CONTENTSIntroduction Chapter I. Approximative methods in the fixed-point theory of multi-valued maps ??1.1. Multi-valued maps and single-valued approximations ??1.2. The rotation of multi-valued vector fields with convex images and fixed-point theorems ??1.3. Obstruction theory and single-valued approximations of multi-valued maps ??1.4. Guide to the literature in Chapter I Chapter II. Homological methods in the fixed-point theory of multi-valued maps. The finite-dimensional case ??2.1. Formulation of a version of the Vietoris-Begle-Sklyarenko theorem ??2.2. The topological characteristic of a multi-valued vector field in a finite-dimensional space ??2.3. The rotation and the topological characteristic of -acyclic and generalized -acyclic multi-valued vector fields ??2.4. Some theorems on the computation of the topological characteristic ??2.5. Fixed-point theorems ??2.6. The Lefschetz theorem ??2.7. Guide to the literature in Chapter II Chapter III. Homological methods in the fixed-point theory of multi-valued maps. The infinite-dimensional case ??3.1. Partitions and the cohomology defined by them ??3.2. The topological characteristic of a multi-valued vector field in a Banach space ??3.3. The rotation of almost acyclic multi-valued vector fields ??3.4. Computation of the topological characteristic and fixed-point theorems ??3.5. Guide to the literature in Chapter III Appendix. Some applications References

https://iopscience.iop.org/article/10.1070/RM1980v035n01ABEH001548/pdf

Topological methods in the fixed-point theory of multi-valued maps

Introduction Part 1. Basic Theory: Preliminaries and outline of Part 1 Tools Partitions of domains in the sphere Part 2. Applications of Basic Theory: Description of main results of Part 2 Outchannels and their properties Fixed points Bibliography Index

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Fixed Point Theorems for Plane Continua with Applications

Each orientation preserving homeomorphism of the plane that is invariant on a nonseparating bounded continuum has a fixed point on the continuum. Introduction. In 1951 Cartwright and Littlewood [1] proved that an orientation preserving homeomorphism h of the Euclidean plane R2 that leaves a compact nonseparating continuum C invariant leaves a point of C fixed. In 1954 Hamilton [2] gave a short proof employing the result of Brouwer [3]: An orientation preserving homeomorphism of R2 that leaves a bounded set invariant has a fixed point (possibly not in the set). We give another short proof also based on Brouwer's theorem. PROOF OF THEOREM. Let F be the fixed point set of h. We shall derive a contradiction from the hypothesis that F n C = 0. Let U be the component of R F that contains C. Then h(U) = U. Let U be the universal cover of U. (U then is homeomorphic to R 2.) Then C lifts to disjoint homeomorphic copies of C in U. (The hypotheses on C insure that there exists a disk D such that C c D c U, and D being simply connected lifts to disjoint homeomorphic copies of D.) Let C be one of these copies of C and let h: U -* U be a homeomorphism covering h IU: U-> U and invariant on C. Since hj U is orientation preserving, so is h. Since hI U has no fixed points neither does h. Thus h is an orientation preserving fixed point free homeomorphism of R2 that is invariant on C. This contradicts Brouwer's theorem. REMARK. Brouwer's theorem is false if h is orientation reversing and neither this proof, Hamilton's, nor Cartwright-Littlewood's works. That the theorem in question remains true is a difficult result recently established by H. Bell [4].

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A short short proof of the Cartwright-Littlewood theorem

Let h be an orientation reversing homeomorphism of the plane onto itself. If X is a plane continuum invariant under h, then h has a fixed point in X. Furthermore, if at least one of the bounded complementary domains of X is invariant under h, then h has at least two fixed points in X.

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Fixed points of orientation reversing homeomorphisms of the plane

exactly two complementary domains, or by Lemma 4, applied inductively, we obtain an orientation reversing homeomor­phism g of the plane having an invariant continuum Z with exactly two complementary domains. By Theorem 1, g has at least two fixed points in Z. Therefore, h has at least two fixed points in X.

Research Announcement: FIXED POINTS OF ORIENTATION REVERSING HOMEOMORPHISMS OF THE PLANE

The purpose of this note is to outline a proof of \"every homeomorphism of the plane into itself that leaves a continuum M invariant has a fixed point in T(M)\". That is, the orientation preserving condition in the Cartwright Littlewood fixed point theorem [3] is unnecessary. All sets will be assumed to be subsets of the plane unless otherwise indicated. DEFINITION. If A is a bounded set then T(A) is the smallest compact set that contains A and has a connected complement.

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A fixed point theorem for plane homeomorphisms

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Harold Bell

Papers

A fixed point theorem for plane homeomorphisms

A fixed point theorem for plane homeomorphisms

Circularity of graphs and continua: topology

Fixed points, index, and degree for some set valued functions