, solution set, compactness, AR, R$, contractibility, acyclicity.