Multi-Valued Mappings of Bounded Generalized Variation

We study the mappings taking real intervals into metric spaces and possessing a bounded generalized variation in the sense of Jordan--Riesz--Orlicz. We establish some embeddings of function spaces, the structure of the mappings, the jumps of the variation, and the Helly selection principle. We show that a compact-valued multi-valued mapping of bounded generalized variation with respect to the Hausdorff metric has a regular selection of bounded generalized variation. We prove the existence of selections preserving the properties of multi-valued mappings that are defined on the direct product of an interval and a topological space, have a bounded generalized variation in the first variable, and are upper semicontinuous in the second variable.     

On the Parametrized Integral of a Multifunction: The Unbounded Case

Integration of set-valued maps (alias multifunctions) depending on a parameter is revisited. Results of Artstein, and of Saint-Pierre and Sajid are extended to the case of set-valued maps whose values may be unbounded. In the general case, this is achieved assuming that the transition probabilities involved in the integration procedure are absolutely continuous with respect to some fixed probability measure. However, when the integrating probability measure does not depend on the parameter this hypothesis is shown to be unnecessary. On the other hand, an alternative proof of a result of Saint-Pierre and Sajid is provided for convex compact-valued multifunctions. An application is given to the control of chattering systems. It is an extension of a result of Artstein to the case of set-valued maps with unbounded values. The proof of the main results is simple and essentially relies on measurable selections arguments.      

Some representations of midconvex set-valued functions

Summary In this note we establish conditions under which every midconvex set-valued function can be represented as sum of an additive function and a convex set-valued function. These results improve some theorems obtained in [8], [10] and [3]. Some results on local Jensen selections of midconvex set-valued functions are also given.     

Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier-Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.     

The Knaster-Kuratowski-Mazurkiewicz theorem and abstract convexities

The Knaster-Kuratowski-Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called “intersection” theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from Rn to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. Math. 11 (1959) 556-575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173-187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285-302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.     

Convergence of Directional Methods under mild differentiability and applications

A semilocal convergence analysis for Directional Methods under mild differentiability conditions is provided in this study. Using our idea of recurrent functions, we provide sufficient convergence conditions as well as the corresponding errors bounds. The results are extended to hold in a Hilbert space setting and a favorable comparison is provided with earlier works [6], [7], [8], [9], [10], [11] and [20]. Numerical examples are also provided in this study.     

A new method for solving singular fourth-order boundary value problems with mixed boundary conditions

In [1], [2], [3], [4], [5], [6], [7] and [8], it is very difficult to get reproducing kernel space of problem (1). This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of fourth-order in the new reproducing kernel space. The numerical results are compared with both the exact solution and its n-order derived functions in the example. It is demonstrated that the new method is quite accurate and efficient for fourth-order problems.     

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.     

On multifunction transforms of probability measures

The paper briefly presents Topsøe's setup and gives two results based on that. A delta theorem for multifunctions improving that of King [2] is given. In particular, the contingent derivative may be a compact-valued instead of a single-valued multi-function. Stochastic optimization is also discussed. Some results are achieved for the limit distribution of optimal values as well as of optimal solutions.     

可行策略对应的图像拓扑下广义博弈Nash平衡的稳定性

以往关于广义博弈Nash平衡的稳定性的研究,均利用可行策略映射之间的一致度量.现考虑在更弱的度量下,利用可行策略映射图像之间的Hausdorff距离定义度量.在此弱图像拓扑下,证明了广义博弈空间的完备性,以及Nash平衡映射的上半连续性和紧性,进而得到广义博弈Nash平衡的通有稳定性.即在Baire分类的意义下,大多数的广义博弈都是本质的.     

A new method for solving a class of mixed boundary value problems with singular coefficient

In [1], [2], [3], [4], [5], [6] and [7], it is very difficult to deal with initial boundary value conditions. In this paper, we give a new method to deal with boundary value conditions, the main contribution of this paper is to put mixed boundary value conditions into reproducing kernel Hilbert space. The numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.     

Characterizations of intervals via continuous selections

We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0,1), [0,1] or the long lineL, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological spaceX is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a)X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b)X is infinite, separable, locally connected and has exactly one continuous selection; (c)X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results.     

On the embedding of some linear spaces in finite projective planes

The problem of embedding of linear spaces in finite projective planes has been examined by several authors ([1], [2], [3], [4], [5], [6]). In particular, it has been proved in [1] that a linear space which is the complement of a projective or affine subplane of order m is embeddable in a unique way in a projective plane of order n. In this article, we give a generalization of this result by embedding linear spaces in a finite projective plane of order n, which are complements of certain regularA-affine linear spaces with respect to a finite projective plane.     

Numerical Analysis of the Stochastic Moving Boundary Problem

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [11], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal [17], and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established     

Existence of a global liapunov functional for certain classes of nonlinear distributed systems : PMM vol. 40, n 6, 1976, pp. 1135–1142

The existence of a global Liapunov functional for nonlinear evolutionary equations in a Hilbert space is investigated as a continuation of paper [1], The results obtained represent a generalization of the results of the theory of absolute stability [2, 3], for the systems with infinite dimensional phase space, and are used for investigation of the nonlocal stability and instability of nonlinear distributed systems. The conditions of existence of the global Liapunov functional obtained are illustrated by an example of a nonlinear parabolic system defined in the interval [0, 1], The concept of a Liapunov functional was first introduced and used with success in [4].     

Martingale solutions and Markov selections for stochastic partial differential equations

We present a general framework for solving stochastic porous medium equations and stochastic Navier–Stokes equations in the sense of martingale solutions. Following Krylov [N.V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 691–708] and Flandoli–Romito [F. Flandoli, N. Romito, Markov selections for the 3D stochastic Navier–Stokes equations, Probab. Theory Related Fields 140 (2008) 407–458], we also study the existence of Markov selections for stochastic evolution equations in the absence of uniqueness.     

Spreading in space–time periodic media governed by a monostable equation with free boundaries,Part 2: Spreading speed

This is Part 2 of our work aimed at classifying the long-time behavior of the solution to a free boundary problem with monostable reaction term in space–time periodic media. In Part 1 (see [2]) we have established a theory on the existence and uniqueness of solutions to this free boundary problem with continuous initial functions, as well as a spreading-vanishing dichotomy. We are now able to develop the methods of Weinberger [15], [16] and others [6], [7], [8], [9], [10] to prove the existence of asymptotic spreading speed when spreading happens, without knowing a priori the existence of the corresponding semi-wave solutions of the free boundary problem. This is a completely different approach from earlier works on the free boundary model, where the spreading speed is determined by firstly showing the existence of a corresponding semi-wave. Such a semi-wave appears difficult to obtain by the earlier approaches in the case of space–time periodic media considered in our work here.     

Convenient Vector Spaces of Smooth Functions

By a convenient vector space is meant a locally convex IR-vector space which is separated, bornological and Mackey-complete. The theory of such spaces, initiated in [Kr 82], [Fr 82], and [FGK 83], has evolved into a book [FK 88]. In the preliminaries below we outline the principal features of this theory relevant to this paper. We are concerned mainly with questions about the reflexiveness of spaces C^∞(X, ?) for various X and matters closely related to this.     

Extenders and ϰ-metrizable compacta

A characterization of ?-metrizable compacta in terms of extension of functions and upper semicontinuous compact-valued retractions to superextensions is established.