http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/79075Contractive maps have nice properties concerning fixed points; a big amount of literature has been devoted to fixed points of nonexpansive maps. The class of shrinking (or strictly contractive) maps is slightly less popular: few specific results on them (not applicable to all nonexpansive maps) appear in the literature and some interesting problems remain open. As an attempt to fill this gap, a condition half way between shrinking and contractive maps has been studied recently. Here we continue the study of the latter notion, solving some open problems concerning these maps.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/36361The topological approaches to find solutions of a coincidence equation f1(x)=f2(x) can roughly be divided into degree and index theories. We describe how these methods can be combined. We are led to a concept of an extended degree theory for function triples which turns out to be natural in many respects. In particular, this approach is useful to find solutions of inclusion problems F(x)∈Φ(x). As a side result, we obtain a necessary condition for a compact AR to be a topological group.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/96737Let X be a closed subset of a Banach space and G an ultimately nonexpansive commutative semigroup of continuous selfmappings. If the G-closure of X is nonempty, then the closure of the orbit of any G-closure point is a commutative topological group.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/81493We are concerned with the study of a multistep iterative scheme with errors involving a finite family of nonexpansive nonself-mappings. We approximate the common fixed points of a finite family of nonexpansive nonself-mappings by weak and strong convergence of the scheme in a uniformly convex Banach space. Our results extend and improve some recent results, Shahzad (2005) and many others.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/42451Using the concept of p-η-proximal mapping, we study the existence and sensitivity analysis of solution of a parametric general variational-like inequality problem in uniformly smooth Banach space. The approach used may be treated as an extension and unification of approaches for studying sensitivity analysis for various important classes of variational inequalities given by many authors in this direction.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/70830The purpose of this paper is to study sufficient and necessary conditions for finite-step iterative sequences with mean errors for a finite family of asymptotically quasi-nonexpansive and type mappings in Banach spaces to converge to a common fixed point. The results presented in this paper improve and extend the recent ones announced by Ghost-Debnath, Liu, Xu and Noor, Chang, Shahzad et al., Shahzad and Udomene, Chidume et al., and all the others.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/35390Let C be a nonempty closed convex subset of a smooth Banach space E and let A be an accretive operator of C into E. We first introduce the problem of finding a point u∈C such that 〈Au,J(v−u)〉≥0 for all v∈C, where J is the duality mapping of E. Next we study a weak convergence theorem for accretive operators in Banach spaces. This theorem extends the result by Gol'shteĭn and Tret'yakov in the Euclidean space to a Banach space. And using our theorem, we consider the problem of finding a fixed point of a strictly pseudocontractive mapping in a Banach space and so on.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/82080We obtain fixed point theorems for nonexpansive mappings defined on unbounded sets. Our assumptions are weaker than the asymptotically contractive condition recently introduced by Jean-Paul Penot.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/74503We prove the existence of coincidence point and common fixed point for mappings satisfying generalized weak contractive condition. As an application, related results on invariant approximation are derived. Our results generalize various known results in the literature.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/24543We obtain common fixed point results for generalized I-nonexpansive Cq-commuting maps. As applications, various best approximation results for this class of maps are derived in the setup of certain metrizable topological vector spaces.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/59692We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let C be a bounded closed convex subset of a uniformly smooth Banach space E. Let {Tn:n∈ℕ} be an infinite family of commuting nonexpansive mappings on C. Let {αn} and {tn} be sequences in (0,1/2) satisfying limntn=limnαn/tnℓ=0 for ℓ∈ℕ. Fix u∈C and define a sequence {un} in C by un=(1−αn)((1−∑k=1ntnk)T1un+∑k=1ntnkTk+1un)+αnu for n∈ℕ. Then {un} converges strongly to Pu, where P is the unique sunny nonexpansive retraction from C onto ∩n=1∞F(Tn).Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/65195We reprove in an extremely simple way the classical theorem that time periodic dissipative systems imply the existence of harmonic periodic solutions, in the case of uniqueness. We will also show that, in the lack of uniqueness, the existence of harmonics is implied by uniform dissipativity. The localization of starting points and multiplicity of periodic solutions will be established, under suitable additional assumptions, as well. The arguments are based on the application of various asymptotic fixed point theorems of the Lefschetz and Nielsen type.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/87657This paper concerns a formula which relates the Lefschetz number L(f) for a map f:M→M′ to the fixed point index I(f) summed with the fixed point index of a derived map on part of the boundary of ∂M. Here M is a compact manifold and M′ is M with a collar attached.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/92429We study Frèchet differentiable stable operators in real Banach spaces. We present the theory of linear and nonlinear stable operators in a systematic way and prove solvability theorems for operator equations with differentiable expanding operators. In addition, some relations to the theory of monotone operators in Hilbert spaces are discussed. Using the obtained solvability results, we formulate the corresponding fixed point theorem for a class of nonlinear expanding operators.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/46052Let f:X→X be a self-map of a compact, connected polyhedron and Φ⊆X a closed subset. We examine necessary and sufficient conditions for realizing Φ as the fixed point set of a map homotopic to f. For the case where Φ is a subpolyhedron, two necessary conditions were presented by Schirmer in 1990 and were proven sufficient under appropriate additional hypotheses. We will show that the same conditions remain sufficient when Φ is only assumed to be a locally contractible subset of X. The relative form of the realization problem has also been solved for Φ a subpolyhedron of X. We also extend these results to the case where Φ is a locally contractible subset.