Lech Górniewicz

Topological fixed point principles for boundary value problems

Preface. Scheme for the relationship of single sections. I: Theoretical Background. I.1. Structure of locally convex spaces. I.2. ANR-spaces and AR-spaces. I.3. Multivalued mappings and their selections. I.4. Admissible mappings. I.5. Special classes of admissible mappings. I.6. Lefschetz fixed point theorem for admissible mappings. I.7. Lefschetz fixed point theorem for condensing mappings. I.8. Fixed point index and topological degree for admissible maps in locally convex spaces. I.9. Noncompact case. I.10. Nielsen number. I.11. Nielsen number: Noncompact case. I.12. Remarks and comments. II: General Principles. II.1 Topological structure of fixed point sets: Aronszajn Browder Gupta-type results. II.2. Topological structure of fixed point sets: inverse limit method. II.3. Topological dimension of fixed point sets. II.4. Topological essentiality. II.5. Relative theories of Lefschetz and Nielsen. II.6. Periodic point principles. II.7. Fixed point index for condensing maps. II.8. Approximation method for the fixed point theory of multivalued mappings. II.9. Topological degree defined by means of approximation methods. II.10. Continuation principles based on a fixed point index. II.11. Continuation principles based on a coincidence index. II.12. Remarks and comments. III: Application to Differential Equations and Inclusions. III.1. Topological approach to differential equations and inclusions. III.2. Topological structure of solution sets: initial value problems. III.3. Topological structure of solution sets: boundary value problems. III.4. Poincare operators. III.5. Existence results. III.6. Multiplicity results. III.7. Wazewski-type results. III.8. Bounding and guiding functions approach. III.9. Infinitely many subharmonics. III.10. Almost-periodic problems. III.11. Some further applications. III.13.Remarks and comments. Appendices. A.1. Almost-periodic single-valued and multivalued functions. A.2. Derivo-periodic single-valued and multivalued functions. A.3. Fractals and multivalued fractals. References. Index.

Boundary value problems on infinite intervals

We present two methods, both based on topological ideas, to the solvability of boundary value problems for differential equations and inclusions on infinite intervals. In the first one, related to the rich family of asymptotic problems, we generalize and extend some statements due to the Florence group of mathematicians Anichini, Cecchi, Conti, Furi, Marini, Pera, and Zecca. Thus, their conclusions for differential systems are as well true for inclusions; all under weaker assumptions (for example, the convexity restrictions in the Schauder linearization device can be avoided). In the second, dealing with the existence of bounded solutions on the positive ray, we follow and develop the ideas of Andres, Górniewicz, and Lewicka, who considered periodic problems. A special case of these results was previously announced by Andres. Besides that, the structure of solution sets is investigated. The case of l.s.c. right hand sides of differential inclusions and the implicit differential equations are also considered. The large list of references also includes some where different techniques (like the Conley index approach) have been applied for the same goal, allowing us to envision the full range of recent attacks on the problem stated in the title.

Handbook of topological fixed point theory

Preface. I. Homological Methods in Fixed Point Theory. 1. Coincidence theory. 2. On the Lefschetz fixed point theorem. 3. Linearizations for maps of nilmanifolds and solvmanifolds. 4. Homotopy minimal periods. 5. Perodic points and braid theory. 6. Fixed point theory of multivalued weighted maps. 7. Fixed point theory for homogeneous spaces - a brief survey. II. Equivariant Fixed Point Theory. 8. A note on equivariant fixed point theory. 9. Equivariant degree. 10. Bifurcations of solutions of SO (2)-symmetric nonlinear problems with variational structure. III. Nielsen Theory. 11. Nielsen root theory. 12. More about Nielsen theories and their applications. 13. Algebraic techniques for calculating the Nielsen number on hyperbolic surfaces. 14. Fibre techniques in Nielsen theory calculations. 15. Wecken theorem for fixed and periodic points. 16. A primer of Nielsen fixed point theory. 17. Nielsen fixed point theory on surfaces. 18. Relative Nielsen theory. IV. Applications. 19. Applicable fixed point principles. 20. The fixed point index of the Poincare translation. 21. On the existence of equilibria and fixed points of maps under constraints. 22. Topological fixed point theory and nonlinear differential equations. 23. Fixed point results based on the Wazeski method.

Controllability of semilinear differential equations and inclusions via semigroup theory in banach spaces

Control problems appear in many branches of physics and technical science. In this paper we investigate the controllability of semilinear differential equations and inclusions via the semigroup theory in Banach spaces. All results are obtained by using fixed point theorems both for single and multivalued mappings.

Existence and Controllability Results for First-and Second-Order Functional Semilinear Differential Inclusions with Nonlocal Conditions

In this paper, we prove existence and controllability results for first- and second-order semilinear differential inclusions in Banach spaces with nonlocal conditions.

Solution sets for differential equations and inclusions

This monograph gives a systematic presentation of classical and recent results obtained in the last couple of years. It comprehensively describes the methods concerning the topological structure of fixed point sets and solution sets for differential equations and inclusions. Many of the basic techniques and results recently developed about this theory are presented, as well as the literature that is disseminated and scattered in several papers of pioneering researchers who developed the functional analytic framework of this field over the past few decades. Several examples of applications relating to initial and boundary value problems are discussed in detail. The book is intended to advanced graduate researchers and instructors active in research areas with interests in topological properties of fixed point mappings and applications; it also aims to provide students with the necessary understanding of the subject with no deep background material needed. This monograph fills the vacuum in the literature regarding the topological structure of fixed point sets and its applications.

On the Banach Contraction Principle for Multivalued Mappings

We give a survey of recent results concerning the Banach contraction principle for multivalued mappings. Nevertheless, this survey contains also some new so far unpublished results. The following main problems are concerned: (i) existence of fixed points (ii) topological structure of the set of fixed points (iii) generalized essentiality. Some applications, mainly to differential inclusions, and open problems are presented as well

Approximation and fixed points for compositions of Rδ-maps

Abstract A set-valued upper semi-continuous map is called an Rβ-map if each of its values is an Rδ-set (we recall that an Rδ-set is a space that can be represented as the intersection of a decreasing sequence of compact AR-spaces). We prove that a compact set-valued map of an AR-space into itself has a fixed point provided it can be factorized by an arbitrary finite number of Rδ-maps through ANR-spaces. This fact is a consequence of a more general result which is the main goal of this paper. The proof relies on a refinement of the approximation technique and does not make use of homological tools.

A generalized Nielsen number and multiplicity results for differential inclusions

Abstract The Nielsen number is defined for a rather general class of multivalued maps on compact connected ANRs, including, e.g., admissible maps (in the sense of Gorniewicz (1976); compare also Gorniewicz (1995)) on tori. Since the Poincare maps generated by the Marchaud vector fields are of this type (see (Andres, 1997)), we can obtain in such a way multiplicity results for differential inclusions. More precisely, the nontrivial Nielsen number gives a lower estimate of coincidence points (in particular, fixed points) corresponding to the desired solutions.

Controllability Results for Nondensely Defined Semilinear Functional Differential Equations

In this paper we investigate the controllability of first order semilinear functional and neutral functional differential equations in Banach spaces.