AN ITERATIVE PROCESS FOR FINDING APPROXIMATE SOLUTIONS TO NONLINEAR EQUATIONS OF STRONGLY ACCRETIVE OPERATORS

In this paper, we investigate the Ishikawa iteration process in a p-uniformly smooth Banach space X. We prove that the Ishikawa iteration process converges strongly to the unique solution of the equation Tx=f when T is a Lipschitzian and strongly accretive operator frow X to X, or to the unique fixed point of T when T is a Lipschitzian and strictly pseudocontractive mapping from a nonempty closed convex subset K of X into itself. Our results are the extension and improvements of the earlier and recent results in this field.

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.

Weighted estimates for strongly singular integral operators with rough kernels

The Fourier transform and the Littlewood-Paley theory are used to give the weighted boundedness of a strongly singular integral operator defined in this paper. The paper shows that the strongly singular integral operator is bounded from the Sobolev space to the Lebesgue space.

New iterative schemes for strongly relatively nonexpansive mappings and maximal monotone operators

In this paper, some new iterative schemes for approximating the common element of the set of fixed points of strongly relatively nonexpansive mappings and the set of zero points of maximal monotone operators in a real uniformly smooth and uniformly convex Banach space are proposed. Some weak convergence theorems are obtained, which extend and complement some previous work.

Strong Converse Inequalities for Certain Mixed Szász-Beta Operators

In this paper,a strong converse inequality of type B in terms of a new Kfunctional Kλα f,t2(0 < α < 2,0 ≤λ≤ 1) for certain mixed Szász-Beta operators is given.By this inequality,the converse theorem can be obtained for the operators.

STRONG CONVERSE INEQUALITY FOR SZASZ-DURRMEYER OPERATORS

For Szasz-Durrmeyer operators L.(f,x),1<p ∞ we prove that, forsome m,where(x)2=x,M>0,is Ditzian-Totik modulus of smoothness.

Strong Converse Inequalities for Szasz-Mirakjan Operators with Weights

In this paper, we give the strong converse inequalities of type B with the new K-functional Kλα(f,t2)w(0 ≤λ≤ 1, 0 < α < 2) on weighted approximation for Sz′asz-Mirakjan operators, which extend the previous results.

Creating unconventional geometric phase gate using squeeze-like operator assisted by strong driving field

Based on the idea that a squeezing process can be thought of as a total cumulative effect of a large number of tiny squeezing processes, we define a squeeze-like operator with a time-dependent squeeze parameter. Applying this operator to and combining with a system which includes a two-photon interaction between two atoms and an initial vacuum cavity field, and resorting to a resonant strong driving classical field, we obtain an unconventional geometric phase gate with a shorter gating time.

ON STRONG APPROXIMATION APPLIED TO POST-WIDDER OPERATORS

We introduce modified Post-Widder operators in polynomial weighted spaces of differentiable functions and we study strong approximation for them.

SOME PROPERTIES OF STRONGLY MONOTONICALLY T2 SPACES

This paper investigates Buck＇s question about which class of spaces is strongly monotonically T2,and if other properties are combined with strongly monotonically T2,which class of spaces could be got. Based on having a cushioned pair-base space and compact strongly monotonically T2 space,some results （Theorems 1--3） are obtained.

Strong Converse Inequality for the Meyer-Knig and Zeller-Durrmeyer Operators

In this paper we give a strong converse inequality of type B in terms of unified K-functional Kλα （f, t2） （0 ≤λ≤ 1, 0 〈 α 〈 2） for the Meyer-Knig and Zeller-Durrmeyer type operators.

CRITERIA OF STRONG TRANSIENCE FOR OPERATOR-SELF-SIMILAR MARKOV PROCESSES

Yamamuro in [1] defines strong and weak transience of Markov processes; gives a criterion for strong transience of Feller processes; and further, discusses strong and weak transience of Ornstein-Uhlenbeck type processes. In this article, the authors weaken the Feller property of the result in [1] to weak Feller property and discuss the strong transience of operator-self-similar Markov processes.

Iterative convergence theorems for maximal monotone operators and relatively nonexpansive mappings

In this paper, some iterative schemes for approximating the common element of the set of zero points of maximal monotone operators and the set of fixed points of relatively nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Some strong convergence theorems are obtained, to extend the previous work.

