Differential Inclusions - the Theory Initiated by Cracow Mathematical School

• Autorzy: Lech Górniewicz
• Języki publikacji: EN

Abstrakty

EN

In that paper, Wazewski demonstrated that each problem of controlling ordinary differential equations of the first order can be articulated with orientor equation terms. That observation served as an essential stimulus to study orientor differential equations and consequently, it contributed to the introduction of the new term, still valid, and that is "differential inclusions". We shall return to the subject of the relations to the control theory in the forthcoming passages of the present lecture. The theory of differential inclusions is located within the mainstream of non-linear analysis - or to put it more precisely - multi-valued analysis. This theory is intensively developed especially in the countries such as France, Germany, Russia, Italy, Canada and USA. In Poland, there is a large group of mathematicians working on these issues. That group is mainly located in the following centres: Gdansk, Torun, Warszawa and Zielona Góra. Moreover, the references listed below - because of the nature of the lecture - are limited to the works by Professor Andrzej Lasota exclusively relating directly or indirectly to the theory of differential inclusions. Rich literature on the subject can be found in the references in the particular monographs. (fragment of text)

Słowa kluczowe

EN

Mathematics; Differential equations

PL

Matematyka; Równania różniczkowe

• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2011
• Tom: 25
• Strony: 7--25

Twórcy

autor

Lech Górniewicz

• Nicolaus Copernicus University in Toruń, Poland

Bibliografia

• Andres J., Górniewicz L., Topological Fixed Point Principles for Boundary Value Problems, Kluwer Academic Publishers, 2003.
• Aubin J.P., Cellina A., Differential Inclusions, Springer-Verlag, Berlin, 1984.
• Aubin J.P., Frankowska H., Set-Valued Analysis, Birkhäuser, Basel, 1990.
• Cellina A., Lasota A., A new approach to the definition of topological degree for multivalued mappings, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 8 (1969), 434-440.
• Chow S.N., Lasota A., On boundary value problems for ordinary differential equations, J. Diff. Equations 14 (1973), 326-337.
• Deimling K., Multivalued Differential Equations, W. De Gruyter, Berlin, 1992.
• Fryszkowski A., Fixed Point Theory for Decomposable Sets, Kluwer Academic Publisher, 2004.
• Górniewicz L., Topological Fixed Point Theory of Multivalued Mappings, Springer-Verlag, Berlin, 2006.
• Kamenski ̆ı M.I., Obukovski ̆ı V.V., Zecca P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, W. De Gruyter, Berlin, 2001.
• Kisielewicz M., Differential Inclusions and Optimal Control, Polish Sc. Publishers and Kluwer Academic Publishers, Dordrecht, 1991.
• Lasota A., Une generalisation du premier theoreme de Fredholm et ses applications a la theorie des equations differentielles ordinarires, Ann. Polon. Math. 18 (1966), 65-77.
• Lasota A., Contingent equations and boundary value problems, reprint (1967), 257-266.
• Lasota A., Applications of generalized functions to contingent equations and control theory, in: Inst. Dynamics Appl. Math. Lecture vol. 51, Univ. of Maryland, 1970-1971, pp. 41-52.
• Lasota A., On the existence and uniqueness of solutions of a multipoint boundary value problem, Ann. Polon. Math. 38 (1980), 205-310.
• Lasota A., Myjak J., Attractors of multifunctions, Bull. Polish Acad. Sci. Math. 48 (2000), 319-334.
• Lasota A., Olech Cz., On the closedness of the set of trajectories of a control system, Bull. Acad. Polon. Sci. 214 (1966), 615-621.
• Lasota A., Olech Cz., An optimal solution of Nicoletti's boundary value problem, Ann. Polon. Math. 18 (1966), 131-139.
• Lasota A., Opial Z., An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. 13 (1965), 781-786.
• Lasota A., Opial Z., Sur la dependance conitue des solutions des equations differentielles ordinaries de leurs secondo members et des coditions aux limites, Ann. Polon. Math. 19 (1967), 13-36.
• Lasota A., Opial Z., On the existence and uniquenness of solutions of a bondary value problem for an ordinary second order differentia equations, Coll. Math. 18 (1967), 1-5.
• Lasota A., Opial Z., Fixed point theorem for multivalued mappings and optimal control problems, Bull. Acad. Polon. Sci. 16 (1968), 645-649.
• Tolstonogov A.A., Differential Inclusions in Banach Spaces (in Russsian), Sc. Acad. of Sciences, Siberian Branch, Novosibirsk, 1986.