Finite Family Of Strict Pseudo-contractions Research Articles

Strong convergence theorem for split monotone variational inclusion with constraints of variational inequalities and fixed point problems

In this paper, inspired by Jitsupa et al. (J. Comput. Appl. Math. 318:293–306, 2017), we propose a general iterative scheme for finding a solution of a split monotone variational inclusion with the constraints of a variational inequality and a fixed point problem of a finite family of strict pseudo-contractions in real Hilbert spaces. Under very mild conditions, we prove a strong convergence theorem for this iterative scheme. Our result improves and extends the corresponding ones announced by some others in the earlier and recent literature.

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.

An iterative algorithm to approximate a common element of the set of common fixed points for a finite family of strict pseudo-contractions and of the set of solutions for a modified system of variational inequalities

In this paper, we introduce a new iterative algorithm for finding a common element of the set of fixed points of a finite family of -strictly pseudo-contractive mappings and the set of solutions of new variational inequalities problems in Hilbert space. By using our main results, we obtain an interesting theorem involving a finite family of κ-strictly pseudo-contractive mappings and two sets of solutions of the variational inequalities problem.

A HYBRID PROJECTION METHOD FOR RELAXED COCOERCIVE MAPPINGS AND STRICTLY PSEUDO-CONTRACTIVE MAPPINGS

The purpose of this paper is to introduce a hybrid projection method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a variational inclusion problem and the set of common fixed points of a finite family of strict pseudo-contractions in Hilbert spaces.

A NEW METHOD FOR A FINITE FAMILY OF PSEUDOCONTRACTIONS AND EQUILIBRIUM PROBLEMS

In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed points of a finite family of strict pseudocontractions and the solution set of pseudomonotone and Lipschitz-type continuous equilibrium problems. The scheme is based on the idea of extragradient methods and fixed point iteration methods. We show that the iterative sequences generated by this algorithm converge strongly to the common element in a real Hilbert space.

Strong convergence of a new iterative scheme for a finite family of strict pseudo-contractions

The purpose of this paper is to introduce a new iterative scheme for finding a common fixed point of a finite family of strict pseudo-contractions. The results obtained in this paper extend and improve the corresponding results of Marino, Colao, Qin and Kang [G. Marino, V. Colao, X. Qin, S. M. Kang, Strong convergence of the modified Mann iterative method for strict pseudo-contractions, Comput. Math. Appl. 57 (2009) 455–465].

Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces

In this paper, we continue to discuss the properties of iterates generated by a strict pseudocontraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336–349 (2007)]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51–60 (2005)] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.

SOME NEW ITERATIVE ALGORITHMS FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS WITH STRICT PSEUDO-CONTRACTIONS AND MONOTONE MAPPINGS

In this paper, we propose some parallel and cyclic algorithms based on the extragradient method (nonextragradient method) for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of a finite family of strict pseudo-contractions and the set of the variational inequality for a monotone, Lipschitz continuous mapping (an inverse strongly monotone mapping). We obtain some weak and strong convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, improve and unify some well-known results in the literature.

Weak convergence theorems for a finite family of strict pseudocontractions

Let E be a uniformly smooth real Banach space which is also uniformly convex and K be a nonempty closed convex subset of E . Let T : K → K be a λ -strict pseudocontraction for some 0 ≤ λ < 1 with x ∗ ∈ F ( T ) : = { x ∈ K : T x = x } ≠ 0̸ . For a fixed x 0 ∈ K , define a sequence { x n } by x n + 1 = ( 1 − α n ) x n + α n T x n , where { α n } is a sequence in [ 0 , 1 ] satisfying the following conditions: (i) ∑ n = 0 ∞ α n = ∞ ; (ii) ∑ n = 0 ∞ α n 2 < ∞ . Then, { x n } converges weakly to a fixed point of T . Furthermore, weak convergence theorems are proved for a common fixed point for a finite family of strict pseudocontractions.

Convergence theorems for strict pseudo-contractions in [formula omitted]-uniformly smooth Banach spaces

In this paper, we discuss some properties of iterates generated by a strict pseudo-contraction or a finite family of strict pseudo-contractions in a real q -uniformly smooth Banach space. We prove the weak convergence and strong convergence of pseudo-contractive mappings. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349], Acedo and Xu [G.L. Acedo, H.K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67 (2007) 2258–2271] and Zhou [H. Zhou, Convergence theorems for λ -strict pseudo-contractions in 2-uniformly smooth Banach spaces, Nonlinear Anal. 69 (2008) 3160–3173].

Strong convergence of the modified Mann iterative method for strict pseudo-contractions

In this paper, we introduce a modified Mann iterative process for approximating a common fixed point of a finite family of strict pseudo-contractions in Hilbert spaces. We establish the strong convergence theorem of the general iteration scheme under some mild conditions. Our results extend and improve the recent ones announced by many others.

Generalized Mann Iterations for Approximating Fixed Points of a Family of Hemicontractions

This paper concerns common fixed points for a finite family of hemicontractions or a finite family of strict pseudocontractions on uniformly convex Banach spaces. By introducing a new iteration process with error term, we obtain sufficient and necessary conditions, as well as sufficient conditions, for the existence of a fixed point. As one will see, we derive these strong convergence theorems in uniformly convex Banach spaces and without any requirement of the compactness on the domain of the mapping. The results given in this paper extend some previous theorems.

Convergence theorems for [formula omitted]-strict pseudo-contractions in 2-uniformly smooth Banach spaces

In this paper, we discuss some properties of iterates generated by a strict pseudo-contraction or a finite family of strict pseudo-contractions in a real 2-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 324 (2007) 336–349]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C . This result extends a recent result of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] both from nonexpansive mappings to λ -strict pseudo-contractions and from Hilbert spaces to 2-uniformly smooth Banach spaces.