Accretive Mappings In Banach Spaces Research Articles

H(., .)-φ-η-mixed accretive mapping with an application

The motive of this article to introduce the notion called H(., .)-φ-η-mixed accretive mappings in Banach spaces. We generalized the idea of proximal-point mappings related with generalized m-accretive mappings to the H(., .)-φ-η-mixed accretive mappings and perusal its aspects single-valued property as well as Lipschitz continuity. Since proximal point mapping plays an important role to solve variational inclusion problems. Therefore, we design an iterative algorithm involving introduced proximal point mapping to solve variational inclusion problem. In last, we discuss its convergence with considerable assumptions.

Newton–Kantorovich regularization method for nonlinear ill-posed equations involving $$m-$$accretive operators in Banach spaces

In this paper, we study nonlinear ill-posed problems involving $$m-$$accretive mappings in Banach spaces. We consider Newton–Kantorovich regularization method for the implementation of Lavrentiev regularization method. Using general Holder type source condition we obtain an error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005) for choosing the regularization parameter.

Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces

The purpose of this article is to propose a splitting algorithm for finding a common zero of a finite family of inclusion problems of accretive operators in Banach space. Under suitable conditions, some strong convergence theorems of the sequence generalized by the algorithm to a common zero of the inclusion problems are proved. Some applications to the convex minimization problem, common fixed point problem of a finite family of pseudocontractive mappings, and accretive variational inequality problem in Banach spaces are presented.

Variational inclusion governed by αβ-H((.,.),(.,.))-mixed accretive mapping

In this paper, we look into a new concept of accretive mappings called ??-H((.,.),(.,.))-mixed accretive mappings in Banach spaces. We extend the concept of proximal-point mappings connected with generalized m-accretive mappings to the ??-H((.,.),(.,.))-mixed accretive mappings and discuss its characteristics like single-valuable and Lipschitz continuity. Some illustration are given in support of ??-H((.,.),(.,.))-mixed accretive mappings. Since proximal point mapping is a powerful tool for solving variational inclusion. Therefore, As an application of introduced mapping, we construct an iterative algorithm to solve variational inclusions and show its convergence with acceptable assumptions.

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.

--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.

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.

A system of generalized variational inclusions involving generalized H(·, ·)-accretive mapping in real q-uniformly smooth Banach spaces

In this paper, we consider a class of accretive mappings called generalized H(·, ·)-accretive mappings in Banach spaces. We prove that the proximal-point mapping of the generalized H(·, ·)-accretive mapping is single-valued and Lipschitz continuous. Further, we consider a system of generalized variational inclusions involving generalized H(·, ·)-accretive mappings in real q-uniformly smooth Banach spaces. Using proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as a refinement and improvement of some known results in the literature.

MULTIVALUED VERSIONS OF A BOLZANO'S THEOREM

The intermediate value theorem for a continuous real valued function is a kind of Bolzano's theorem. Similar results also hold for compact, monotone or accretive mappings in Banach spaces. In this paper we give multivalued versions of Bolzano's theorem.

A HYBRID PROXIMAL POINT ALGORITHM AND STABILITY FOR SET-VALUED MIXED VARIATIONAL INCLUSIONS INVOLVING (A, ${\eta}$)-ACCRETIVE MAPPINGS

Abstract. A new class of nonlinear set-valued mixed variational inclu-sions involving (A;)-accretive mappings in Banach spaces is introducedand studied, which includes many kind of variational inclusion (inequal-ity) and complementarity problems as special cases. By using the resol-vent operator associated with (A;)-accretive operator due to Lan-Cho-Verma, the existence of solution for this kind of variational inclusion isproved, and a new hybrid proximal point algorithm is established and sug-gested, the convergence and stability theorems of iterative sequences gen-erated by new iterative algorithms are also given in q-uniformly smoothBanach spaces. 1. IntroductionThe variational inclusion, which was introduced and studied by Hassouniand Mouda [7], is a useful and important generalization of the variational in-equality. Various variational inclusions have been intensively studied in recentyears. Many authors (see, [1], [3], [4], [5], [6], [10], [11], [12], [14], [15], [19])introduced the concepts of -subdi erential operators, maximal -monotoneoperators, H-monotone operators, A-monotone operators, (H;)-monotone op-erators, (A;)-accretive mappings, (G;)-monotone operators, and de ned re-solvent operators associated with them, respectively.Moreover, by using the resolvent operator technique, many authors con-structed some approximation algorithms for some nonlinear variational inclu-sions in Hilbert spaces or Banach spaces. Recently, Verma [16] has developeda hybrid version of the Eckstein-Bertsekas [2] proximal point algorithm, intro-duced the algorithm based on the (A;)-maximal monotonicity framework andstudied convergence of the algorithm.

A Hybrid Proximal Point Three-Step Algorithm for Nonlinear Set-Valued Quasi-Variational Inclusions System Involving -Accretive Mappings

The main purpose of this paper is to introduce and study a new class of generalized nonlinear set-valued quasi-variational inclusions system involving -accretive mappings in Banach spaces. By using the resolvent operator due to Lan-Cho-Verma associated with -accretive mappings and the matrix analysis method, we prove the convergence of a new hybrid proximal point three-step iterative algorithm for this system of set-valued variational inclusions and an existence theorem of solutions for this kind of the variational inclusions system. The results presented in this paper generalize, improve, and unify some recent results in this field.

