ON SELECTIONS OF SET-VALUED EULER-LAGRANGE INCLUSIONS WITH APPLICATIONS

We discuss the set-valued dynamics related to the theory of functional equations.We look for selections of convex set-valued functions satisfying set-valued Euler-Lagrange inclusions.We improve and extend upon some of the results in[13,20],but under weaker assumptions.Some applications of our results are also provided.

Continuous Homogeneous Selections of Set-Valued Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper, continuous homogeneous selections for the set-valued metric generalized inverses T^ of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces. Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T~ are given. The results are an answer to the problem posed by Nashed and Votruba.

CONTINUOUS SELECTIONS OF THE SET-VALUED METRIC GENERALIZED INVERSE IN 2-STRICTLY CONVEX BANACH SPACES

In this paper,we prove that if X is an almost convex and 2-strictly convex space,linear operator T:X→Y is bounded,N(T)is an approximative compact Chebyshev subspace of X and R(T)is a 3-Chebyshev hyperplane,then there exists a homogeneous selection T^(σ)of T^(■)such that continuous points of T^(σ)and T^(■)are dense on Y.

SET-VALUED EXTENSION OF OPERATORS VIA STEINER SELECTIONS (Ⅰ)-THEORETICAL RESULTS

A way to extend operators in spaces of continuous functions to spaces of continuous set_valued functions is proposed. This extension is developed through the Steiner selections of the set_valued functions. Their properties and characteristics of the convergence of sequences of operators of this class are studied. In Part Ⅱ of this series some applications to approximation theory will be shown.

SET-VALUED EXTENSION OF OPERATORS VIA STEINER SELECTIONS(Ⅱ)-APPLICATIONS TO APPROXIMATION

On the basis of Part (Ⅰ) of this series some applications to the approximation of set_valued functions are obtained: Korovkin type theorems, a method to extend classical approximation operators to the set_valued setting and a Jackson type estimate.

EXISTENCE PROBLEMS OF ADDITIVE SELECTION MAPS FOR ANOTHER TYPE SUBADDITIVE SET-VALUED MAP

In this paper, we consider the following subadditive set-valued map F ： X→P0（Y） ：where r and s are two natural numbers. And we discuss the existence and unique problem of additive selection maps for the above set-valued map.

The Unique Existence of Additive Selection Maps for a Weak Subadditive Set-valued Map

In this paper, we discuss the unique existent problems of a additive selection map for the following map F:X→P0(Y):F(r∑i=1αixi)■r∑i=1αiF(xi),■xi∈K,i=1, ··· , r,where αi and αi (i = 1, ··· , r) are all non-negative real numbers.

An Improvement of Existence Theorems of Additive Selection Maps for Sub-additive Set-valued Maps

In this paper, we will construct a new class of subadditive set-valued maps and use Cantor theorem to prove that the set-valued map has an unique additive selection map when the set-valued map satisfies some certain conditions, and then compare the obtained result with the well-known results.

基于SELECT理念腹腔镜下保留脾脏的胰体尾切除

腹腔镜下胰体尾切除术目前已经成为治疗胰腺体尾部肿瘤的标准术式。而腹腔镜下保留脾脏的胰体尾切除术由于可保留脾脏功能,减少脾切除术后并发症,越来越受到重视。然而,胰腺肿瘤发病隐匿、解剖位置复杂,这些对其诊断和保留脾脏的胰体尾切除术带来了挑战。近年来,我们团队积累了丰富的胰腺肿瘤诊疗经验,创新提出“SELECT”(S-Single-Operator Cholangiopancreatoscopy,E-ERCP,L-Laparoscopy,E-Endoscopic ultrasound,C-Choledochoscopy/Confocal laser endomicroscopy,T-Traditional Chinese medicine)多镜组合中西医微创诊治理念。根据胰腺肿瘤的类型及特点,应用多种内镜和腹腔镜技术,多种不同微创诊疗方法的优选组合,围手术期采用中医药,进行中西医结合加速康复。将SELECT理念充分应用于腹腔镜下保留脾脏的胰体尾切除术,有利于术前精准诊断、术中精准切除、术后并发症预判和治疗,一站式诊治胰腺肿瘤,使病人利益最大化。

CONE-DIRECTED CONTINGENT DERIVATIVES AND GENERALIZED PREINVEX SET-VALUED OPTIMIZATION

By using cone-directed contingent derivatives, the unified necessary and sufficient optimality conditions are given for weakly and strongly minimal elements respectively in generalized preinvex set-valued optimization.

