A Remark on the Generalized Roper-Suffridge Extension Operator for Spirallike Mappings of Type β and Order α

Suppose f is a spirallike function of type β and order α on the unit disk D.Let Ωn,p1,p2,…,pn={z=（z2,z2,…,zn）′∈C^n：∑j=1^n｜zj）^Pj〈1},where 1≤p1≤2,pj≥1,j=2,…,n,are real numbers.In this paper,we will prove that Φn,β2,γ2,…βn,γn（f）（z）=（f（z1）, preserves spirallikeness of type β and order α on Ωn,p1,p2,…,Pn.

Some New Theorems of Lipschitz Type Mappings in Cone Metric Spaces

In this paper, some new existence and uniqueness of common fixed points for three mappings of Lipschitz type are obtained. The conditions are greatly weaker than the classic conditions in cone metric spaces. These results improve and generalize several wellknown comparable results in the literature. Moreover, our results are supported by some examples.

Convergence Theorems of φ-pseudo Contractive Type Mappings in Normed Linear Spaces

In this paper, by virtue of an inequality and sane analysis techniques, we prove sane convergence theorems cm the iterative process for nonlinear mappings of-pseudo contractive type in named linear spaces, which extend and improve the corresponding results obtained by others recently.

Mapping Detailed Soil Property Using Small Scale Soil Type Maps and Sparse Typical Samples

Soil type maps at the scale of I Z 1 000 000 are used extensively to provide soil spatial distribution information for soil erosion assessment and watershed management models in China. However, the soil property maps produced through conventional direct linking method usually suffer low accuracy as well as the lack of spatial details within a soil type polygon. This paper presents an effective method to produce detailed soil property map based on representative samples which were extracted from each polygon on the 1 ： 1 000 000 soil type map. The representative sample of each polygon is defined as the location that can represent the largest area within the polygon. The representativeness of a candidate sample is determined by calculating the soil-forming environment condition similarities between the sample and other locations. Once the representative sample of each polygon has been chosen, the property values of the existing typical samples are assigned to the corresponding representative samples with the same soil type. Finally, based on these representative samples, the detailed soil property map could be produced by using existing digital soil mapping methods. The case study in XuanCheng City, Anhui Province of China, demonstrated the proposed method could produce soil property map at a higher level of spatial details and accuracy： 1） The soil organic matter （SOM） map produced based on the representative samples can not only depict the detailed spatial distribution of SOM within a soil type polygon but also largely reduce the abrupt change of soil property at the boundaries of two adjacent polygons. 2） The Root Mean Squared Error （RMSE） of the SOM map based on the representative samples is 1.61, and it is 1.37 for the SOM map produced by using conventional direct linking method. Therefore, the proposed method is an effective approach to produce spatial detailed soil property map with higher accuracy for environment simulation models.

SHARP ESTIMATES OF ALL HOMOGENEOUS EXPANSIONS FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS OF TYPE B AND ORDER α IN SEVERAL COMPLEX VARIABLES

In this article, first, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order B on the unit ball in complex Ba- nach spaces are given. Second, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in （in are also established. In particular, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings （include quasi-convex mappings of type A and quasi-convex mappings of type B） in several complex variables are get accordingly. Our results state that a weak version of the Bieber- bach conjecture for quasi-convex mappings of type B and order a in several complex variables is proved, and the derived conclusions are the generalization of the classical results in one complex variable.

ASYMPTOTIC BEHAVIOR FOR COMMUTATIVE SEMIGROUPS OF ALMOST ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS

This article introduces the concept of commutative semigroups of almost asymptotically nonexpansive-type mappings in a Banach space X which has the Opial property and whose norm is UKK, and establishes the weak convergence theorems for almostorbits of this class of commutative semigroups. The author improves, extends and develops some recent and earlier results.

ON THE CONVERGENCE PROBLEMS OF ISHIKAWA AND MANN ITERATIVE PROCESSES WITH ERROR FOR Φ-PSEUDO CONTRACTIVE TYPE MAPPINGS

The purpose of this paper is to introduce the concept of Φ_pseudo contractive type mapping and to study the convergence problem of Ishikawa and Mann iterative processes with error for this kind of mappings. The results presented in this paper improve and extend many authors'recent results.

A SUBCLASS OF QUASI-CONVEX MAPPINGS ON A REINHARDT DOMAIN IN C^n

Let D p 1,p 2,⋯,p n={z∈C n:∑l=1 n|z l|p l<1},p l>1,l=1,2,⋯,n.In this article,we first establish the sharp estimates of the main coefficients for a subclass of quasi-convex mappings(including quasi-convex mappings of type A and quasi-convex mappings of type B)on D p 1,p 2,⋯,p n under some weak additional assumptions.Meanwhile,we also establish the sharp distortion theorems for the above mappings.The results that we obtain reduce to the corresponding classical results in one dimension.

