Accretive Mappings Research Articles

Extragradient Method for Solutions of Variational Inequality Problems in Banach Spaces

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.

Strong convergence theorems for the approximation of fixed points of demicontinuous pseudocontractive mappings

In this paper, we introduce a new explicit iterative scheme for approximation of fixed points of demicontinuous pseudocontractive mappings in uniformly smooth Banach spaces and prove strong convergence of our proposed iterative scheme. Furthermore, we modify our explicit iterative scheme for approximation of zeroes of bounded demicontinuous accretive mappings in uniformly smooth Banach spaces. Our result improves, extends and unifies most of the results that have been proved for this class of mappings.

Convergence of a Hybrid Iterative Scheme for Fixed Points of Nonexpansive Maps, Solutions of Equilibrium, and Variational Inequalities Problems

Let be a closed, convex, and nonempty subset of a real -uniformly smooth Banach space , which is also uniformly convex. For some , let and be family of nonexpansive maps and -inverse strongly accretive map, respectively. Let be a bifunction satisfying some conditions. Let be a nonexpansive projection of onto . For some fixed real numbers , , and arbitrary but fixed vectors , let and be sequences generated by , , , , , where is fixed, and are sequences satisfying appropriate conditions. If , under some mild conditions, we prove that the sequences and converge strongly to some element in .

Ishikawa-hybrid proximal point algorithm for NSVI system

A nonlinear set-valued inclusions system framework for an Ishikawa-hybrid proximal point algorithm is developed and studied using the notion of an (A, η)-accretive mapping. Convergence analysis for the algorithm of solving the nonlinear set-valued inclusions system and existence analysis of solution for the system are explored along with some results on the resolvent operator corresponding to the (A, η)-accretive mapping in a Banach space. The result that the sequence generated by the algorithm converges linearly to a solution of the system with the convergence rate � � is proved. MSC: 49J40; 47H06

A general iteration scheme for variational inequality problem and common fixed point problems of nonexpansive mappings in q-uniformly smooth Banach spaces

In this paper, we introduce a general iterative algorithm for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of systems of variational inequalities for two inverse strongly accretive mappings in a q-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as improvement, supplementation, development and extension of the corresponding results in some references to a great extent.

1-set-contractive perturbations of accretive operators in Menger PN-spaces

Based on the topological degree for 1-set-contractive fields established in [11], we discuss the 1-set-contractive perturbation and the existence of zero points for nonlinear equations with accretive mappings in Menger PN-spaces and obtain some new results.

Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type with Lipschitz and Bounded Nonlinear Operators

Let E be a reflexive real Banach space with uniformly Gâteaux differentiable norm and F, K : be Lipschitz accretive maps with Suppose that the Hammerstein equation has a solution. An explicit iteration method is shown to converge strongly to a solution of the equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorems are significant improvements on important recent results (e.g., (Chiume and Djitte, 2012)).

Approximation of solutions of generalized equations of Hammerstein type

Let E be a 2-uniformly smooth real Banach space. For each i=1,2,…,m, let Fi,Ki:E→E be bounded and accretive mappings. Assume that the generalized Hammerstein equation u+∑i=1mKiFiu=0 has a solution in E. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein equation. Our result improves, generalizes and extends the results of Chidume and Ofoedu [C.E. Chidume, E.U. Ofoedu, Solution of nonlinear integral equations of Hammerstein type, Nonlinear Anal. 74 (2011) 4293–4299] and Chidume and Shehu [C.E. Chidume, Y. Shehu, Approximation of solutions of generalized equations of Hammerstein type, Comput. Math. Appl. 63 (2012) 966–974].

Strong convergence of an new iterative method for a zero of accretive operator and nonexpansive mapping

Let E be a Banach space and A an m-accretive operator with a zero. Consider the iterative method that generates the sequence {xn} by the algorithm xn+1 = αn γφ (xn )+( I − αnF)Jrn xn, where {an} and {rn} are two sequences satisfying certain conditions, Jrn denotes the resolvent (I + rnA) -1 for rn >0 ,F be a strongly positive bounded linear operator on E is 0 <γ < ¯ γ, and j be a MKC on E. Strong convergence of the algorithm {xn} is proved assuming E either has a weakly continuous duality map or is uniformly smooth. MSC: 47H09; 47H10

Strong convergence theorems for variational inequality problems and fixed point problems in uniformly smooth and uniformly convex Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings and the set of common fixed points for a nonexpansive mapping in a uniformly smooth and uniformly convex Banach space. We obtain a strong convergence theorem under some suitable conditions. Our results improve and extend the recent ones announced by many others in the literature.

