An iterative algorithm for finding a common solution of fixed points and a general system of variational inequalities for two inverse strongly accretive operators (original) (raw)

Viscosity iterative method for a new general system of variational inequalities in Banach spaces

Journal of Inequalities and Applications, 2013

In this paper, we study a new iterative method for finding a common element of the set of solutions of a new general system of variational inequalities for two different relaxed cocoercive mappings and the set of fixed points of a nonexpansive mapping in real 2-uniformly smooth and uniformly convex Banach spaces. We prove the strong convergence of the proposed iterative method without the condition of weakly sequentially continuous duality mapping. Our result improves and extends the corresponding results announced by many others. MSC: 46B10; 46B20; 47H10; 49J40 Keywords: a new general system of variational inequalities; relaxed cocoercive mapping; strong convergence Ax-Ay ≤ L xy , ∀x, y ∈ C; (ii) accretive if there exists j(xy) ∈ J(xy) such that Ax-Ay, j(xy) ≥ , ∀x, y ∈ C; (iii) α-inverse strongly accretive if there exist j(xy) ∈ J(xy) and α >  such that Ax-Ay, j(xy) ≥ α Ax-Ay  , ∀x, y ∈ C;

Relaxed and composite viscosity methods for variational inequalities, fixed points of nonexpansive mappings and zeros of accretive operators

Fixed Point Theory and Applications, 2014

In this paper, we present relaxed and composite viscosity methods for computing a common solution of a general systems of variational inequalities, common fixed points of infinitely many nonexpansive mappings and zeros of accretive operators in real smooth and uniformly convex Banach spaces. The relaxed and composite viscosity methods are based on Korpelevich's extragradient method, the viscosity approximation method and the Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for relaxed and composite viscosity algorithms not only in the setting of a uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results given in the literature.

Strong convergence theorems for solving a general system of finite variational inequalities for finite accretive operators and fixed points of nonexpansive semigroups with weak contraction mappings

Fixed Point Theory and Applications, 2012

In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to find the solutions of the class of k-strictly pseudocontractive mappings and apply a general system of finite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Ceng et al. (2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yao et al. (2010) and many other authors. MSC: Primary 47H05; 47H10; 47J25

Extragradient Method for Solutions of Variational Inequality Problems in Banach Spaces

Abstract and Applied Analysis, 2013

We introduce an iterative process which converges strongly to solutions of a certain variational inequity problem for-inverse strongly accretive mappings in the set of common fixed points of finite family of strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

A New Extragradient-Viscosity Algorithm for Finite Families of Asymptotically Nonexpansive Mappings and Variational Inequality Problems in Banach Spaces

IJNNA

ln this paper, a new approach for finding common element of the set of solutions of the variational inequality problem for accretive mappings and the set of fixed points for asymptotically nonexpansive mappings is introduced and studied. Consequently, strong convergence results for finite families of asymptotically nonexpansive mappings and variational inequality problems are established in the setting of uniformly convex Banach space and 2-uniformly smooth Banach space. Furthermore, we prove that a slight modification of our novel scheme could be applied in finding common element of solution of variational inequality problems in Hilbert space. Our results improve, extend and generalize several recently announced results in literature.. . .

Strong convergence for solving a general system of variational inequalities and fixed point problems in Banach spaces

Journal of Inequalities and Applications, 2013

In this paper, we propose and analyze some iterative algorithms by hybrid viscosity approximation methods for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex Banach space which has a uniformly Gâteaux differentiable norm, and we prove some strong convergence theorems under appropriate conditions. The results presented in this paper improve, extend, supplement and develop the corresponding results recently obtained in the literature. MSC: 49J30; 47H09; 47J20

Strong convergence theorems for variational inequalities and fixed point problems in Banach spaces

Carpathian Journal of Mathematics

In this paper, using a sunny generalized non-expansive retraction which is different from the metric projection and generalized metric projection in Banach spaces, we present a retractive iterative algorithm of Krasnosel’skii-type, whose sequence approximates a common solution of a mono-variational inequality of a finite family of η-strongly-pseudo-monotone-type maps and fixed points of a countable family of generalized non-expansive-type maps. Furthermore, some new results relevant to the study are also presented. Finally, the theorem proved complements, improves and extends some important related recent results in the literature.