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In recent years, the definition of relatively nonexpansive multivalued mapping and the definition of weak relatively nonexpansive multivalued mapping have been presented and studied by many authors. In this paper, we give some results about weak relatively nonexpansive multivalued mappings and give two examples which are weak relatively nonexpansive multivalued mappings but not relatively nonexpansive multivalued mappings in Banach space 𝑙2 and 𝐿𝑝[0,1](1

The authors have obtained the following results: (1) the definition of uniformly closed countable family of nonlinear mappings, (2) strong convergence theorem by the monotone hybrid algorithm for two countable families of hemirelatively nonexpansive mappings in a Banach space with new method of proof, (3) two examples of uniformly closed countable families of nonlinear mappings and applications, (4) an example which is hemirelatively nonexpansive mapping but not weak relatively nonexpansive mapping, and (5) an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping. Therefore, the results of this paper improve and extend the results of Plubtieng and Ungchittrakool (2010) and many others.

Saewan and Kumam (2010) have proved the convergence theorems for finding the set of solutions of a general equilibrium problems and the common fixed point set of a family of closed and uniformly quasi-𝜙-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. In this paper, authors prove the convergence theorems and do not need the Kadec-Klee property of Banach space and some other conditions used in the paper of S. Saewan and P. Kumam. Therefore, the results presented in this paper improve and extend some recent results.

The purpose of this paper is to prove strong convergence theorems for common fixed points of two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized -projection operator. In order to get the strong convergence theorems, a new iterative scheme by monotone hybrid method is presented and is used to approximate the common fixed points. Then, two examples of countable families of uniformly closed nonlinear mappings are given. The results of this paper modify and improve the results of Li et al. (2010), the results of Takahashi and Zembayashi (2008), and many others.

Let 𝑋 be a uniformly convex Banach space and 𝒮={𝑇(𝑠)∶0≤𝑠0𝐹(𝑇(𝑠))≠∅. Consider the iterative method that generates the sequence {𝑥𝑛} by the algorithm 𝑥𝑛+1=𝛼𝑛𝑓(𝑥𝑛)+𝛽𝑛𝑥𝑛+(1−𝛼𝑛−𝛽𝑛)(1/𝑠𝑛)∫𝑠𝑛0𝑇(𝑠)𝑥𝑛𝑑𝑠,𝑛≥0, where {𝛼𝑛}, {𝛽𝑛}, and {𝑠𝑛} are three sequences satisfying certain conditions, 𝑓∶𝐶→𝐶 is a contraction mapping. Strong convergence of the algorithm {𝑥𝑛} is proved assuming 𝑋 either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.

The purpose of this paper is to present the notion of weak relatively nonexpansive multi-valued mapping and to prove the strong convergence theorems of fixed point for weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper, the notion of multivalued resolvent of maximal monotone operator has been also presented which is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of maximal monotone operator.

The following main results have been given. (1) Let E be a p-uniformly convex Banach space and let T:E→E* be a (p-1)-L-Lipschitz mapping with condition 0

In this paper, strong convergence theorem is obtained for the modified Noor iterations in the framework of uniformly smooth Banach spaces. Our results extend and improve the recent ones announced by Wittman, Kim, Xu, and some others.

Saewan and Kumam (2010) have proved the convergence theorems for finding the set of solutions of a general equilibrium problems and the common fixed point set of a family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. In this paper, authors prove the convergence theorems and do not need the Kadec-Klee property of Banach space and some other conditions used in the paper of S. Saewan and P. Kumam. Therefore, the results presented in this paper improve and extend some recent results. [ABSTRACT FROM AUTHOR]