ITERATIVE SOLUTIONS FOR SYSTEMS OF NONLINEAR OPERATOR EQUATIONS IN BANACH SPACE

By using partial order method, the existence, uniqueness and iterative approximation of solutions for a class of systems of nonlinear operator equations in Banach space are discussed. The results obtained in this paper extend and improve recent results.

A NEW PERTURBATION THEOREM FOR MOORE-PENROSE METRIC GENERALIZED INVERSE OF BOUNDED LINEAR OPERATORS IN BANACH SPACES

In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.

A New System of Variational Inclusions with （H,η）-accretive Operators in Banach Spaces

In this paper, we first introduce a new class of generalized accretive operators named （H,η）-accretive in Banach space. By studying the properties of （H,η）-accretive, we extend the concept of resolvent operators associated with m-accretive operators to the new （H,η）-accretive operators. In terms of the new resolvent operator technique, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm.

A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES

This article is committed to deal with measure of non-compactness of operators in Banach spaces.Firstly,the collection C(X)(consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication)is a normed semigroup,and the mapping J from C(X)onto F(Ω)is a fully order-preserving positively linear surjective isometry,whereΩis the closed unit ball of X^*and F(Ω)the collection of all continuous and w^*-lower semicontinuous sublinear functions on X^*but restricted toΩ.Furthermore,both EC=JC-JC and EK=JK-JK are Banach lattices and EK is a lattice ideal of EC.The quotient space EC/EK is an abstract M space,hence,order isometric to a sublattice of C(K)for some compact Haudorspace K,and(FQJ)C which is a closed cone is contained in the positive cone of C(K),where Q:EC→EC/EK is the quotient mapping and F:EC/EK→C(K)is a corresponding order isometry.Finally,the representation of the measure of non-compactness of operators is given:Let BX be the closed unit ball of a Banach space X,thenμ(T)=μ(T(BX))=||(F QJ)T(BX)||C(K),∀T∈B(X).

The Generalized Roper-suffridge Operator in Banach Spaces

In this paper, by the definition of spirallike mapping of type β and order α ,we discuss that the generalized Roper-Suffridge extension operator preserves spirallikeness of type β and order α in complex Banach spaces.

One Roper-Suffridge Extension Operator in Complex Banach Spaces

In this paper, by the definition of almost spirallike mappings of type β and order α and the geometric property of the spirallike mapping of type β, we prove that the generalized Roper-Suffridge extension operator preserves almost spirallikeness of type β and order α in complex Banach spaces. Key words：

System of set-valued mixed quasi-variational-like inclusions involving H-η-monotone operators in Banach spaces

A new system of the set-valued mixed quasi-variational-like inclusions （SSMQVLI） involving H-η-monotone operators is studied in general Banach spaces without uniform smoothness. By using the resolvent operator technique of H-η-monotone operators, a new iterative algorithm for finding approximate solutions to SSMQVLI is proposed. It is shown that the iterative sequences generated by the algorithm converge strongly to the exact solution of SSMQVLI under appropriate assumptions. These obtained new results have extended and improved previous results.

Weighted Composition Operators from the Bloch Space to Weighted Banach Spaces on Bounded Homogeneous Domains

We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient conditions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide ＂computable＂ estimates on the operator norm.

A New System of Generalized Variational Inclusions Involving H-η-monotone Operators in Uniformly Smooth Banach Spaces

In this paper, we introduce and study a new system of generalized vari- ational inclusions involving H-η-monotone operators in uniformly smooth Banach spaces. Using the resolvent operator technique associated with H-η-monotone opera- tors, we prove the approximation solvability of solutions using an iterative algorithm. The results in this paper extend and improve some known results from the literature.

Hyperreflexivity of Operator Algebras of the Banach Space

In this paper, we introduced the hyperreflexivity of operator algebra A on Banach space, discussed the necessary, and sufficient condition that A is hyperreflexive, the estimate of hyperreflexive constant and the invariance of hyperreflexivety under the similarity transformation.

Double coupled fixed point for multi-valued mixed increasing operators in ordered Banach spaces

In this paper, a new concept of double coupled fixed point for multi-valued mixed increasing operators is given and some new double coupled fixed point theorems for multi-valued mixed increasing operators in ordered Banach spaces are also given. These results extend and generalize some results of Huang and Fang.