Path Convergence and Approximation of Common Zeroes of a Finite Family of m‐Accretive Mappings in Banach Spaces

Let E be a real Banach space which is uniformly smooth and uniformly convex. Let K be a nonempty, closed, and convex sunny nonexpansive retract of E, where Q is the sunny nonexpansive retraction. If E admits weakly sequentially continuous duality mapping j, path convergence is proved for a nonexpansive mapping T : K → K. As an application, we prove strong convergence theorem for common zeroes of a finite family of m‐accretive mappings of K to E. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings from K to E under certain mild conditions.

General Nonlinear Random Equations with Random Multivalued Operator in Banach Spaces

We introduce and study a new class of general nonlinear random multivalued operator equations involving generalized -accretive mappings in Banach spaces. By using the Chang's lemma and the resolvent operator technique for generalized -accretive mapping due to Huang and Fang (2001), we also prove the existence theorems of the solution and convergence theorems of the generalized random iterative procedures with errors for this nonlinear random multivalued operator equations in -uniformly smooth Banach spaces. The results presented in this paper improve and generalize some known corresponding results in iterature.

Stability of iterative processes with errors for a system of nonlinear [formula omitted]-accretive variational inclusions in Banach spaces

In this paper, by using the concept of ( A , η ) -accretive mappings and the new resolvent operator technique associated with ( A , η ) -accretive mappings, we introduce and study a system of general mixed quasivariational inclusions involving ( A , η ) -accretive mappings in Banach spaces, and construct a new perturbed iterative algorithm with mixed errors for this system of nonlinear ( A , η ) -accretive variational inclusions in q -uniformly smooth Banach spaces. Our results improve and generalize the corresponding results of recent works.

A new system of general nonlinear variational inclusions involving (A,η)-accretive mappings in Banach spaces

In this paper, a new systemof general nonlinear variational inclusions involving (A,η) accretive mappings in Banach spaces is introduced and studied, which includes many variational inequality (inclusion) problems as special cases. By using the resolvent operator technique for (A,η) -accretive mapping due to Lan-Cho-Verma, an existence and uniqueness theorem of solutions for this system of variational inclusion is proved. A new iterative algorithm for finding approximate solution of this system variational inclusion is suggested and discussed, the convergence and stability of iterative sequence generated by new iterative algorithm is also given. The theorems presented in this paper improve and unify many known results variational inequalities and variational inclusions. Mathematics subject classification (2000): 49J40, 47H17, 47H10.

Generalized nonlinear random [formula omitted]-accretive equations with random relaxed cocoercive mappings in Banach spaces

In this paper, we introduce and study a new class of generalized nonlinear random ( A , η ) -accretive equations with random relaxed cocoercive mappings in Banach spaces. By using the Chang’s lemma and the resolvent mapping technique for ( A , η ) -accretive mappings due to Lan et al. [H.Y. Lan, Y.J. Cho, R.U. Verma, Nonlinear relaxed cocoercive variational inclusions involving ( A , η ) -accretive mappings in Banach spaces, Comput. Math. Appl. 51 (2006) 1529–1538], we also prove the existence theorems of the solution and convergence theorems of the generalized random iterative procedures with errors for these nonlinear random equations in q -uniformly smooth Banach spaces. The results presented in this paper improve and generalize some known corresponding results in the literature.

Nonlinear parametric multi-valued variational inclusion systems involving [formula omitted]-accretive mappings in Banach spaces

In this paper, by using the new parametric resolvent operator technique associated with ( A , η ) -accretive mappings, we analyze and establish an existence theorem for new nonlinear parametric multi-valued variational inclusion systems involving ( A , η ) -accretive mappings in Banach spaces. Our results generalize sensitivity analysis results of other recent works on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions and nonlinear mixed quasi-variational inclusion systems.

Iterative algorithms for a new system of nonlinear variational inclusions with [formula omitted]-accretive mappings in Banach spaces

In this paper we introduce and study a new system of nonlinear variational inclusions with ( A , η ) -accretive mappings in Banach spaces. By using the resolvent operator associated with ( A , η ) -accretive mappings, we construct some new iterative algorithms 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.

Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively Φ-pseudocontractive mappings without Lipschitzian assumptions

In this paper, the equivalence of the strong convergence between the modified Mann and Ishikawa iterations with errors in two different schemes by Xu [Y.G. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998) 91–101] and Liu [L.S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114–125] respectively is proven for the generalized strongly successively Φ-pseudocontractive mappings without Lipschitzian assumption. Our results generalize the recent results of the papers [Zhenyu Huang, F. Bu, The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption, J. Math. Anal. Appl. 325 (1) (2007) 586–594; B.E. Rhoades, S.M. Soltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289 (2004) 266–278; B.E. Rhoades, S.M. Soltuz, The equivalence between Mann–Ishikawa iterations and multi-step iteration, Nonlinear Anal. 58 (2004) 219–228] by extending to the most general class of the generalized strongly successively Φ-pseudocontractive mappings and hence improve the corresponding results of all the references given in this paper by providing the equivalence of convergence between all of these iteration schemes for any initial points u 1 , x 1 in uniformly smooth Banach spaces.

Nonlinear relaxed cocoercive variational inclusions involving ( A, η)-accretive mappings in banach spaces

In this paper, we introduce a new concept of ( A, η)-accretive mappings, which generalizes the existing monotone or accretive operators. We study some properties of ( A, η)-accretive mappings and define resolvent operators associated with ( A, η)-accretive mappings. By using the new resolvent operator technique, we also construct a new perturbed iterative algorithm with mixed errors for a class of nonlinear relaxed Cocoercive variational inclusions involving ( A, η)-accretive mappings and study applications of ( A, η)-accretive mappings to the approximation-solvability of this class of nonlinear relaxed Cocoercive variational inclusions in q-uniformly smooth Banach spaces. Our results improve and generalize the corresponding results of recent works.