M-accretive Operators In Banach Spaces Research Articles

Hybrid Viscosity Approximation Method for Zeros of m-Accretive Operators in Banach Spaces

Let X be a uniformly smooth Banach space and A be an m-accretive operator on X with A −1(0) ≠ ∅. Assume that F: X → X is δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1. This article proposes hybrid viscosity approximation methods which combine viscosity approximation methods with hybrid steepest-descent methods. For each t ∈ (0, 1) and each integer n ≥ 0, let {x t, n } be defined by x t, n = tf(x t, n ) + (1 − t)[J r n x t, n − θ t F(J r n x t, n )] where f: X → X is a contractive map, {r n } ⊂ [ϵ, ∞) for some ϵ > 0 and {θ t : t ∈ (0, 1)} ⊂ [0, 1) with . We deduce that as t → 0, {x t, n } converges strongly to a zero p of A, which is a unique solution of some variational inequality. On the other hand, given a point x 0 ∈ X and given sequences {λ n }, {μ n } in [0, 1], {α n }, {β n } in (0, 1], let the sequence {x n } be generated by It is proven that under appropriate conditions {x n } converges strongly to the same zero p of A. The results presented here extend, improve and develop some very recent theorems in the literature to a great extent.

Convergence of composite iterative methods for finding zeros of accretive operators

Strong convergence theorems on composite iterative schemes by the viscosity approximation methods for finding a zero of an accretive operator are established in Banach spaces. The main results generalize the recent corresponding results of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007) 2350–2360], Ceng et al. [L.C. Ceng, A.R. Khan, Q.H. Ansari, J.C. Yao, Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces, Nonlinear Anal. 70 (2009) 1830–1840], Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643] to viscosity methods in a strictly convex and reflexive Banach space having a uniformly Gâteaux differentiable norm. Our results also improve the corresponding results of [T.D. Benavides, G.L. Acedo, H.K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr. 248–249 (2003) 62–71; R. Chen, Z. Zhu, Viscosity approximation fixed points for nonexpansive and m -accretive operators, Fixed Point Theory Appl. 2006 (2006) 1–10; S. Kamimura, W. Takahashi, Approximation solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory 106 (2000) 226–240; P.E. Maingé, Viscosity methods for zeroes of accretive operators, J. Approx. Theory 140 (2) (2006) 127–140; K. Nakajo, Strong convergence to zeros of accretive operators in Banach spaces, J. Nonlinear Convex Anal. 7 (2006) 71–81].

Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces

We introduce a new composite iterative scheme to approximate a zero of an m -accretive operator A defined on uniform smooth Banach spaces and a reflexive Banach space having a weakly continuous duality map. It is shown that the iterative process in each case converges strongly to a zero of A . The results presented in this paper substantially improve and extend the results due to Ceng et al. [L.C. Ceng, H.K. Xu, J.C. Yao, Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators, Taiwanese J. Math. (in press)], Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643]. Our work provides a new approach for the construction of a zero of m -accretive operators.

NONLINEAR EVOLUTION GOVERNED BY ACCRETIVE OPERATORS IN BANACH SPACES: ERROR CONTROL AND APPLICATIONS

Nonlinear evolution equations governed by m-accretive operators in Banach spaces are discretized via the backward or forward Euler methods with variable stepsize. Computable a posteriori error estimates are derived in terms of the discrete solution and data, and shown to converge with optimal order [Formula: see text]. Applications to scalar conservation laws and degenerate parabolic equations (with or without hysteresis) in L1, as well as to Hamilton–Jacobi equations in C0 are given. The error analysis relies on a comparison principle, for the novel notion of relaxed solutions, which combines and simplifies techniques of Benilan and Kružkov. Our results provide a unified framework for existence, uniqueness and error analysis, and yield a new proof of the celebrated Crandall–Liggett error estimate.

Compact perturbations of 𝑚-accretive operators in Banach spaces

This paper continues a discussion that arose twenty years ago, concerning the perturbation of an m m -accretive operator by a compact mapping in Banach spaces. Indeed, if A A is m m -accretive and g g is compact, then the boundary condition t x ∉ A ( x ) g ( x ) tx \notin A(x)g(x) for x ∈ ∂ G ∩ D ( A ) x \in \partial G \cap D(A) and t > 0 t>0 implies that 0 0 is in the closure of the range of A + g A+g . Perhaps the most interesting aspect of this result is the proof itself, which does not appeal to the classical degree theory argument used for this type of problem.

Existence theorems for m-accretive operators in Banach spaces

In 1985, the second author proved a surjective result for m-accretive and ϕ-expansive mappings for uniformly smooth Banach spaces. However, in this case, we have been able to remove the uniform smoothness of the Banach space, without any additional assumption.

Zeros and mapping theorems for perturbations of m-accretive operators in Banach spaces

Let X be a real Banach space and T : D( T) ⊂ X → 2 X be an m-accretive operator. Let C : D( T) ⊂ X → X be a bounded operator (not necessarily continuous) such that C( T + I) −1 is compact. Suppose that for every x ∈ D( T) with ∥ x∥ > r, there exists jx ∈ Jx such that 〈 u + C x , j x 〉 ≥ 0 , for all u ∈ Tx. Then, we have 0 ∈ ( T + C ) ( D ( T ) ∩ B r ( 0 ) ) , ¯ where B r (0) denotes the open ball of X with centre at zero and radius r > 0. Assume, furthermore, that T : D( T) → 2 X is strongly accretive. Then, 0 ∈ ( T + C)( D( T) ∩ B r (0)). As applications of the above zero theorem, we derive many new mapping theorems for perturbations of m-accretive operators in Banach spaces. When, T and C are odd operators, we also obtain some new mapping theorems.