THE OPTIMALITY CONDITIONS OF NONCONVEXSET-VALUED VECTOR OPTIMIZATION

The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz John optimality condition of set-valued vector optimization is established. Finally, under the assumption of pseudoconvexity, the optimality condition is proved to be sufficient.

Common Fixed Point Theorems for Sequences of φ-type Contraction Set-valued Mappings

In complete metric spaces, the common fixed point theorems for sequences of φ-type contraction set-valued mappings are established, and the corresponding random com- mon fixed point theorems for set-valued mappings are also obtained.

On super efficiency in set-valued optimization

The set-valued optimization problem with constraints is considered in the sense of super efficiency in locally convex linear topological spaces. Under the assumption of iccone-convexlikeness, by applying the seperation theorem, Kuhn-Tucker＇s, Lagrange＇s and saddle points optimality conditions, the necessary conditions are obtained for the set-valued optimization problem to attain its super efficient solutions. Also, the sufficient conditions for Kuhn-Tucker＇s, Lagrange＇s and saddle points optimality conditions are derived.

ON THE GENERALIZED FRITZ JOHN OPTIMALITY CONDITIONS OF VECTOR OPTIMIZATION WITH SET-VALUED MAPS UNDER BENSON PROPER EFFICIENCY

A kind of tangent derivative and the concepts of strong and weak * pseudoconvexity for a set-valued map are introduced. By the standard separation theorems of the convex sets and cones the optimality Fritz John condition of set-valued optimization under Benson proper efficiency is established, its sufficience is discussed. The form of the optimality conditions obtained here completely tally with the classical results when the set-valued map is specialized to be a single-valued map.

KUHN-TUCKER CONDITION AND WOLFE DUALITY OF PREINVEX SET-VALUED OPTIMIZATION

The optimality Kuhn-Tucker condition and the wolfe duality for the preinvex set-valued optimization are investigated. Firstly, the concepts of alpha-order G-invex set and the alpha-order S-preinvex set-valued function were introduced, from which the properties of the corresponding contingent cone and the alpha-order contingent derivative were studied. Finally, the optimality Kuhn-Tucker condition and the Wolfe duality theorem for the alpha-order S-preinvex set-valued optimization were presented with the help of the alpha-order contingent derivative.

Benson proper efficiency for vector optimization of generalized subconvexlike set-valued maps in ordered linear spaces

Several equivalent statements of generalized subconvexlike set-valued map are established in ordered linear spaces. Using vector closure, we introduce Benson proper efficient solution of vector optimization problem. Under the assumption of generalized subconvexlikeness, scalarization, multiplier and saddle point theorems are obtained in the sense of Benson proper efficiency.

EKELAND'S PRINCIPLE FOR SET-VALUED VECTOR EQUILIBRIUM PROBLEMS

In this paper, we introduce a concept of quasi C-lower semicontinuity for setvalued mapping and provide a vector version of Ekeland＇s theorem related to set-valued vector equilibrium problems. As applications, we derive an existence theorem of weakly efficient solution for set-valued vector equilibrium problems without the assumption of convexity of the constraint set and the assumptions of convexity and monotonicity of the set-valued mapping. We also obtain an existence theorem of ε-approximate solution for set-valued vector equilibrium problems without the assumptions of compactness and convexity of the constraint set.

Some Dynamical Properties in Set-valued Discrete Systems

1 Introduction A discrete dynamical system can be expressed as xn+1 ?f(xn), n = 0,1,2,... where X is a metric space and f : X →X is a continuous map. The study of it tells us how the points in the base space X moved. Nevertheless, this is not enough for the researches of biological species, demography, numerical simulation and attractors (see [1], [2]). It is necessary to know how the subsets of X moved. In this direction, we consider the set-valued discrete system associated to f, An+1 = (f|-)(An), n = 0,1,2,... where (f|-) is the natural extension of f to K(X) (the class of all compact subsets of X).

Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities

An auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities is extended. The existence and uniqueness of the solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities are proved, a novel and innovative three-step iterative algorithm to compute approximate solution is constructed, and the existence of the solution of the generalized set-valued strongly nonlinear mixed variational-like inequality is shown using the auxiliary principle iterative sequences generated by the algorithm technique. The convergence of three-step is also proved.

ε-super Effcient Solutions of Optimization Problems with Set-valued Maps in Locally Convex Space

In this paper,the ε-super effcient solution for set-valued map vector optimization in locally convex space is introduced.And under the assumption of the nearly generalized cone-subconvexlikeness for set-valued maps,used new methods,the scalarization theorem and Lagrange multiplier theorem for ε-super effcient solution are established.