THE INVARIANCE OF STRONG AND ALMOST SPIRALLIKE MAPPINGS OF TYPE β AND ORDER α

The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallike-hess of type β and order α on the unit ball B^n in C^n and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirllikeness of type β, strong and almost starlikeness of order α, strong starlikeness on the corresponding domains.Thus we can construct more subclasses of spirallike mappings in several complex variables.

The Spirallike Mapping of Type a and the Generalized Roper-Suffridge Operator

In this paper, we get the fact that generalized Roper-Suffridge operators act on spirallike mapping of type α on different domains is invariable, these results include some results known.

INTERATIVE APPROXIMATION OF FIXED POINTS FOR ALMOST ASYMPTOTICALLY NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES

A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced,which includes a number of known classes of nonlinear Lipschitzian mappings and non_Lipschitzian mappings in Banach spaces as special cases; for example,the known classes of nonexpansive mappings,asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings.The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered.Not only S.S.Chang's inequality but also H.K.Xu's one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one.Moreover,Zhang Shi_sheng's method(Applied Mathematics and Mechanics(English Edition),2001,22(1):25-34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also,the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria.The above results unify,improve and generalize Zhang Shi_sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors.

DEGREE THEORY FOR MULTIVALUED (S) TYPE MAPPINGS AND FIXED POINT THEOREMS

The main purpose of this paper is devoted to generalizing the results of Browder[1,2]This paper consists of four parts. In the first part, we introduce the concepts of multivalued (S) and (S), type mappings and the concepts of the limits of multivalued (S) and (S) + type mappings. These kinds of mappings contain many monotone type mappings, such as maximal monotone mapping, bounded pseudo-monotone mapping and bounded generalized pseudo-monotone mapping, as its special cases. In the second part we define the pseudo-degree for (S) type mapping and the degree for (S)+ type mapping. These two kinds of degrees are all the generalizations of the degree defined by Browder[1,2] As applications, we utilize the degree theory presented in part 2 to study the existence of solutions for the multivalued operator equations (see part 3) and to obtain some new fixed point theorems in part 4.

Strong Convergence Theorems for Mixed Type Asymptotically Nonexpansive Mappings

The purpose of this paper is to study a new two-step iterative scheme with mean errors of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.

Fixed Point Theorem of{a, b, c}Contraction and Nonexpansive Type Mappings in Weakly Cauchy Normed Spaces

Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following： 1. If T is {a,b,c}-ntype mapping, then inf{ ｜｜ T（x）-x｜｜ ：x C C} =0, accordingly T has a unique fixed point. Moreover, any sequence {Xn}n∈NN in C with limn→∞｜｜T（xn） - Xn｜｜ = 0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any x∈C the sequence of iterates {Tn （x）}n∈N has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4, 7] and [13].

A Common Fixed Point Theorem for Compatible Mappings of Type （K） in Intuitionistic Fuzzy Metric space

The purpose of our paper is to obtain a common fixed point theorem for two pairs of self-mappings of compatible of type （K） in a complete intuitionistic fuzzy Metric space with example. Our result generalized and improves similar other results in literature.

A FIXED POINT APPROACH TO THE STABILITY OF A GENERALIZED APOLLONIUS TYPE QUADRATIC FUNCTIONAL EQUATION

Using the fixed point method, this article proves the Hyers-Ulam-Rassias stability of a generalized Apollonius type quadratic functional equation in Banach spaces. The conditions of these results are demonstrated by the quadratic functional equation of Apollonius type.

SYSTEM OF GENERALIZED VECTOR VARIATIONAL INEQUALITIES

In this paper, the authors introduce and study system of generalized vector variational inequalities. Under suitable conditions, the existence of solutions for system of generalized vector variational inequalities is presented by Kakutani-Fan-Glicksberg fixed point theorem.

FIXED POINT RESULTS ON METRIC-TYPE SPACES

In this paper we obtain fixed point and common fixed point theorems for self- mappings defined on a metric-type space, an ordered metric-type space or a normal cone metric space. Moreover, some examples and an application to integral equations are given to illustrate the usability of the obtained results.

Loewner Chains Associated with the Generalized Roper-Suffridge Extension Operator on the Unit Ball in Hilbert Spaces

In this paper, we prove that the generalized Roper-Suffridge extension operator can be embeded in Loewner chains on the unit ball in Hibert spaces, and obtain the fact that the operator keeps the properties of almost spirallike mapping of typeβ and order α, almost starlikeness of order α, spirallikeness of type ofβ and starlikeness.