Pseudo-contractions In Banach Spaces Research Articles

Iterative Methods for Obtaining an Infinite Family of Strict Pseudo-Contractions in Banach Spaces

Iterative Methods for Obtaining an Infinite Family of Strict Pseudo-Contractions in Banach Spaces

Explicit and implicit iterative algorithms for strict pseudo-contractions in Banach spaces

Explicit and implicit iterative algorithms for strict pseudo-contractions in Banach spaces

A Note on the Mann Iteration for k-Strict Pseudocontractions in Banach Spaces

ABSTRACTWe show that a variant of a previously defined function can be used to characterize 2-uniform smoothness. We then obtain greatly simplified proofs of two convergence theorems in the literature using a generalization of a lemma of and the aforementioned characterization. We also more easily obtain rates of asymptotic regularity corresponding to the studied iterations. Finally, we derive a way to relate two constants which are characteristic to 2-uniformly smooth spaces.

Strong convergence of some algorithms for $$\lambda $$ λ -strict pseudo-contractions in Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of common fixed points of an infinite family of strictly pseudo-contractive mappings in a real 2-uniformly smooth Banach space. Then we proved a strong convergence theorem under some suitable conditions. Our results generalize and improve several recent results.

Weak convergence theorems for strict pseudo-contractions in Banach spaces

In this paper, it is shown that there is a gap in the paper [Chidume, C. E., Shahzad, N.: Weak convergence theorems for a finite family of strict pseudo-contractions. Nonlinear Anal., 72, 1257–1265 (2010)], consequently, the main results of the paper do not hold in uniformly smooth Banach spaces. Meanwhile, it is also shown that the main results (Lemma 3.4, Theorems 3.5–3.6, 3.8–3.9) in the paper [Cholamjiak, P., Suantai, S.: Weak convergence theorems for a countable family of strict pseudo-contractions in Banach spaces. Fixed Point Theory Appl., 2010, Article ID 632137, 16 pages (2010)] do not hold in L p for p > 3. Finally, some modified results are presented in the setting of uniformly smooth and uniformly convex Banach spaces which include L p for p ≥ 2 as special cases. Furthermore, our arguments are also different from the ones given by the authors above.

Approximating fixed points of a countable family of strict pseudocontractions in Banach spaces

We prove the strong convergence of the modified Mann-type iterative scheme for a countable family of strict pseudocontractions in q-uniformly smooth Banach spaces. Our results mainly improve and extend the results announced in (Y. Yao, H. Zhou, Y.-C. Liou, Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J. Appl. Math. Comput. 29 (2009), 383-389).

Weak and strong convergence theorems for a countable family of strict pseudocontractions in Banach spaces

We first investigate the weak convergence of the Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly smooth Banach space which is uniformly convex or satisfies Opial's condition. Then the strong convergence theorems of the modified Mann-type iterative scheme are also proved in a uniformly smooth Banach space. As a consequence, several weak and strong convergence theorems for an infinite family of pseudocontractions are established.

Strong convergence for total asymptotically pseudocontractive semigroups in Banach spaces

In this article, the demiclosedness principle for total asymptotically pseudocontractions in Banach spaces is established. The strong convergence to a common fixed point of total asymptotically pseudocontractive semigroups in Banach spaces is established based on the demiclosedness principle, the generalized projective operator, and the hybrid method. The main results presented in this article extend and improve the corresponding results of many authors. MSC:47H05, 47H09, 49M05.

Strong convergence for asymptotical pseudocontractions with the demiclosedness principle in banach spaces

The aim of this article is to give an answer to an interesting question proposed in Zhou. At the end of his article, he remarked that it was of great interest to extend his results to certain Banach spaces. So in this article, we extend the demiclosedness principle from Hilbert spaces to Banach spaces. A strong convergence theorem for asymptotical pseudo-contractions in Banach spaces is established. The approaches are based on the extended demiclosedness principle, and the generalized projective operator, and the hybrid method in mathematical programming. Our results extend the previous known results from Hilbert spaces to Banach spaces. MSC: 47H10; 47H09; 47H05.

New Iterative Manner Involving Sunny Nonexpansive Retractions for Pseudocontractive Mappings

Iterative methods for pseudocontractions have been studied by many authors in the literature. In the present paper, we firstly propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.

Hybrid methods for accretive variational inequalities involving pseudocontractions in Banach spaces

We use strongly pseudocontractions to regularize a class of accretive variational inequalities in Banach spaces, where the accretive operators are complements of pseudocontractions and the solutions are sought in the set of fixed points of another pseudocontraction. In this paper, we consider an implicit scheme that can be used to find a solution of a class of accretive variational inequalities.Our results improve and generalize some recent results of Yao et al. (Fixed Point Theory Appl, doi:10.1155/2011/180534, 2011) and Lu et al. (Nonlinear Anal, 71(3-4), 1032-1041, 2009).2000 Mathematics subject classification 47H05; 47H09; 65J15

Strong Convergence Theorems of Modified Ishikawa Iterative Method for an Infinite Family of Strict Pseudocontractions in Banach Spaces

We introduce a new modified Ishikawa iterative process and a new W‐mapping for comput‐ ing fixed points of an infinite family of strict pseudocontractions mapping in the framework of q‐uniformly smooth Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions. The results obtained in this paper extend and improve the recent results of Cai and Hu 2010, Dong et al. 2010, Katchang and Kumam 2011 and many others in the literature.