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/73530Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/29470Let f:X→X be a map of a compact, connected Riemannian manifold, with or without boundary. For ∈>0 sufficiently small, we introduce an ∈-Nielsen number N∈(f) that is a lower bound for the number of fixed points of all self-maps of X that are ∈-homotopic to f. We prove that there is always a map g:X→X that is ∈-homotopic to f such that g has exactly N∈(f) fixed points. We describe procedures for calculating N∈(f) for maps of 1-manifolds.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/75848We consider various problems regarding roots and coincidence points for maps into the Klein bottle K. The root problem where the target is K and the domain is a compact surface with non-positive Euler characteristic is studied. Results similar to those when the target is the torus are obtained. The Wecken property for coincidences from K to K is established, and we also obtain the following 1-parameter result. Families fn,g:K→K which are coincidence free but any homotopy between fn and fm, n≠m, creates a coincidence with g. This is done for any pair of maps such that the Nielsen coincidence number is zero. Finally, we exhibit one such family where g is the constant map and if we allow for homotopies of g, then we can find a coincidence free pair of homotopies.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/57950In the appendix to the book by F. F. Bonsal, Lectures on Some Fixed Point Theorems of Functional Analysis (Tata Institute, Bombay, 1962) a proof by Singbal of the Schauder-Tychonoff fixed point theorem, based on a locally convex variant of Schauder mapping method, is included. The aim of this note is to show that this method can be adapted to yield a proof of Kakutani fixed point theorem in the locally convex case. For the sake of completeness we include also the proof of Schauder-Tychonoff theorem based on this method. As applications, one proves a theorem of von Neumann and a minimax result in game theory.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/63939We prove that N(f)=|L(f)| for any continuous map f of a given infranilmanifold with Abelian holonomy group of odd order. This theorem is the analogue of a theorem of Anosov for continuous maps on nilmanifolds. We will also show that although their fundamental groups are solvable, the infranilmanifolds we consider are in general not solvmanifolds, and hence they cannot be treated using the techniques developed for solvmanifolds.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/23582We obtain common fixed point results for generalized I-nonexpansive R-subweakly commuting maps on nonstarshaped domain. As applications, we establish noncommutative versions of various best approximation results for this class of maps in certain metrizable topological vector spaces.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/68513We study Nielsen coincidence theory for maps between manifolds of same dimension regardless of orientation. We use the definition of semi-index of a class, review the definition of defective classes, and study the occurrence of defective root classes. We prove a semi-index product formula for lifting maps and give conditions for the defective coincidence classes to be the only essential classes.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/84093In classical fixed point and coincidence theory, the notion of Nielsen numbers has proved to be extremely fruitful. Here we extend it to pairs (f1,f2) of maps between manifolds of arbitrary dimensions. This leads to estimates of the minimum numbers MCC (f1,f2) (and MC (f1,f2), resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are (f1,f2). Furthermore we deduce finiteness conditions for MC (f1,f2). As an application, we compute both minimum numbers explicitly in four concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E(f1,f2) into path components. Its higher-dimensional topology captures further crucial geometric coincidence data. An analoguous approach can be used to define also Nielsen numbers of certain link maps.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/10673We introduce a new class of asymptotically nonexpansive mappings and study approximating methods for finding their fixed points. We deal with the Krasnosel'skii-Mann-type iterative process. The strong and weak convergence results for self-mappings in normed spaces are presented. We also consider the asymptotically weakly contractive mappings.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/96012We consider a general quasi-variational inequality problem involving nonlinear, nonconvex and nondifferentiable term in uniformly smooth Banach space. Using retraction mapping and fixed point method, we study the existence of solution of general quasi-variational inequality problem and discuss the convergence analysis and stability of a three-step iterative algorithm for general quasi-variational inequality problem. The theorems presented in this paper generalize, improve, and unify many previously known results in the literature.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/69758Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E, and let T:K→K be a uniformly continuous pseudocontraction. If f:K→K is any contraction map on K and if every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers {αn}, {μn}, that the iteration process z1∈K, zn+1=μn(αnTzn+(1−αn)zn)+(1−μn)f(zn), n∈ℕ, strongly converges to the fixed point of T, which is the unique solution of some variational inequality, provided that K is bounded.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/25897We will consider the number of fixed points of homeomorphisms composed of finitely many slide homeomorphisms on closed oriented nonprime 3-manifolds. By isotoping such homeomorphisms, we try to reduce their fixed point numbers. The numbers obtained are determined by the intersection information of sliding spheres and sliding paths of the slide homeomorphisms involved.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/34143In classical Lefschetz-Nielsen theory, one defines the Lefschetz invariant L(f) of an endomorphism f of a manifold M. The definition depends on the fundamental group of M, and hence on choosing a base point ∗∈M and a base path from ∗ to f(∗). At times, it is inconvenient or impossible to make these choices. In this paper, we use the fundamental groupoid to define a base-point-free version of the Lefschetz invariant.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/53286We apply the KKM technique to study fixed point theory, minimax inequality and coincidence theorem. Some new results on Fan-Browder fixed point theorem, Fan's minimax theorem and coincidence theorem are obtained.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/FPTA/2006/17563Let X be an H-space of the homotopy type of a connected, finite CW-complex, f:X→X any map and pk:X→X the kth power map. Duan proved that pkf:X→X has a fixed point if k≥2. We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spaces X whose rational cohomology is the tensor product of an exterior algebra on odd dimensional generators with the tensor product of truncated polynomial algebras on even dimensional generators. The role of the power map is played by a θ-structure μθ:X→X as defined by Hemmi-Morisugi-Ooshima. The conclusion is that μθf and fμθ each has a fixed point.Hindawi Publishing Corporation© 2006, Hindawi Publishing Corporation. All rights reserved.