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Periodic problems for ODEs via multivalued Poincaré operators
Full text linksummary:We shall consider periodic problems for ordinary differential equations of the form where satisfies suitable assumptions. To study the above problem we shall follow an approach based on the topological degree theory. Roughly speaking, if on some ball of , the topological degree of, associated to (), multivalued Poincaré operator turns out to be different from zero, then problem () has solutions. Next by using the multivalued version of the classical Liapunov-Krasnoselskǐ guiding potential method we calculate the topological degree of the Poincaré operator . To do it we associate with a guiding potential which is assumed to be locally Lipschitzean (instead of ) and hence, by using Clarke generalized gradient calculus we are able to prove existence results for (), of the classical type, obtained earlier under the assumption that is . Note that using a technique of the same type (adopting to the random case) we are able to obtain all of above mentioned results for the following random periodic problem: where is a random operator satisfying suitable assumptions. This paper stands a simplification of earlier works of F. S. De Blasi, G. Pianigiani and L. Górniewicz (see: [gor7], [gor8]), where the case of differential inclusions is considered
Solving equations by topological methods
No full textTyt. z nagł.Abstract. In this this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in diffrential equations. Keywords: Lefschetz number, fixed points, CAC-maps, condensing maps, ANR-spaces, fixed point index
Differential inclusions - the theory initiated by Cracow Mathematical School
Full text linkAnnual Lecture dedicated to the memory of Professor Andrzej Lasota
Topological fixed point theory of multivalued mappings
No full textThis volume presents a broad introduction to the topological fixed point theory of multivalued (set-valued) mappings, treating both classical concepts as well as modern techniques. A variety of up-to-date results is described within a unified framework. Topics covered include the basic theory of set-valued mappings with both convex and nonconvex values, approximation and homological methods in the fixed point theory together with a thorough discussion of various index theories for mappings with a topologically complex structure of values, applications to many fields of mathematics, mathematical economics and related subjects, and the fixed point approach to the theory of ordinary differential inclusions. The work emphasises the topological aspect of the theory, and gives special attention to the Lefschetz and Nielsen fixed point theory for acyclic valued mappings with diverse compactness assumptions via graph approximation and the homological approach. Audience: This work will be of interest to researchers and graduate students working in the area of fixed point theory, topology, nonlinear functional analysis, differential inclusions, and applications such as game theory and mathematical economics
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