VISCOSITY APPROXIMATION METHODS FOR THE SPLIT EQUALITY COMMON FIXED POINT PROBLEM OF QUASI-NONEXPANSIVE OPERATORS

Let H;, H;, H;be real Hilbert spaces, let A : H;→ H;, B : H;→ H;be two bounded linear operators. The split equality common fixed point problem（SECFP） in the infinite-dimensional Hilbert spaces introduced by Moudafi（Alternating CQ-algorithm for convex feasibility and split fixed-point problems. Journal of Nonlinear and Convex Analysis）is to find x ∈ F（U）, y ∈ F（T） such that Ax = By,（1）where U : H;→ H;and T : H;→ H;are two nonlinear operators with nonempty fixed point sets F（U） = {x ∈ H;: Ux = x} and F（T） = {x ∈ H;: Tx = x}. Note that,by taking B = I and H;= H;in（1）, we recover the split fixed point problem originally introduced in Censor and Segal. Recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms with weak convergence for the SECFP（1） of firmly quasi-nonexpansive operators. In this paper, we introduce two viscosity iterative algorithms for the SECFP（1） governed by the general class of quasi-nonexpansive operators. We prove the strong convergence of algorithms. Our results improve and extend previously discussed related problems and algorithms.

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.

ROPER-SUFFRIDGE EXTENSION OPERATOR ON A REINHARDT DOMAIN

Let pj ∈ N and pj≥ 1, j = 2, …, k, k ≥ 2 be a fixed positive integer. We introduce a Roper-Suffridge extension operator on the following Reinhardt domain ΩN ={z =（z1, z′2,…, z′k）′∈ C × C^n2×…× Cnk： ｜z1｜^2＋ ｜｜z2｜｜2^p2＋ … ＋ ｜｜zk ｜｜k^pk〈 1} given〈1} give by F（z）=（f（z1）＋f＇（z1）∑j=2 kPj（zj,（f＇（z1））1/p2 z2＇,…,（f＇（z1））1/pkzk＇）＇, where f is a normaljized biholomorphic function k（z） =（f（z1） ＋ f′（z1）=2 on the unit disc D, and for 2 ≤ j ≤ k, Pj ： C^nj→ C is a homogeneous polynomial of degree pj and zj =（zj1, …, zjnj）′∈ C^nj, nj ≥ 1, pj ≥ 1,｜｜zj｜｜j =（∑l=1 nj｜zjl｜^pj）1/pj. In this paper, some conditions for Pjare found under which the loperator p ｜zjl｜pj=1reserves the properties of almost starlikeness of order α, starlikeness of order αand strongly starlikeness of order α on ΩN, respectively.

APPROXIMATING SOLUTION OF 0 ∈T(x) FOR AN H-MONOTONE OPERATOR IN HILBERT SPACES

The purpose of this paper is to study the solution of 0∈ T（x） for an H- monotone operator introduced in [Fang and Huang, Appl. Math. Comput. 145（2003）795- 803] in Hilbert spaces, which is the first proposal of it＇s kind. Some strong and weak convergence results are presented and the relations between maximal monotone operators and H-monotone operators are analyzed. Simultaneously, we apply these results to the minimization problem for T = δf and provide some numerical examples to support the theoretical findings.

Method of accurately calculating mean field operator in multi-configuration time-dependent Hartree-Fock frame

The accurate theoretical expressions of the mean field operator associated with the multi-configuration time-dependent Hartree-Fock （MCTDHF） method are presented in this paper. By using a theoretical formula, derived without approxima- tion, we can study the multi-electron correlation dynamics accurately. Some illustrative calculations are carried out to check the accuracy of the expression of the mean field operator. The results of illustrative calculations indicate the reliability of the accurate expression of the mean field operator. This theoretical calculation method for the mean field operator may be of considerable help in future studies of the correlated dynamics of many-electron systems in strong laser fields.

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.

New Result for Strongly Starlike Functions

In this paper, using Salagean differential operator, we define and investigate a new subclass of univalent functions . We also establish a characterization property for functions belonging to the class .