Approximating stationary points of stochastic optimization problems in Banach space

In this paper, we present a uniform strong law of large numbers for random set-valued mappings in separable Banach space and apply it to analyze the sample average approximation of Clarke stationary points of a nonsmooth one stage stochastic minimization problem in separable Banach space. Moreover, under Hausdorff continuity, we show that with probability approaching one exponentially fast with the increase of sample size, the sample average of a convex compact set-valued mapping converges to its expected value uniformly. The result is used to establish exponential convergence of stationary sequence under some metric regularity conditions.     

On a class of nonlinear stochastic integral equations

We prove some existence and uniqueness results for a nonlinear stochastic integral equation using fixed-point theory methods to ensure the convergence of the successive approximations to the unique random solution.     

On normal crossings in a Banach space

We show that if A is a closed complex analytic subset of a Banach space X with an unconditional basis (e.g., X=?₂) that has only normal crossings for singularities, then the structure and ideal sheaves of A are cohesive sheaves over X, and consequently, they are acyclic over any pseudoconvex open subset of X.     

Iterative approximation for a zero of accretive operator and fixed points problems in Banach space

In this paper we introduced an iteration scheme for viscosity approximation for a zero of accretive operator and fixed points problems in a reflexive Banach space with weakly continuous duality mapping. A new iterative sequence is introduced and strong convergence of the algorithm x_n is proved. The results improve and extend the results of Hu and Liu [L. Hu and L. Liu, A new iterative algorithm for common solutions of a finite family of accretive operators, Nonlinear Anal. 70 (2009) 2344-2351] and some others.     

On the rate of convergence in the central limit theorem and the type of the Banach space

Let E be a Banach space. Let ξ be a sequence of which goes to zero. Let X be a centered E-valued random variable, which is bounded. Let S_n be the sum of n independent copies of X. Assume that whenever X satisfies the CLT, we have.

$\text{Sup}\text{t ?R}\text{|P(}\text{6S}{}_{\mathrm{n}}\text{√n | ? t}\text{) ? μ ({x;6x6 ?})| ? ξ}{}_{\mathrm{n}}$

where μ is the (Gaussian) limit of the laws of S_n. Then E is type 2.     

Solvability of a class of boundary value problems in the space of convergent sequences

We consider two types of infinite systems of second-order differential equations with boundary conditions in the Banach sequence space c. For each system, sufficient conditions are provided for the existence of solutions. Our approach is based on the use of measure of noncompactness concept and the application of Darbo’s theorem for condensing operators. Some examples are presented to illustrate the obtained results.     

Positive solutions of a nonlinear four-point boundary value problems in Banach spaces

In this paper, by using the fixed points of strict-set-contractions, we study the existence of at least one or two positive solutions to the four-point boundary value problem

     

On growth and covering theorems of quasi-convex mappings in the unit ball of a complex banach space

A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ℂⁿ, the “quasi-convex mapping” is exactly the “quasi-convex mapping of type A” introduced by K. A. Roper and T. J. Suffridge.     

Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space

We show that any fixed point of a Lipschitzian, strictly pseudocontractive mapping on a closed, convex subset of a Banach space is necessarily unique, and may be norm approximated by an iterative procedure. Our argument provides a convergence rate estimate and removes the boundedness assumption on , generalizing theorems of Liu.

     

On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces

In vector optimization, topological properties of the set of efficient and weakly efficient points are of interest. In this paper, we study the connectedness of the setE ^w of all weakly efficient points of a subsetZ of a locally convex spaceX with respect to a continuous mappingp:X Y,Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions thatZ is convex and closed and thatp is a pointwise quasiconvex mapping (i.e., a generalized quasiconvex concept), the setE ^w is connected, if the lower level sets ofp are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of Refs. 1–3. Moreover, some examples in vector approximation are given.The author is grateful to Dr. D. T. Luc and to a referee for pointing out an error in an earlier version of this paper.     

Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Jensen's inequality f(EX) ≤ Ef(X) for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of (possibly) infinite-dimensional linear topological spaces. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. Two different methods of proof are given, one based on geometric properties of convex sets and the other based on the Strong Law of Large Numbers. Various conditions under which Jensen's inequality becomes strict are studied. The relation between Jensen's inequality and Fatou's Lemma is examined.     

