REGULARIZATION METHODS FOR NONLINEAR ILL-POSED PROBLEMS WITH ACCRETIVE OPERATORS

This article is devoted to the regularization of nonlinear ill-posed problems with accretive operators in Banach spaces. The data involved are assumed to be known approximately. The authors concentrate their discussion on the convergence rates of regular solutions.

On the Characteristics of the Convergence of Ishikawa Type Iterative Sequences for Strong Pseudocontractions and Strongly Accretive Operators

In this paper, the results characterize the convergence of Ishikawa type iterative sequences (with errors) for constructing the solutions of strongly accretive operator equations, the solutions of rn-accretive operator equations, and the fixed points of strong pseudocontractions. These results extend and improve Theorems 1-3 of Chidume and Osilike (Nonlinear Anal. TMA, 1999, 36(7): 863-872).

Remarks on the Mapping Theorems Involving Compact Perturbations of M-Accretive Operators in Banach Spaces

In the present paper, by virtue of new approch techniques, we obtain several mapping theorems involving compact perturbations of m—accretive operators.These results improve and extend the corresponding those obtained by Kartsatos,Zhu, and Kartsatos and Mabry.

Viscosity Approximations by Generalized Contractions for Resolvents of Accretive Operators in Banach Spaces

In this paper, we prove a strong convergence theorem for resolvents of accretive operators in a Banach space by the viscosity approximation method with a generalized contraction mapping. The proximal point algorithm in a Banach space is also considered. The results extend some very recent theorems of W. Takahashi.

ISHIKAWA TYPE ITERATIVE SEQUENCES WITH ERRORS FOR LIPSCHITZIAN φ-STRONGLY ACCRETIVE OPERATOR EQUATIONS IN ARBITRARY BANACH SPACES

In this paper, we investigate the problem of approximating solutions of the equations of Lipschitzian ψ-strongly accretive operators and fixed points of Lipschitzian ψ-hemicontractive operators by lshikawa type iterative sequences with errors. Our results unify, improve and extend the results obtained previously by several authors including Li and Liu (Acta Math. Sinica 41 (4)(1998), 845-850), and Osilike (Nonlinear Anal. TMA, 36(1)(1999), 1-9), and also answer completely the open problems mentioned by Chidume (J. Math. Anal. Appl. 151 (2)(1990), 453-461).

A Note on Iterative Solutions of Nonlinear Accretive Operator Equations

Suppose that X is a real Banach space, H: X→X is a Lipschitz operator, T: X→X is a uniformly continuous operator with bounded range, and H+T is strongly accretive. Then the Ishikawa iteration process converges strongly to the unique solution of the equation Hx+Tx=f . This conclusion extends the corresponding results in recent papers.

ITERATIVE PROCESS TO -HEMICONTRACTIVE OPERATOR AND-STRONGLY ACCRETIVE OPERATOR EQUATIONS

Let E be an arbitrary real Banach space and K be a nonempty closed convex subsets of E. Let T:K→K be a uniformly continuous _hemicontractive operator with bounded range and a n,b n,c n,a ′ n,b ′ n,c ′ n be sequences in [0,1] satisfying:ⅰ) a n+b n+c n=a ′ n+b ′ n+c ′ n=1. n≥0; ⅱ) lim b n= lim b ′ n= lim c ′ n= 0; ⅲ)∑∞n=0b n=∞; ⅳ) c n=o(b n). For any given x 0,u 0,v 0∈K, define the Ishikawa type iterative sequence x n as follows: x n+1 =a nx n+b nTy n+c nu n, y n=a ′ nx n+b ′ nTx n+c ′ nv n (n≥0), where u n and v n are bounded sequences in K. Then x n converges strongly to the unique fixed point of T. Related result deals with the convergence of Ishikawa type iterative sequence to the solution of _strongly accretive operator equations.

ITERATIVE PROCESS FOR CERTAIN NONLINEAR MAPPINGS WITH LIPSCHITZ CONDITION

Using the new analysis techniques, the problem of iterative approximation of solutions of the equation for Lipschitz phi-strongly accretive operators and of fixed points for Lipschitz phi-strongly pseudo-contractive mappings are discussed. The main results of this paper improve and extend the corresponding results obtained by Chang, Chidume, Deng, Ding, Tan-Xu and Osilike.

A Note on Modified Mann Accretive Iterations for Zero Points of Operators

A more general form of modified Mann iterations which converges strongly to a zero point of an m-accretive operator is given. The work in this paper is an extension and complement of the corresponding result of Kim T.H. and Xu H.K in 2005

Approximation of Fixed Point and Solution for -Hemicontraction and -Strongly Quasi-Accretive Operator without Lipschitz Assumption

Let X be a real uniformly smooth Banach space and let T:D(T)(?)X→Xbe (?)-hemicontractive and locally bounded at its fixed point q∈F(T).Under somesuitable assumptions on the iteration parameters {αn}and{βn},we have proved thatthe Mann and Ishikawa iteration processes for T converge strongly to the unique fixedpoint q of T.Several related results deal with iterative solutions of nonlinear equationsinvolving (?)-strongly quasi-accretive operators.Our results extend and generalize thosecorresponding ones by Xu and Roach,Zhou and Jia and others.

Generalized Krasnoselskii-Mann-Type Iteration for Nonexpansive Mappings in Banach Spaces

The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings,and it is well known that the clas-sic Krasnoselskii-Mann iteration is weakly convergent in Hilbert spaces.The weak convergence is also known even in Banach spaces.Recently,Kanzow and Shehu pro-posed a generalized Krasnoselskii-Mann-type iteration for nonexpansive mappings and established its convergence in Hilbert spaces.In this paper,we show that the generalized Krasnoselskii-Mann-type iteration proposed by Kanzow and Shehu also converges in Banach spaces.As applications,we proved the weak convergence of generalized proximal point algorithm in the uniformly convex Banach spaces.

Accretivity of the General Second Order Linear Differential Operator

For the general second order linear differential operator ■ with complex-valued distributional coefficients a_(jk), b_j, and c in an open set Ω ? R^n(n ≥ 1), we present conditions which ensure that-L^0 is accretive, i.e., Re<-L_0φ, φ >≥ 0 for all φ∈ C_0~∞(Ω).