The numbers of periodic orbits hidden at fixed points of 2-dimensional holomorphic mappings

Let △n be the ball |x| 1 in the complex vector space C n , let f :△n→ C n be a holomorphic mapping and let M be a positive integer. Assume that the origin 0 = (0, . . . , 0) is an isolated fixed point of both f and the M-th iteration f M of f. Then the (local) Dold index P M (f, 0) at the origin is well defined, which can be interpreted to be the number of virtual periodic points of period M of f hidden at the origin: any holomorphic mapping f 1 :△n→ C n sufficiently close to f has exactly P M (f, 0) distinct periodic points of period M near the origin, provided that all the fixed points of f M 1 near the origin are simple. Therefore, the number O M (f, 0) = P M (f, 0)/M can be understood to be the number of virtual periodic orbits of period M hidden at the fixed point. According to the works of Shub-Sullivan and Chow-Mallet-Paret-Yorke, a necessary condition so that there exists at least one virtual periodic orbit of period M hidden at the fixed point, i.e., O M (f, 0)≥1, is that the linear part of f at the origin has a periodic point of period M. It is proved by the author recently that the converse holds true. In this paper, we will study the condition for the linear part of f at 0 so that O M (f, 0)≥2. For a 2 × 2 matrix A that is arbitrarily given, the goal of this paper is to give a necessary and sufficient condition for A, such that O M (f, 0)≥2 for all holomorphic mappings f :△2 → C 2 such that f(0) = 0, Df(0) = A and that the origin 0 is an isolated fixed point of f M .     

On the indices of fixed points of mappings in cones and applications

Assume that K is a cone in a Banach space and A:K → K is completely continuous. We obtain a formula for the index in K of a fixed point of A under the assumption that a linearization exists and satisfies an invertibility condition. We then use this formula to generalize some results of Amann on the number of fixed points of A to the case where K has empty interior.     

Approximating fixed points of Suzuki-generalized nonexpansive mappings

In this paper, we prove weak and strong convergence theorems for Ishikawa iteration of Suzuki-generalized nonexpansive mappings in uniformly convex Banach spaces. Furthermore, we extend the results to CAT(0) spaces. Our work extends the results of Suzuki [T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mapping, J. Math. Anal. Appl. 340 (2008) 1088–1095] and Takahashi and Kim [W. Takahashi, G.E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Jpn. 48 (1998) 1–9].     

Lifting fixed points of completely positive mappings

Let 1?n?∞, and let be a row contraction on some Hilbert space H. Let F(T) be the space of all X∈B(H) such that . We show that, if non-zero, this space is completely isometric to the commutant of the Cuntz part of the minimal isometric dilation of .     

On piecewise-monotone mappings with closed set of periodic points on dendrites

In this paper, piecewise-monotone dynamical systems with closed set of periodic points given on dendrites are considered.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 8 , Suzdal Conference-2, 2003.This revised version was published online in April 2005 with a corrected cover date.     

Fixed points, selections and common fixed points for nonexpansive-type mappings

We study the existence of fixed points in the context of uniformly convex geodesic metric spaces, hyperconvex spaces and Banach spaces for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity. Besides, we use the fixed point theorems proved here to give common fixed point results for commuting mappings.     

On the stability of iterative approximations to fixed points of nonexpansive mappings

We study iterative retraction approximations to fixed points of the nonexpansive self-mapping given on the closed convex set G in a Banach space B. The conditions which guarantee weak and strong convergence and stability of these approximations with respect to perturbations of both operator A and constraint set G are considered. The results of this paper are new even in a Hilbert space for the iterative projection approximations.     

Opial's modulus and fixed points of semigroups of mappings

If is a Banach space with the non-strict Opial property and and is a nonempty convex weakly compact subset of , then every semigroup of asymptotically regular selfmappings of with has a common fixed point.

     

Iterative approximation of fixed points of nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm and be a nonexpansive mapping with F(T):={x∈K:Tx=x}≠∅. For a fixed δ∈(0,1), define by Sx:=(1−δ)x+δTx, ∀x∈K. Assume that {zt} converges strongly to a fixed point z of T as t→0, where zt is the unique element of K which satisfies zt=tu+(1−t)Tzt for arbitrary u∈K. Let {αn} be a real sequence in (0,1) which satisfies the following conditions: ; . For arbitrary x₀∈K, let the sequence {xn} be defined iteratively by

xn+1=αnu+(1−αn)Sxn.     

On some algorithms in Banach spaces finding fixed points of nonlinear mappings

By using a modification of the hybrid method in mathematical programming, we give some new algorithm to find out a fixed point of mappings which are in some sense non-expansive. Our results hold in reflexive, strictly convex, smooth Banach spaces with the Kadec?-Klee property.     

An extension of multi-valued contraction mappings and fixed points

A class of new multi-valued contraction mappings is presented. Under the new contractive conditions, some fixed point theorems for multi-valued self-mappings and nonself-mappings are proved.

     

Fixed point indices and periodic points of holomorphic mappings

Let Δ ⁿ be the ball |x| <  1 in the complex vector space , let be a holomorphic mapping and let M be a positive integer. Assume that the origin is an isolated fixed point of both f and the Mth iteration f ^M of f. Then for each factor m of M, the origin is again an isolated fixed point of f ^m and the fixed point index of f ^m at the origin is well defined, and so is the (local) Dold’s index [Invent. Math. 74(3), 419–435 (1983)] at the origin:

Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces

In this paper, we introduce the new iterative sequences with errors approximating the common fixed point for a couple of quasi-contractive mappings and show the stability of the iterative procedures with errors in q-uniformly smooth Banach spaces. Our results extend, improve and unify the corresponding results of Chidume, Osilike, Liu, and others.