Accretive Mappings Research Articles

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.

Convergence of hybrid steepest descent method for variational inequalities in Banach spaces

Let E be a q-uniformly smooth real Banach space with constant d q , q > 1. Let T i : E → E, i = 1, 2, … , r be a finite family of nonexpansive mappings with K ≔ ∩ i = 1 r Fix ( T i ) ≠ ∅ and K = Fix( T r T r−1 … T 1) = Fix( T 1 T r … T 2) = ⋯ = Fix( T r−1 T r−2 … T r ). Let G : E → E be an η-strongly accretive map which is also κ-Lipschitzian. A hybrid steepest descent method introduced by Yamada [25] and studied by various authors is proved to converge strongly to the unique solution of the variational inequality problem VI( G, K) in q-uniformly smooth real Banach space, in particular, in L p spaces 1 < p < ∞.

The Over-Relaxed A-Proximal Point Algorithm for General Nonlinear Mixed Set-Valued Inclusion Framework

The purpose of this paper is (1) a general nonlinear mixed set-valued inclusion framework for the over-relaxed -proximal point algorithm based on the ( , )-accretive mapping is introduced, and (2) it is applied to the approximation solvability of a general class of inclusions problems using the generalized resolvent operator technique due to Lan-Cho-Verma, and the convergence of iterative sequences generated by the algorithm is discussed in -uniformly smooth Banach spaces. The results presented in the paper improve and extend some known results in the literature.

A general iterative algorithm for nonexpansive mappings in Banach spaces

Let $E$ be a real $q$-uniformly smooth Banach space whose duality map is weakly sequentially continuous. Let $T:E\to E$ be a nonexpansive mapping with $F(T)\neq\emptyset.$ Let $A:E\to E$ be an $\eta$-strongly accretive map which is also $\kappa$-Lipschitzian. Let $f:E\to E$ be a contraction map with coefficient $0\lt \alpha\lt1.$ Let a sequence $\{y_{n}\}$ be defined iteratively by $y_{0}\in E,~~ y_{n+1}=\alpha_n\gamma f(y_n)+(I-\alpha_n\mu A)Ty_n,n\geq0,$ where $\{\alpha_n\},~~\gamma$ and $\mu$ satisfy some appropriate conditions. Then, we prove that $\{y_{n}\}$ converges strongly to the unique solution $x^{*} \in F(T)$ of the variational inequality $\langle(\gamma f-\mu A)x^{*},j(y-x^{*})\rangle\leq0,~\forall~y\in F(T).$ Convergence of the correspondent implicit scheme is also proved without the assumption that $E$ has weakly sequentially continuous duality map. Our results are applicable in $l_{p}$ spaces, $1\lt p \lt \infty$.

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.

New perturbed finite step iterative algorithms for a system of extended generalized nonlinear mixed quasi-variational inclusions

This paper introduces a new system of extended generalized nonlinear mixed quasi-variational inclusions involving A -maximal m -relaxed η -accretive (so called ( A , η ) -accretive (Lan et al. (2006) [37] )) mappings in q -uniformly smooth Banach spaces. By using the resolvent operator technique for A -maximal m -relaxed η -accretive mappings due to Lan et al., we establish the existence and uniqueness of solution for this system of extended generalized nonlinear mixed quasi-variational inclusions and construct a new perturbed N -step iterative algorithm with mixed errors for solving the mentioned system. We also prove the convergence of the sequences generated by our algorithms in q -uniformly smooth Banach spaces. The results presented in this paper extend and improve some known results in the literature.

Ishikawa Iteration Process for Nonlinear Φ-strongly Pseudocontractive and Strongly Accretive Mappings &lt;I&gt;q&lt;/I&gt;-Uniformly Smooth Banach Spaces

Ishikawa Iteration Process for Nonlinear Φ-strongly Pseudocontractive and Strongly Accretive Mappings &lt;I&gt;q&lt;/I&gt;-Uniformly Smooth Banach Spaces

SENSITIVITY ANALYSIS FOR A NEW SYSTEM OF VARIATIONAL INEQUALITIES

In this paper, we study the behavior and sensitivity analysis of the solution set for a new system of generalized parametric multi-valued variational inclusions with (A,·)-accretive mappings in q-uniformly smoo- th Banach spaces. The present results improve and extend many known results in the literature. techniques. By using the projection method, Dafermos (2), Yen (12), Mukherjee and Verma (7), Noor (9) and Pan (10) studied the sensitivity analysis of solutions of some variational inequalities with single-valued mappings in finite-dimensional spaces or Hilbert spaces. By using the resolvent operator technique, Agarwal et al. (1), Jeong (3) stud- ied a new system of parametric generalized nonlinear mixed quasi-variational inclusions in a Hilbert space and in Lp(p ‚ 2) spaces, respectively. In 2008, using the concept and technique of resolvent operators, Lan (4) introduced and studied the behavior and sensitivity analysis of the solution set for a system of generalized parametric (A,·)-accretive variational inclusions in Banach spaces. Motivated and inspired by the research work going on this field, in this paper, we study the behavior and sensitivity analysis of the solution set for a new system of generalized parametric multi-valued variational inclusions with (A,·)-accretive mappings in q-uniformly smooth Banach spaces. The present results improve and extend many known results in the literature.