Banach空间中一类二阶非线性脉冲积分-微分方程边值问题解的存在性

本文利用一个新的比较结果和Mnch不动点定理,研究了Banach空间中一类二阶非线性脉冲积分-微分方程边值问题解的存在性,改进和推广了相关文献的结果.     

Auxiliary principle and iterative algorithm for a new system of generalized mixed equilibrium problems in Banach spaces

In this paper, we study a new system of generalized mixed equilibrium problems involving skew-symmetric bifunctions (SGMEP) in reflexive Banach spaces. A system of auxiliary mixed equilibrium problems (SAMEP) for solving the SGMEP is introduced and the existence and uniqueness of the solutions of the SAMEP is first proved. Next, by using the auxiliary principle technique, a new iterative algorithm to compute the approximate solutions of the SGMEP is suggested and analyzed. Finally, the strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild conditions. These results improve, unify and generalize some known results in recent literature.     

The auxiliary principle and an algorithm for a system of generalized set-valued mixed variational-like inequality problems in Banach spaces

In this paper, we introduce and study a new system of generalized set-valued mixed variational-like inequality problems (SGSMVLIP) and its related auxiliary problems in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and an iterative algorithm of solutions for the system of generalized set-valued mixed variational-like inequality problems. At first, the existence and uniqueness of solutions of the auxiliary problems for (SGSMVLIP) is shown. Next, an iterative algorithm for solving (SGSMVLIP) is constructed by using the existence and uniqueness result. Finally, we prove the existence of solutions of (SGSMVLIP) and discuss the convergence analysis of the algorithm. These results improve, unify and generalize many corresponding known results given in the literature.     

On existence and uniqueness of stationary points in semi-infinite optimization

The paper deals with semi-infinite optimization problems which are defined by finitely many equality constraints and infinitely many inequality constraints. We generalize the concept of strongly stable stationary points which was introduced by Kojima for finite problems; it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed problem, where perturbations up to second order are allowed. Under the extended Mangasarian-Fromovitz constraint qualification we present equivalent conditions for the strong stability of a considered stationary point in terms of first and second derivatives of the involved functions. In particular, we discuss the case where the reduction approach is not satisfied. Received June 30, 1995 / Revised version received October 9, 1998? Published online June 11, 1999     

Iterative algorithm of solutions for a system of generalized mixed implicity equilibrium problems in reflexive Banach spaces

In this paper, a new system of generalized mixed implicity equilibrium problems involving non-monotone set-valued mappings is introduced and studied in real reflexive Banach spaces. Following the idea of Moudafi, we consider a system of generalized equations problems and show its equivalence with the system of generalized mixed implicity equilibrium problems. By using a fixed point formulation of the system of generalized equation problems, a new iterative algorithm for solving the system of generalized mixed implicity equilibrium problems is suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithm is proved under suitable conditions. These results are new and unify and generalize some recent results in this field.     

Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces

In this paper, we introduce a general iteration scheme for a finite family of asymptotically quasi-nonexpansive mappings. The new iterative scheme includes the modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor and Khan and Takahashi scheme as special cases. Our results are generalizations as well as refinement of several known results in the current literature.     

On Lipschitz and D.C. surfaces of finite codimension in a Banach space

Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated σ-ideals are studied. These σ-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.     

Global solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces

In this paper, the global solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces are investigated. The existence and uniqueness of solutions and their iterative approximation are obtained by using Mönch fixed point theorem and new comparison results. The results presented here essentially improve, generalize and unify many well-known results.     

On a proximal point method for convex optimization in banach spaces

We analyze the behavior of a parallel proximal point method for solving convex optimization problems in reflexive Banach spaces. Similar algorithms were known to converge under the implicit assumption that the norm of the space is Hilbertian. We extend the area of applicability of the proximal point method to solving convex optimization problems in Banach spaces on which totally convex functions can be found. This includes the class of all smooth uniformly convex Banach spaces. Also, our convergence results leave more flexibility for the choice of the penalty function involved in the algorithm and, in this way, allow simplification of the computational procedure.