Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces

The purpose of the paper is to introduce modified Halpern and Ishikawa iteration for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in Banach spaces. We also consider two strong convergence theorems for relatively nonexpansive mappings with some proper restriction.     

Strong convergence theorems for a new iterative method of -strictly pseudo-contractive mappings in Hilbert spaces

In this paper, we introduce a new iterative method of a k-strictly pseudo-contractive mapping for some 0≤k<1 and prove that the sequence {x_n} converges strongly to a fixed point of T, which solves a variational inequality related to the linear operator A. Our results have extended and improved the corresponding results of Y.J. Cho, S.M. Kang and X. Qin [Some results on k-strictly pseudo-contractive mappings in Hilbert spaces, Nonlinear Anal. 70 (2008) 1956–1964], and many others.     

Halpern-type iterations for strongly relatively nonexpansive mappings in Banach spaces

In this paper, we note that the main convergence theorem in Zhang et al. (2011) [21] is incorrect and we prove a correction. We also modify Halpern’s iteration for finding a fixed point of a strongly relatively nonexpansive mapping in a Banach space. Consequently, two strong convergence theorems for a relatively nonexpansive mapping and for a mapping of firmly nonexpansive type are deduced. Using the concept of duality theorems, we obtain analogue results for strongly generalized nonexpansive mappings and for mappings of firmly generalized nonexpansive type. In addition, we study two strong convergence theorems concerning two types of resolvents of a maximal monotone operator in a Banach space.     

Mann and Ishikawa iterations with errors for non-lipschitzian mappings in Banach spaces

Let C be a nonempty closed convex subset of a uniformly convex Banach space and let T : C → C be completely continuous asymptotically nonexpansive in the intermediate sense. In this paper, we prove that the Ishikawa (and Mann) iteration process with errors converges strongly to some fixed point of T, which generalizes the recent results due to Huang [1].     

A perturbed iterative procedure for multivalued pseudo-contractive mappings and multivalued accretive mappings in Banach spaces

The main purpose of this paper is to introduce and study a new perturbed iteration method for multivalued mappings. Some strong convergence theorems of perturbed iteration sequences for multivalued pseudo-contractive mappings and strongly accretive mappings are obtained.     

Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings

In a recent paper, Rhoades [1] presented some generalizations of Schu [2] on the convergence of the Mann and Ishikawa iterations of asymptotically nonexpansive mappings in uniformly convex Banach spaces. We continue the study on the Ishikawa (and Mann) iteration process with errors and prove that if X is a uniformly convex Banach space, ø ≠ E ⊂ X closed bounded and convex, and T : E → E is an asymptotically nonexpansive mapping, then the Ishikawa (and Mann) iteration process with errors converges strongly to some fixed point of T.     

Strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in Banach spaces

In this paper, we introduce a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of a general variational inequality and the set of common fixed points of an infinite family of k_i-strict pseudocontractions in a Banach space. We obtain some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others.     

Hybrid Ishikawa iterative methods for a nonexpansive semigroup in Hilbert space

In this paper, on the base of the Ishikawa iteration method and the hybrid method in mathematical programming, we give two new strong convergence methods for finding a point in the common fixed point set of a nonexpansive semigroup in Hilbert space.     

Strong convergence theorems for three-step iterations with errors for non-lipschitzian nonself-mappings in Banach spaces

Suppose C is a nonempty closed convex subset of a real uniformly convex Banach space X with P is a nonexpansive retraction of X onto C. Let T : C → X be an asymptotically nonexpansive in the intermediate sense nonself-mapping. In this paper, we introduced the three-step iterative sequence for such map with errors. Moreover, we prove that, if T is completely continuous, then the three-step iterative sequences converges strongly to a fixed point of T.     

Existence and iteration for a mixed equilibrium problem and a countable family of nonexpansive mappings in Banach spaces

We prove the existence of a solution of the mixed equilibrium problem (MEP) by using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solutions set of a mixed equilibrium problem and the fixed points set of a countable family of nonexpansive mappings in the frameworks of Banach spaces. By using a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solutions set of MEP and common fixed points set of nonexpansive mappings. Moreover, some applications concerning the equilibrium and the convex minimization problems are obtained.     

Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings

In convex metric spaces, the Ishikawa iteration process and the Ishikawa iteration process with errors is defined for asymptotically quasi-nonexpansive mappings. It is proved some sufficient and necessary conditions for the iterative scheme converges to the fixed point of the asymptotically quasi-nonexpansive mappings. These results generalize and unify many important known results in recent literature.     

Fourth-order convergence theorem by using majorizing functions for super-Halley method in Banach spaces

In this paper, we study the semilocal convergence of a multipoint fourth-order super-Halley method for solving nonlinear equations in Banach spaces. We establish the Newton–Kantorovich-type convergence theorem for the method by using majorizing functions. We also get the error estimate. In comparison with the results obtained in Wang et al. [X.H. Wang, C.Q. Gu, and J.S. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces, Numer. Algorithms 56 (2011), pp. 497–516], we can provide a larger convergence radius. Finally, we report some numerical applications to demonstrate our approach.     

Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces

In this paper, we study a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces. We prove some weak and strong convergence theorems for this new iteration process. The results of this paper improve and extend the corresponding results of Chidume et al. (2003) [10], Osilike and Aniagbosor (2000) [3], Schu (1991) [4], Takahashi and Kim (1998) [9], Tian et al. (2007) [18], Wang (2006) [11], Yang (2007) [17] and others.     

Set-contractive mappings and difference equations in Banach spaces

In this paper, we consider the difference equation on an arbitrary Banach space (X, ∥·∥_x), Δ(q_nΔx_n + f_n(x_n) = 0, where {q_n} is a positive sequence and f_n is X-valued. We shall give conditions so that for a given x ϵ X, there exists a solution of this equation asymptotically equal to x.     

Accumulation and control of random errors in the Ishikawa iterative process in arbitrary Banach space

The purpose of this note is to study directly the relation between random errors and two parameter sequences in the Ishikawa iterative process and to estimate the accumulation of errors in the iterative process. The obtained results show that when one of the sequences converges to zero, the accumulation of errors is bounded if the other sequence is non-summable; while the accumulation of errors is controllable in a permissible range if the sequence is summable.     

Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces

We introduce three-step iterative schemes with errors for two and three nonexpansive maps and establish weak and strong convergence theorems for these schemes. Mann-type and Ishikawa-type convergence results are included in the analysis of these new iteration schemes. The results presented in this paper substantially improve and extend the results due to [S.H. Khan, H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal. 8 (2005) 1295–1301], [N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005) 1031–1039], [W. Takahashi, T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5 (1995) 45–58], [K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993) 301–308] and [H.F. Senter, W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974) 375–380].