Fuzzy nonlinear set-valued variational inclusions

The purpose of this paper is to study a new class of fuzzy nonlinear set-valued variational inclusions in real Banach spaces. By using the fuzzy resolvent operator techniques for m -accretive mappings, we establish the equivalence between fuzzy nonlinear set-valued variational inclusions and fuzzy resolvent operator equation problem. Applying this equivalence and Nadler’s theorem, we suggest some iterative algorithms for solving fuzzy nonlinear set-valued variational inclusions in real Banach spaces. By using the inequality of Petryshyn, the existence of solutions for these kinds of fuzzy nonlinear set-valued variational inclusions without compactness is proved and convergence criteria of iterative sequences generated by the algorithm are also discussed.

Necessary and Sufficient Condions for Strong Convergence of Ishikawa Iterative Sequence of Variational Inclusions with &lt;I&gt;φ&lt;/I&gt;-Strongly Accretive Type Mappings

Necessary and Sufficient Condions for Strong Convergence of Ishikawa Iterative Sequence of Variational Inclusions with &lt;I&gt;φ&lt;/I&gt;-Strongly Accretive Type Mappings

Iterative Solution of Strongly Accretive Operator Equation in Normed Linear Space

Under the condition of removing the restriction bounder range {(I-A)xn},We study the problems of the convergence of the sequence defined by the steepest descent method for the zero of strongly accretive mapping in normed linear space by using a new analytical method,which improve and extend some related results obtained by LI Xiaoling and LIU Liwei.

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.

An Ishikawa-Hybrid Proximal Point Algorithm for Nonlinear Set-Valued Inclusions Problem Based on -Accretive Framework

A general nonlinear framework for an Ishikawa-hybrid proximal point algorithm using the notion of -accretive is developed. Convergence analysis for the algorithm of solving a nonlinear set-valued inclusions problem and existence analysis of solution for the nonlinear set-valued inclusions problem are explored along with some results on the resolvent operator corresponding to -accretive mapping due to Lan-Cho-Verma in Banach space. The result that sequence generated by the algorithm converges linearly to a solution of the nonlinear set-valued inclusions problem with the convergence rate is proved.

Approximation of Solutions to a System of Variational Inclusions in Banach Spaces

The purpose of this paper is to introduce an iterative method for finding solutions of a general system of variational inclusions with inverse-strongly accretive mappings. Strong convergence theorems are established in uniformly convex and 2-uniformly smooth Banach spaces.

SENSITIVITY ANALYSIS FOR A SYSTEM OF GENERALIZED NONLINEAR MIXED QUASI-VARIATIONAL INCLUSIONS WITH (A, η)-ACCRETIVE MAPPINGS IN BANACH SPACES

In this paper, we study the behavior and sensitivity analysis of the solution set for a new system of parametric generalized nonlinear mixed quasi-variational inclusions with (A, <TEX>${\eta$</TEX>)-accretive mappings in quniformly smooth Banach spaces. The present results improve and extend many known results in the literature.

The generalized relaxed proximal point algorithm involving [formula omitted]-maximal-relaxed accretive mappings with applications to Banach spaces

A general design for the relaxed proximal point algorithm using the notion of A -maximal-relaxed accretiveness is developed, and then the convergence analysis for this algorithm in the context of solving a general class of nonlinear inclusion problems is examined along with some other results of interest involving A -maximal-relaxed accretive mappings in a real Banach space setting.

Applications of perturbations on accretive mappings to nonlinear elliptic systems involving (p, q)-laplacian

Using perturbation results on sums of ranges of nonlinear accretive mappings of Calvert and Gupta, we present some abstract results about the existence of solutions of nonlinear Neumann elliptic systems involving the (p, q)-Laplacian. The systems discussed in this paper and the method used extend and complement some previous works.

General proximal point algorithm involving η-maximal accretiveness framework in Banach spaces

First, based on η-maximal accretiveness, a generalization to Rockafellar’s theorem (1976) in the context of approximating a solution to a general inclusion problem involving a multivalued η-maximal accretive mapping using the proximal point algorithm in a q-uniformly smooth Banach space setting is considered. Then an application to a minimization problem of a functional is examined. The general framework for η-maximal accretiveness generalizes the general theory of multivalued maximal monotone mappings.