Implicit Variational Inclusions and Algorithms Involving （A, η）-monotone Operators in 2-uniformly Smooth Banach Spaces

This paper deals with a new class of nonlinear set valued implicit variational inclusion problems involving （A, η）-monotone mappings in 2-uniformly smooth Banach spaces. Semi-inner product structure has been used to study the （A, η）-monotonicity. Using the generalized resolvent operator technique and the semi-inner product structure, the approximation solvability of the proposed problem is investigated. An iterative algorithm is constructed to approximate the solution of the problem. Convergence analysis of the proposed algorithm is investigated. Similar results are also investigated for variational inclusion problems involving （H, η）-monotone mappings.

ITERATIVE APPROXIMATION OF FIXED POINTS FOR MULTIVALUED OPERATORS OF THE MONOTONE TYPE IN UNIFORMLY SMOOTH BANACH SPACES

In this paper, let K be a nonempty subset of a uniformly smooth Banach space X, and T:K→2~k be a multivalued operator of the monotone type. The iterative sequence which converges strongly to the unique fixed point of T is given. Our results are the extension and improvements of the results obtained previously by several authors including Dunn, Chidume, Deng and Ding.

The Index of Invariant Subspaces of Bounded below Operators on Banach Spaces

For an operator on a Banach space , let be the collection of all its invariant subspaces. We consider the index function on and we show, amongst others, that if is a bounded below operator and if , , then If in addition are index 1 invariant subspaces of , with nonzero intersection, we show that . Furthermore, using the index function, we provide an example where for some , holds .

IMPROVED GRADIENT METHOD FOR MONOTONE AND LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {An}n∈N be a family of monotone and Lipschitz continuos mappings of C into E＊. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [i0] for solving the variational inequality problem for {AN} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.

A system of general nonlinear variational inclusions in Banach spaces

A new system of general nonlinear variational inclusions is introduced and studied in Banach spaces. An iterative algorithm is developed and analyzed by use of the resolvent operator techniques to find the approximate solutions of the system of general nonlinear variational inclusions involving different nonlinear operators in uniformly smooth Banach spaces.

Stable Perturbed Algorithms for a New Class of Generalized Nonlinear Implicit Quasi Variational Inclusions in Banach Spaces

In this work, a new class of variational inclusion involving T-accretive operators in Banach spaces is introduced and studied. New iterative algorithms for stability for their class of variational inclusions and its convergence results are established.

ITERATIVE APPROXIMATION WITH ERRORS OF FIXED POINT FOR A CLASS OF NONLINEAR OPERATOR WITH A BOUNDED RANGE

Let X be a uniformly smooth real Banach space. tri T:X→X be a con-tinuous and strongly accretive operator. For a given f∈X,define S: X→X by 1)satisfying: Moreover, suppose that {Sxn} and {Syn} are bounded, then {Xn} converges strongly to the unique fixed point of S.

STABILITY AND CONVERGENCE OF STEPSIZE-DEPENDENT LINEAR MULTISTEP METHODS FOR NONLINEAR DISSIPATIVE EVOLUTION EQUATIONS IN BANACH SPACE

Stability and global error bounds are studied for a class of stepsize-dependent linear multistep methods for nonlinear evolution equations governed by ω-dissipative vector fields in Banach space.To break through the order barrier p≤1 of unconditionally contractive linear multistep methods for dissipative systems,strongly dissipative systems are introduced.By employing the error growth function of the methods,new contractivity and convergence results of stepsize-dependent linear multistep methods on infinite integration intervals are provided for strictly dissipative systems(ω<0)and strongly dissipative systems.Some applications of the main results to several linear multistep methods,including the trapezoidal rule,are supplied.The theoretical results are also illustrated by a set of numerical experiments.

Generalized regular points of a C^1 map between Banach spaces

Let f be a C^1 map between two Banach spaces E and F. It has been proved that the concept of generalized regular points of f, which is a generalization of the notion of regular points of f, has some crucial applications in nonlinearity and global analysis. We characterize the generalized regular points of f using the three integer-valued （or infinite） indices M（x0）, Mc（x0） and Mr（x0） at x0 ∈ E generated by f and by analyzing generalized inverses of bounded linear operators on Banach spaces, that is, iff ＇（x0） has a generalized inverse in the Banach space B（E, F） of all bounded linear operators on E into F and at least one of the indices M（x0）, Mc（x0） and Mr（x0） is finite, then xo is a generalized regular point off if and only if the multi-index （M（x）, Me（x）, Mr（x）） is continuous at X0.