# Handbook of Topological Fixed Point Theory

## byRobert F. BrownEditorMassimo Furi, L. Gorniewicz

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Fixed point theory concerns itself with a very simple, and basic, mathematical setting. For a functionf that has a setX as bothdomain and range, a ?xed point off isa pointx ofX for whichf(x)=x. Two fundamental theorems concerning ?xed points are those of Banach and of Brouwer. In Banach's theorem, X is a complete metric space with metricd andf:X?X is required to be a contraction, that is, there must existL<_20_120_such20_thatd28_f28_x29_2c_f28_y29_29_3f_ld28_x2c_y29_20_for20_allx2c_y3f_x.20_theconclusion20_is20_thatf20_has20_a20_3f_xed20_point2c_20_in20_fact20_exactly20_one20_of20_them.20_brouwer27_27_stheorem20_requiresx20_to20_betheclosed20_unit20_ball20_in20_a20_euclidean20_space20_and20_f3a_x3f_x20_to20_be20_a20_map2c_20_that20_is2c_20_a20_continuous20_function.20_again20_we20_can20_conclude20_that20_f20_has20_a20_3f_xed20_point.20_but20_in20_this20_case20_the20_set20_of3f_xed20_points20_need20_not20_be20_a20_single20_point2c_20_in20_fact20_every20_closed20_nonempty20_subset20_of20_the20_unit20_ball20_is20_the20_3f_xed20_point20_set20_for20_some20_map.20_themetriconx20_in20_banach27_27_stheorem20_is20_used20_in20_the20_crucialhypothesis20_about20_the20_function2c_20_that20_it20_is20_a20_contraction.20_the20_unit20_ball20_in20_euclidean20_space20_is20_also20_metric2c_20_and20_the20_metric20_topology20_determines20_the20_continuity20_of20_the20_function2c_20_but20_the20_focus20_of20_brouwer27_27_s20_theorem20_is20_on20_topological20_characteristics20_of20_the20_unit20_ball2c_20_in20_particular20_that20_it20_is20_a20_contractible20_3f_nite20_polyhedron.20_the20_theorems20_of20_banach20_and20_brouwer20_illustrate20_the20_di3f_erence20_between20_the20_two20_principal20_branches20_of20_3f_xed20_point20_theory3a_20_metric20_3f_xed20_point20_theory20_and20_topological20_3f_xed20_point20_theory. 1="" such="" _thatd28_f28_x29_2c_f28_y29_29_3f_ld28_x2c_y29_="" for="" _allx2c_y3f_x.="" theconclusion="" is="" thatf="" has="" a="" ed="" _point2c_="" in="" fact="" exactly="" one="" of="" them.="" _brouwer27_27_stheorem="" requiresx="" to="" betheclosed="" unit="" ball="" euclidean="" space="" and="" _f3a_x3f_x="" be="" _map2c_="" that="" _is2c_="" continuous="" function.="" again="" we="" can="" conclude="" f="" point.="" but="" this="" case="" the="" set="" _of3f_xed="" points="" need="" not="" single="" every="" closed="" nonempty="" subset="" point="" some="" map.="" themetriconx="" _banach27_27_stheorem="" used="" crucialhypothesis="" about="" _function2c_="" it="" contraction.="" also="" _metric2c_="" metric="" topology="" determines="" continuity="" focus="" _brouwer27_27_s="" theorem="" on="" topological="" characteristics="" _ball2c_="" particular="" contractible="" ite="" polyhedron.="" theorems="" banach="" brouwer="" illustrate="" _di3f_erence="" between="" two="" principal="" branches="" _theory3a_="" theory="">
Title:Handbook of Topological Fixed Point TheoryFormat:HardcoverDimensions:972 pagesPublished:April 13, 2006Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:1402032218

ISBN - 13:9781402032219

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Table of Contents

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 Poincaré 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.