Strong convergence theorems for finitely many nonexpansive mappings

Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gêteaux differentiable and which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, Ceng–Cubiotti–Yao [Strong convergence theorems for finitely many nonexpansive mappings and applications, Nonlinear Analysis 67 (2007) 1464–1473] introduced a new iterative scheme for a finite commuting family of nonexpansive mappings, and proved strong convergence theorems about this iteration. In this paper, only under the hypothesis that E$E$ is a reflexive Banach space which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, and several control conditions about the iterative coefficient are removed, we present a short and simple proof of the above theorem.     

An implicit iterative scheme for monotone variational inequalities and fixed point problems

In this paper we introduce an implicit iterative scheme for finding a common element of the set of common fixed points of N$N$ nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The implicit iterative scheme is based on two well-known methods: extragradient and approximate proximal. We obtain a weak convergence theorem for three sequences generated by this implicit iterative scheme. On the basis of this theorem, we also construct an implicit iterative process for finding a common fixed point of N+1$N+1$ mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the other N$N$ mappings are nonexpansive.     

A new iteration process for finite families of generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings

A new iteration process is introduced and proved to converge strongly to a common fixed point for a finite family of generalized Lipschitz nonlinear mappings in a real reflexive Banach space E$E$ with a uniformly Gâteaux differentiable norm if at least one member of the family is pseudo-contractive. It is also proved that a slight modification of the process converges to a common zero for a finite family of generalized Lipschitz accretive operators defined on E$E$. Results for nonexpansive families are obtained as easy corollaries. Finally, the new iteration process and the method of proof are of independent interest.     

Fixed point theorem for generalized Φ -pseudocontractive mappings

Let E$E$ be a real Banach space, C$C$ be a nonempty closed convex subset of E$E$ and T:C→C$T:C\to C$ be a continuous generalized Φ$\Phi$-pseudocontractive mapping. It is proved that T$T$ has a unique fixed point in C$C$.     

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.     

Contractive mappings and existence of cycle times for a monotone and homogeneous function

Given a monotone and homogeneous self-mapping f$f$ of the n$n$-dimensional positive cone, a family of contractive mappings is used to define an equivalence relation in the index set, as well as a total order among the equivalence classes. Then, it is shown (i) that the cycle times are well-defined at each index belonging to the maximal and minimal classes, and (ii) that the cycle times of f$f$ exist at every index whenever a weak convexity condition is satisfied.     

Iterative approximation of solutions of nonlinear equations of Hammerstein type

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are Lipschitz ?$?$-strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Approximation of solutions of Hammerstein equations with bounded strongly accretive nonlinear operators

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are bounded strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. A new explicit coupled iteration process is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Fixed points for generalized contractions and applications to control theory

The concepts of “weak/strong topological contraction” and a generalization of Banach contraction mappings called “p$p$-contraction” are introduced and used to prove fixed point theorems for self-mappings from a topological/metric space into itself satisfying topological contraction/metric p$p$-contraction, respectively. Certain non-linear integral equations defined on C[a,b]$C[a,b]$ satisfying generalized Lipschitzian conditions can easily be solved by applying these theorems. In the sequel, we shall study the possibility of optimally controlling the solution of the ordinary differential equation via dynamic programming.     

Nadler’s type principle with high order of convergence

Recently Proinov [P.D. Proinov, A generalization of the Banach contraction principle with high order of convergence of successive approximations, Nonlinear Analysis (2006), doi:10.1016/j.na.2006.09.008] generalized Banach contraction principle with high order of convergence. We extend some results of Proinov to the case of multi-valued maps from a complete metric space X$X$ into the space of all nonempty proximinal closed subsets of X$X$. Our results not only generalize Nadler’s fixed-point theorem (in the case when T$T$ is a mapping from a complete metric space X$X$ into the space of all nonempty proximinal closed subsets of X$X$) but also gives high order of convergence. As an application, we obtain an existence theorem for first-order initial value problem.     

Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

We develop a notion of nonlinear expectation–G$G$-expectation–generated by a nonlinear heat equation with infinitesimal generator G$G$. We first study multi-dimensional G$G$-normal distributions. With this nonlinear distribution we can introduce our G$G$-expectation under which the canonical process is a multi-dimensional G$G$-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our G$G$-Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G$G$-expectation.     

Ergodic theorems for hybrid sequences in a Hilbert space with applications

In this paper, we introduce the notion of generalized hybrid sequences, extending the notion of nonexpansive sequences introduced and studied in our previous work Djafari Rouhani (1981, 1990, 1990, 1997, 2002, 2004, 2002)  , , , , ,  and , and prove ergodic and convergence theorems for such sequences in a Hilbert space H$H$. Subsequently, we apply our results to prove new fixed point theorems for generalized hybrid mappings, first introduced in Kocourek et al. (2010)  [14], Takahashi and Takeuchi (2011)  [20], defined on arbitrary nonempty subsets of H$H$.     

Strong convergence theorem for approximation of solutions of equations of Hammerstein type

Let H$H$ be a real Hilbert space. Let K,F:H→H$K,F:H\to H$ be bounded, continuous and monotone mappings. Suppose that u^∗∈H$u^{\ast }\in H$ is a solution to the Hammerstein equation u+KFu=0$u+KFu=0$. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the Hammerstein equation. Furthermore, we give some examples to show that our result is interdisciplinary in nature, covers a large variety of areas and should be of much interest to a wide audience.     

The approximate fixed point property in product spaces

In this paper we generalize to unbounded   convex subsets C$C$ of hyperbolic   spaces results obtained by W.A. Kirk and R. Espínola on approximate fixed points of nonexpansive mappings in product spaces _(C×M)∞${(C\times M)}_{\infty }$, where M$M$ is a metric space and C$C$ is a nonempty, convex, closed and bounded subset of a normed or a CAT(0)-space. We extend the results further, to families _(Cu)u∈M${\left(C_u\right)}_{u\in M}$ of unbounded convex subsets of a hyperbolic space. The key ingredient in obtaining these generalizations is a uniform quantitative version of a theorem due to Borwein, Reich and Shafrir, obtained by the authors in a previous paper using techniques from mathematical logic. Inspired by that, we introduce in the last section the notion of uniform approximate fixed point property   for sets C$C$ and classes of self-mappings of C$C$. The paper ends with an open problem.     

Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory

In this paper, we introduce the concept of a Q$Q$-function defined on a quasi-metric space which generalizes the notion of a τ$\tau$-function and a w$w$-distance. We establish Ekeland-type variational principles in the setting of quasi-metric spaces with a Q$Q$-function. We also present an equilibrium version of the Ekeland-type variational principle in the setting of quasi-metric spaces with a Q$Q$-function. We prove some equivalences of our variational principles with Caristi–Kirk type fixed point theorems for multivalued maps, the Takahashi minimization theorem and some other related results. As applications of our results, we derive existence results for solutions of equilibrium problems and fixed point theorems for multivalued maps. We also extend the Nadler’s fixed point theorem for multivalued maps to a Q$Q$-function and in the setting of complete quasi-metric spaces. As a consequence, we prove the Banach contraction theorem for a Q$Q$-function and in the setting of complete quasi-metric spaces. The results of this paper extend and generalize many results appearing recently in the literature.     

Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces

We introduce a new composite iterative scheme to approximate a zero of an m$m$-accretive operator A$A$ defined on uniform smooth Banach spaces and a reflexive Banach space having a weakly continuous duality map. It is shown that the iterative process in each case converges strongly to a zero of A$A$. The results presented in this paper substantially improve and extend the results due to Ceng et al. [L.C. Ceng, H.K. Xu, J.C. Yao, Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators, Taiwanese J. Math. (in press)], Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643]. Our work provides a new approach for the construction of a zero of m$m$-accretive operators.