Theorems of strong convergence of mixed iterative methods for obtaining strict pseudocontractions in Banach spaces

In this work, theorems of strong convergence of a mixed iteration scheme within the viscosity approximation methods for obtaining λ -strict pseudocontractions are established in p -uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. The main result extends various results existing in the current literature.

Strong convergence of an iterative algorithm for an infinite family of strict pseudo-contractions in Banach spaces

In this paper, we introduce a new W-mapping and present an iterative algorithm for an infinite family of strict pseudo-contractions. Strong convergence theorems are proved in Banach spaces. Our results improve and extend the corresponding result announced by Cai and Hu [G. Cai, C. Hu, Strong convergence theorems of modified Ishikawa iterative process with errors for an infinite family of strict pseudo-contractions, Nonlinear Anal. 71(12) (2009) 6044–6053].

Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces

We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Frechet differentiable norm. Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad (2010) is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.

Strong Convergence Theorems of Viscosity Iterative Methods for a Countable Family of Strict Pseudo-contractions in Banach Spaces

AbstractFor a countable family "Equation missing" of strictly pseudo-contractions, a strong convergence of viscosity iteration is shown in order to find a common fixed point of "Equation missing" in either a p-uniformly convex Banach space which admits a weakly continuous duality mapping or a p-uniformly convex Banach space with uniformly Gâteaux differentiable norm. As applications, at the end of the paper we apply our results to the problem of finding a zero of accretive operators. The main result extends various results existing in the current literature.

An iterative method of fixed points for closed and quasi-strict pseudo-contractions in Banach spaces

The purpose of this article is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for closed and quasi-strict pseudo-contractions. Its results hold in reflexive, strictly convex and smooth Banach spaces with the property (K). The results of this paper improve and extend the corresponding results of Matsushita and Takahashi (J. Approx. Theory 134:257–266, 2005), Qin and Su (Nonlinear Anal. 67:1958–1965, 2007), Marino and Xu (J. Math. Anal. Appl. 329:336–346, 2007) and others.

Strong convergence theorems for the implicit iteration process for a finite family of hemicontractive mappings in Banach space

The purpose of this work is to study the following implicit iteration scheme x n = α n x n − 1 + ( 1 − α n ) T n x n , n ≥ 1 , where T n = T n m o d N , and to prove several strongly convergent theorems of the iteration for a finite family of hemicontractive mappings in Banach space. Our results extend a recent result of Haiyun Zhou [Haiyun Zhou, Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces, Nonlinear Anal. 68 (2008) 2977–2983] and Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773], and we have proved that the sequence { x n } converges strongly to a common fixed point of a finite family of hemicontractive mappings { T i } i = 1 N .

Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space X whose norm is uniformly Gâteaux differentiable and T : C → C be a continuous pseudo-contraction with a nonempty fixed point set F ( T ) . For arbitrary given element u ∈ C and for t ∈ ( 0 , 1 ) , let { y t } be the unique continuous path such that y t = ( 1 − t ) T y t + t u . Assume that y t → p ∈ F ( T ) as t → 0 . Let { α n } , { β n } and { γ n } be three real sequences in (0, 1) satisfying the following conditions: (i) α n + β n + γ n = 1 ; (ii) lim n → ∞ α n = lim n → ∞ β n = 0 ; (iii) lim n → ∞ β n 1 − γ n = 0 ; or (iii)′ ∑ n = 0 ∞ α n 1 − γ n = ∞ . Let { ϵ n } be a summable sequence of positive numbers. For arbitrary initial datum x 0 = x 0 0 ∈ C and a fixed n ≥ 0 , construct elements { x n m } as follows: x n m + 1 = α n u + β n x n + γ n T x n m , m = 0 , 1 , 2 , … . Suppose that there exists a least positive integer N ( n ) satisfying the following condition: ‖ T x n N ( n ) + 1 − T x n N ( n ) ‖ ≤ γ n − 1 ( 1 − γ n ) ϵ n . Define iteratively a sequence { x n } in an explicit manner as follows: x n + 1 = x n + 1 0 = x n N ( n ) + 1 = α n u + β n x n + γ n T x n N ( n ) , n ≥ 0 . Then { x n } converges strongly to a fixed point of T . For all the continuous pseudo-contractive mappings for which is possible to construct the sequence x n , this result improves and extends a recent result of Yao et al. [Yonghong Yao, Yeong-Cheng Liou, Rudong Chen, Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear Anal., 67 (2007) 3311–3317].

Strong Convergence of Viscosity Methods for Continuous Pseudocontractions in Banach Spaces

We define a viscosity method for continuous pseudocontractive mappings defined on closed and convex subsets of reflexive Banach spaces with a uniformly Gâteaux differentiable norm. We prove the convergence of these schemes improving the main theorems in the work by Y. Yao et al. (2007) and H. Zhou (2008).