Remarks on perturbation theory of maximal monotone and m-accretive operators in Banach spaces

In this paper we get and improve some results in the perturbation theory of maximal monotone and m-accretive operators having compact resolvents in Banach spaces, in which the composition of resolvent and perturbation is assumed compact.

Iterative processes with mixed errors for nonlinear equations with perturbed m-accretive operators in Banach spaces

Let X be a real uniformly smooth Banach space of which the dual X ∗ has a Fréchet differentiable norm. Let A: D( A)⊂ X→2 X be an m-accretive operator with closed domain D( A) and bounded range R( A) and S: X→ X a continuous and α-strongly accretive operator with bounded range R( I− S). It is proved that the Ishikawa and Mann iterative processes with mixed errors converge strongly to the unique solution of the equation z∈ Sx+ λAx for given z∈ X and λ>0.

Iterative approximation of solutions to nonlinear equations involving m-accretive operators in Banach spaces

Let E be an arbitrary real Banach space and T :E→E be a Lipschitz continuous accretive operator. Under the lack of the assumption lim n→∞ α n =lim n→∞ β n =0, we prove that the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation x+ Tx= f. Moreover, this result provides a convergence rate estimate for some special cases of such a sequence. Utilizing this result, we imply that if T :E→E is a Lipschitz continuous strongly accretive operator then the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation Tx= f. Our results improve, generalize and unify the ones of Liu, Chidume and Osilike, and to some extent, of Reich.

The Mann process for perturbed m-accretive operators in Banach spaces

This work was carried out while this author was visiting the University ofAlabama in Huntsville underthe nancial support by the LG Yonam Foundation, 1998, and he would like to thank Professor Claudio H.Morales for his hospitality in the Department of Mathematics.0362-546X/01/$-see front matter ? 2001 Elsevier Science Ltd. All rights reserved.PII: S0362-546X(00)00115-2

Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involving m-accretive operators in Banach spaces

Ishikawa-type and Mann-type iterative processes with errors for constructing solutions of nonlinear equations involving m-accretive operators in Banach spaces

On perturbations of M-accretive operators in Banach spaces

In this paper, we consider the solvability of nonlinear equations of the form Au+Cu 3 p where A is an m-accretive operator on a Banach space X, C is a mapping on X and p ∈ X.

Random Continuous Perturbations of Random m-Accretive Operators in Banach Spaces

Random Continuous Perturbations of Random m-Accretive Operators in Banach Spaces

Ranges of Perturbed Maximal Monotone and m-Accretive Operators in Banach Spaces

Ranges of Perturbed Maximal Monotone and m-Accretive Operators in Banach Spaces

Nonlinear equations involving compact perturbations of m-accretive operators in Banach spaces

Nonlinear equations involving compact perturbations of m-accretive operators in Banach spaces

Sets in the Ranges of Sums for Perturbations of Nonlinear m-Accretive Operators in Banach Spaces

Sets in the Ranges of Sums for Perturbations of Nonlinear m-Accretive Operators in Banach Spaces

Ranges of perturbed maximal monotone and 𝑚-accretive operators in Banach spaces

A more comprehensive and unified theory is developed for the solvability of the inclusions S ⊂ R ( A + B ) ¯ S \subset \overline {R(A + B)} , int S ⊂ R ( A + B ) S \subset R(A + B) , where A : X ⊃ D ( A ) → 2 Y A:X \supset D(A) \to {2^Y} , B : X ⊃ D ( B ) → Y B:X \supset D(B) \to Y and S ⊂ X S \subset X . Here, X X is a real Banach space and Y = X Y = X or Y = X ∗ Y = {X^*} . Mainly, A A is either maximal monotone or maccretive, and B B is either pseudo-monotone or compact. Cases are also considered where A A has compact resolvents and B B is continuous and bounded. These results are then used to obtain more concrete sets in the ranges of sums of such operators A A and B B . Various results of Browder, Calvert and Gupta, Gupta, Gupta and Hess, and Kartsatos are improved and/or extended. The methods involve the application of a basic result of Browder, concerning pseudo-monotone perturbation of maximal monotone operators, and the Leray-Schauder degree theory.

Sets in the ranges of sums for perturbations of nonlinear 𝑚-accretive operators in Banach spaces

Several results are given involving nonlinear range inclusions of the types B + D ⊂ R ( T + C ) ¯ B + D \subset \overline {R(T + C)} and int ⁡ ( B + D ) ⊂ R ( T + C ) \operatorname {int} (B + D) \subset R(T + C) , where B, D are subsets of a real Banach space X, the operator T : X ⊃ D ( T ) → 2 X T:X \supset D(T) \to {2^X} is at least m-accretive, and the perturbation C : X ⊃ D ( C ) → X C:X \supset D(C) \to X is at least compact, or demicontinuous, or m-accretive. Leray-Schauder degree theory is used in most of the results, and extended versions of recent results of Calvert and Gupta, Morales, Reich, and the author are shown to be possible by using mainly homotopies of compact transformations.

Iterative Solution of Nonlinear Equations Involving m-Accretive Operators in Banach Spaces

Iterative Solution of Nonlinear Equations Involving m-Accretive Operators in Banach Spaces