Index formulas for countably k-set contractive operators

We give an index formula for countably k$k$-set contractive bounded linear operators in a real Banach space, by using a degree theory for countably condensing operators. As a consequence, we present another index formula for differentiable countably k$k$-set contractive operators.     

A fixed point theorem for weakly Zamfirescu mappings

In [5], Zamfirescu (1972) gave a fixed point theorem that generalizes the classical fixed point theorems by Banach, Kannan, and Chatterjea. In this paper, we follow the ideas of Dugundji and Granas to extend Zamfirescu’s fixed point theorem to the class of weakly Zamfirescu maps. A continuation method for this class of maps is also given.     

On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces

We prove a strong convergence theorem for multivalued nonexpansive mappings which includes Kirk’s convergence theorem on CAT(0) spaces. The theorem properly contains a result of Jung for Hilbert spaces. We then apply the result to approximate a common fixed point of a countable family of single-valued nonexpansive mappings and a compact valued nonexpansive mapping.     

A fixed point theorem for pointwise eventually nonexpansive mappings in nearly uniformly convex Banach spaces

The purpose of this paper is to prove the existence of a fixed point for a pointwise eventually nonexpansive mapping in a nearly uniformly convex Banach space. This provides an affirmative answer to a question given by Kirk and Xu [W.A. Kirk, Hong-Kun Xu, Asymptotic pointwise contraction, Nonlinear Anal. 69 (2008), 4706-4712].     

Lipschitz constants for iterates of mean lipschitzian mappings

We present a sharp evaluation of the Lipschitz constant for every iterate for mean lipschitzian mappings, solving a problem posed by K. Goebel and M. A. Japón Pineda.     

The common minimal-norm fixed point of a finite family of nonexpansive mappings

Consider a finite family of nonexpansive mappings which are defined on a closed convex subset of a Hilbert space H. Suppose the set of common fixed points of this family is nonempty. We then address the problem of finding the minimum-norm element of this common fixed point set. We introduce both cyclic and parallel iteration methods to find this minimal-norm element.     

Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings

In this paper, we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. These implicit iterative algorithms are based on two well-known methods: extragradient and approximate proximal methods. We obtain some weak convergence theorems for these implicit iterative algorithms. Based on these theorems, we also construct some implicit iterative processes for finding a common fixed point of two mappings, such that one of these two mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive.     

Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces

In this paper, we introduce a modified Ishikawa iterative process for approximating a fixed point of nonexpansive mappings in Hilbert spaces. We establish some strong convergence theorems of the general iteration scheme under some mild conditions. The results improve and extend the corresponding results of many others.     

Fixed point property for nonexpansive mappings versus that for nonexpansive semigroups

The main result of this paper is that a closed convex subset of a Banach space has the fixed point property for nonexpansive mappings if and only if it has the fixed point property for nonexpansive semigroups.     

Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces

In an infinite-dimensional Hilbert space, the normal Mann’s iteration algorithm has only weak convergence, in general, even for nonexpansive mappings. In order to get a strong convergence result, we modify the normal Mann’s iterative process for an infinite family of nonexpansive mappings in the framework of Banach spaces. Our results improve and extend the recent results announced by many others.     

Strong convergence theorems for finitely many nonexpansive mappings and applications

Let E$E$ be a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. Recently, Takahashi and Shimoji [W. Takahashi, K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Modelling 32 (2000) 1463–1471] introduced an iterative scheme given by finitely many nonexpansive mappings in E$E$ and proved weak convergence theorems which are connected with the problem of image recovery. In this paper we introduce a new iterative scheme which includes their iterative scheme as a special case. Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gâteaux differentiable and which has a weakly continuous duality mapping, we prove strong convergence theorems which are connected with the problem of image recovery. Using the established results, we consider the problem of finding a common fixed point of finitely many nonexpansive mappings.     

Weak and strong convergence theorems for countable Lipschitzian mappings and its applications

We use Mann’s iteration and the hybrid method in mathematical programming to obtain weak and strong convergence to common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium problems and variational inequalities for continuous monotone mappings.     

The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces

A recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique. The advantage of this technique is that one can find a particular solution to the associated problems, and in most cases this particular solution solves some variational inequality. In this paper, we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a Banach space which is reflexive and has a weakly continuous duality map. Both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved.     

A general iterative method for a finite family of nonexpansive mappings

We study the approximation of common fixed points of a finite family of nonexpansive mappings and suggest a modification of the iterative algorithm without the assumption of any type of commutativity. Also we show that the convergence of the proposed algorithm can be proved under some types of control conditions.     

Approximation methods for common fixed points for a countable family of nonexpansive mappings

Let E=L_p or l_p space, 1<p<∞. Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ∈(0,1), define a family of nonexpansive maps by S_i?(1−δ)I+δT_i where I is the identity map of K. Let . It is proved that the iterative sequence {x_n} defined by: x₀∈K,x_n+1=α_nu+∑_i≥1σ_i,tnS_ix_n,n≥0 converges strongly to a common fixed point of the family where {α_n} and {σ_i,tn} are sequences in (0,1) satisfying appropriate conditions, in each of the following cases: (a) E=l_p,1<p<∞, and (b) E=L_p,1<p<∞ and at least one of the maps T_i’s is demicompact. Our theorems extend the results of [P. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 325 (2007) 469-479] from Hilbert spaces to l_p spaces, 1<p<∞.     

Implicit iteration process for a finite family of asymptotically hemi-contractive mappings in Banach spaces

The purpose of this paper is to study the strong convergence of a process of implicit iteration to a common fixed point for a finite family of asymptotically hemi-contractive mappings. Our results extend a recent result of M.O. Osilike and B.G. Akuchu [Common fixed points of a finite family of asymptotically pseudocontractive maps, Fixed Point Theory Appl. 2 (2004) 81–88] from Hilbert spaces to p$p$-uniformly convex Banach spaces with p>1$p>1$.     

Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in CAT(0) spaces

We establish △-convergence and strong convergence theorems for an iterative process for a finite family of generalized nonexpansive multivalued mappings in a CAT(0) space. Moreover, we present a fixed point theorem for a pair consisting of a finite family of generalized nonexpansive single valued mappings, and a generalized nonexpansive multivalued mapping in CAT(0) spaces.     

Fixed points of semigroups of Lipschitzian mappings defined on nonconvex domains

Certain fixed point theorems are established for nonlinear semigroups of Lipschitzian mappings defined on nonconvex domains in Hilbert and Banach spaces. Some known results are thus generalized.