A New System of Random Generalized Variational Inclusions with Random Fuzzy Mappings and Random --Accretive Mappings in Banach Spaces

We introduce a new notion of random --accretive mappings and prove the Lipschitz continuity of the random resolvent operator associated with the random --accretive mappings. We introduce and study a new system of random generalized variational inclusions with random --accretive mappings and random fuzzy mappings in Banach spaces. By using the random resolvent operator, an iterative algorithm for solving such system of random generalized variational inclusions is constructed in Banach spaces. Under some suitable conditions, we prove the convergence of the iterative sequences generated by the algorithm.

Modified extragradient methods for variational inequality problems and fixed point problems for an infinite family of nonexpansive mappings in Banach spaces

In this paper, we introduce a new general iterative algorithm for finding a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general variational inequality for two inverse-strongly accretive mappings in Banach space. We obtain some strong convergence theorems by a modified extragradient method under suitable conditions. Our results extend the recent results announced by many others.

Graphical approximation of common solutions to generalized nonlinear relaxed cocoercive operator equation systems with (A, η)-accretive mappings

Abstract In this paper, we develop a new perturbed iterative algorithm framework with errors based on the variational graphical convergence of operator sequences with (A, η)-accretive mappings in Banach space. By using the generalized resolvent operator technique associated with (A, η)-accretive mappings, we also prove the existence of solutions for a class of generalized nonlinear relaxed cocoercive operator equation systems and the variational convergence of the sequence generated by the perturbed iterative algorithm in q-uniformly smooth Banach spaces. The obtained results improve and generalize some well-known results in recent literatures. 2000 Mathematics Subject Classification: 47H05; 49J40

ITERATIVE ALGORITHM FOR A NEW SYSTEM OF GENERALIZED SET-VALUED QUASI-VARIATIONAL-LIKE INCLUSIONS WITH (A, ${\eta}$)-ACCRETIVE MAPPINGS IN BANACH SPACES

In this paper, we introduce and study a new system of generalized set-valued quasi-variational-like inclusions with (A, )-accretive mapping in Banach spaces. By using the resolvent operator associated with (A, )-accretive mappings, we construct a new iterative algorithm for approximating the solution of this system of variational inclusions. We also prove the existence of solutions and the convergence of the sequences generated by the algorithm in Banach spaces. The results presented in this paper extend and improve some known results in the literature.

The Meir‐Keeler Type for Solving Variational Inequalities and Fixed Points of Nonexpansive Semigroups in Banach Spaces

The aim of this paper is to introduce a new iterative scheme for finding common solutions of the variational inequalities for an inverse strongly accretive mapping and the solutions of fixed point problems for nonexpansive semigroups by using the modified viscosity approximation method associate with Meir‐Keeler type mappings and obtain some strong convergence theorem in a Banach spaces under some parameters controlling conditions. Our results extend and improve the recent results of Li and Gu (2010), Wangkeeree and Preechasilp (2012), Yao and Maruster (2011), and many others.

Study on the Existence and Uniqueness of Solution of Generalized Capillarity Problem

By using the perturbation theories on sums of ranges of nonlinear accretive mappings of Calvert and Gupta (1978), the abstract result on the existence and uniqueness of the solution in Lp(Ω) of the generalized Capillarity equation with nonlinear Neumann boundary value conditions, where 2N/(N + 1) &lt; p &lt; +∞ and N(≥1) denotes the dimension of RN, is studied. The equation discussed in this paper and the methods here are a continuation of and a complement to the previous corresponding results. To obtain the results, some new techniques are used in this paper.

An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth 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 notself strict pseudocontractions and the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings in q-uniformly smooth Banach space. We obtain some strong convergence theorems under suitable conditions. Our results improve and extend the recent results announced by Qin et al. (J Comput Appl Math 233:231–240, 2009), Yao et al. (Acta Appl Math 110:1211–1244, 2010) and many others.

--Accretive Mappings and a New System of Generalized Variational Inclusions with --Accretive Mappings in Banach Spaces

We introduce a new class of generalized accretive mappings, named --accretive mappings, in Banach spaces. We define a resolvent operator associated with --accretive mappings and show its Lipschitz continuity. We also introduce and study a new system of generalized variational inclusions with --accretive mappings in Banach spaces. By using the resolvent operator technique associated with --accretive mappings, we construct a new iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We also prove the existence of solutions for the generalized variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.

Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces

In this paper, we study a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of variational inequalities for two inverse strongly accretive mappings and the set of common fixed points of an infinite family of nonexpansive mappings in a real 2-uniformly smooth and uniformly convex Banach space. We prove some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the corresponding results announced by many others.

Another weak convergence theorems for accretive mappings in banach spaces

AbstractWe present two weak convergence theorems for inverse strongly accretive mappings in Banach spaces, which are supplements to the recent result of Aoyama et al. [Fixed Point Theory Appl. (2006), Art. ID 35390, 13pp.].2000 MSC: 47H10; 47J25.