A New Class of Banach Spaces with Uniform Normal Structure

Let X be a Banach space, S(X) be the unit sphere of X, φ be a function: S(X)→ S(X *) such that φ(x)∈ x, and v φ(ε) =inf 1-12x+y: x,y∈S(X), and 〈φ(x), x-y 〉≥ε, 0≤ε≤2, where x is the set of norm 1 supporting functionals of S(X) at x. A geometric concept, modulus of V convexity V(ε)= sup {V φ(ε), for all φ: S(X)→S(X *)}, is introduced; the properties of V(ε) and the relationship between V(ε) and other geometric concepts are discussed. The main result is that V12>0 implies normal structure.

ISHIKAWA ITERATIVE PROCESS IN UNIFORMLY SMOOTH BANACH SPACES

Let E be a uniformly smooth Banach space, K be a nonempty closed convex subset of E, and suppose: T: K --> K is a continuous Phi-strongly pseudocontractive operator with a bounded range. Using a new analytical method, under general cases, the Ishikawa iterative process {x(n)} converges strongly to the unique fixed point x* of the operator T were proved. The paper generalizes and extends a lot of recent corresponding results.

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.

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

The fixed points of set-valued operators satisfying a condition of monotonicitytype in s-uniformly smooth Banach spaces are approximated by recursive averagingprocess in the present paper.

Banach Space Valued Martingales with Respect to Complex Measure and Their Inequalities

Let d μ=ψ d ν be a complex valued measure where ν is a non negative measure and ψ is a complex valued function which satisfies b + p or b + ∞∩a 1 condition. We prove some basic martingale inequalities such as B G inequalities, weak ( p,p) and strong (p,p) type inequalities for Banach space valued martingale with respect to complex measure μ .

STRONG CONVERGENCE OF APPROXIMATED SEQUENCES FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES

This paper studies the convergence of the sequence defined by x0 ∈ C, xn＋l =αnu＋（1-αn）Txn, n=0, 1,2,..., where 0 ≤αn ≤ 1, limn→∞ αn = 0, ∑n=0^∞ αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.

FURTHER INVESTIGATION INTO APPROXIMATION OF A COMMON SOLUTION OF FIXED POINT PROBLEMS AND SPLIT FEASIBILITY PROBLEMS

The purpose of this paper is to study and analyze an iterative method for finding a common element of the solution set ~ of the split feasibility problem and the set F（T） of fixed points of a right Bregman strongly nonexpansive mapping T in the setting of p- uniformly convex Banach spaces which are also uniformly smooth. By combining Mann＇s iterative method and the Halpern＇s approximation method, we propose an iterative algorithm for finding an element of the set F（T）∩Ω moreover, we derive the strong convergence of the proposed algorithm under appropriate conditions and give numerical results to verify the efficiency and implementation of our method. Our results extend and complement many known related results in the literature.

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.

REMARK ON STABILITY OF ISHIKAWA ITERATIVE PROCEDURES

The stability of the Ishikawa iteration procedures was studied for one class of continuity strong pseudocontraction and continuity strongly accretive operators with bounded range in real uniformly smooth Banach space. Under parameters satisfying certain conditions, the convergence of iterative sequences was proved. The results improve and extend the recent corresponding results, and supply the basis of theory for further discussing convergence of iteration procedures with errors.

ISHIKAWA ITERATIVE PROCESS FOR CONSTRUCTING SOLUTIONS OF m-ACCRETIVE OPERATOR EQUATIONS

The convergence of the Ishikawa iteration sequences with errors for constructing solutions of m-accretive operator equations is characterized. Moreover, the error estimates of approximate solutions for locally Lipschitzian and m-accretive operator equations are established.

Viscosity Iterative Algorithm for Variational Inequality Problems and Fixed Point Problems of Strict Pseudo-contractions in Uniformly Smooth Banach Spaces

The purpose of this paper is to study a new viscosity iterative algorithm based on a generalized contraction for finding a common element of the set of solutions of a general variational inequality problem for finite inversely strongly accretive mappings and the set of common fixed points for a countable family of strict pseudo-contractions in uniformly smooth Banach spaces. We prove some strong convergence theorems under some suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.

Best Simultaneous Approximation to Totally Bounded Sequences in Banach Spaces

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth.

Strong Convergence Theorems of Viscosity Approximation for Accretive Operators

Let E be a real Banach space and let A be an m-accretive operator with a zero. Define a sequence {x_n} as follows:x_n+1=α_nf(x_n)+(1-α_n)J_r_n x_n,where {α_n},{r_n} are sequences satisfying certain conditions,and J_r denotes the resolvent(I+rA)^(-1)for r>1.Strong convergence of the algorithm {x_n} is obtained provided that E either has a weakly continuous duality map or is uniformly smooth.

Three-step Iterations with Errors for Nonlinear Strongly Accretive Operator Equations

In this paper, we suggest and analyse a three-step iterative scheme with errors for solving nonlinear strongly accretive operator equation Tx = f without the Lipshitz condition. The results presented in this paper improve and extend current results in the more general setting.

Operator-valued martingale transforms in rearrangement invariant spaces and applications

In this paper the operator-valued martingale transform inequalities in rearrangement invariant function spaces are proved.Some well-known results are generalized and unified.Applications are given to classical operators such as the maximal operator and the p-variation operator of vector-valued martingales,then we can very easily obtain some new vector-valued martingale inequalities in rearrangement invariant function spaces.These inequalities are closely related to both the geometrical properties of the underlying Banach spaces and the Boyd indices of the rearrangement invariant function spaces.Finally we give an equivalent characterization of UMD Banach lattices,and also prove the Fefferman-Stein theorem in the rearrangement invariant function spaces setting.