Weak convergence of random <Emphasis Type="Bold">p</Emphasis>-mappings and the exploration process of inhomogeneous continuum random trees

We study the asymptotics of the p-mapping model of random mappings on [n] as n gets large, under a large class of asymptotic regimes for the underlying distribution p. We encode these random mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees, this exploration process being derived (Aldous-Miermont-Pitman 2004) from a bridge with exchangeable increments. Our setting generalizes previous results by allowing a finite number of “attracting points” to emerge.Research supported by NSF Grant DMS-0203062.Research supported by NSF Grant DMS-0071468.     

A random weak ergodic property of infinite products of operators in metric spaces

ABSTRACT

We study the random weak ergodic property of infinite products of mappings acting on complete metric spaces. Our results describe an aspect of the asymptotic behaviour of random infinite products of such mappings. More precisely, we show that in appropriate spaces of sequences of operators there exists a subset, which is a countable intersection of open and everywhere dense sets, such that each sequence belonging to this subset has the random weak ergodic property. Then we show that several known results in the literature can be deduced from our general result.     

A Note on a Composition of Two Random Integral Mappings β and Some Examples

Abstract

The method of random integral representation, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note, we show that a composition of two random integral mappings ^β is again a random integral mapping. We illustrate our results with some examples.     

Random Coincidence Point Theorem in Fréchet Spaces with Applications

Abstract

We proved a random coincidence point theorem for a pair of commuting random operators in the setup of Fréchet spaces. As applications, we obtained random fixed point and best approximation results for *-nonexpansive multivalued maps. Our results are generalizations or stochastic versions of the corresponding results of Shahzad and Latif [Shahzad, N.; Latif, A. A random coincidence point theorem. J. Math. Anal. Appl. 2000, 245, 633–638], Khan and Hussain [Khan, A.R.; Hussain, N. Best approximation and fixed point results. Indian J. Pure Appl. Math. 2000, 31 (8), 983–987], Tan and Yaun [Tan, K.K.; Yaun, X.Z. Random fixed point theorems and approximation. Stoch. Anal. Appl. 1997, 15 (1), 103–123] and Xu [Xu, H.K. On weakly nonexpansive and *-nonexpansive multivalued mappings. Math. Japon. 1991, 36 (3), 441–445].     

Brownian bridge asymptotics for random mappings

Uniform random mappings of an n-element set to itself have been much studied in the combinatorial literature. We introduce a new technique, which starts by specifying a coding of mappings as walks with ± 1 steps. The uniform random mapping is thereby coded as a nonuniform random walk, and our main result is that as n→∞ the random walk rescales to reflecting Brownian bridge. This result encompasses a large number of limit theorems for “global” characteristics of uniform random mappings. © 1994 John Wiley & Sons, Inc.     

Rigidity of H–type groups

On H–type groups N the left invariant horizontal vector fields span a subbundle of the tangent bundle, called the horizontal bundle HN. Generalized contact mappings f on N are smooth mappings which preserve HN. The question is: how many such mappings exist? In the case of the Heisenberg group these are the contact mappings in the classical sense and they exist in abundance. In this paper it is shown that if the dimension of the center of N is at least three, then the generalized contact mappings are in the automorphism group of a finite dimensional Lie algebra g. The elements in are the infinitesimal generators of local one parameter subgroups of generalized contact transformations. Rigidity is defined as the property that is finite dimensional. For the case of the complexified Heisenberg group, i.e. the case when the dimension of the center of N is two, it has been shown [RR] that g is infinite dimensional. Received January 4, 2000; in final form March 20, 2000 / Published online April 12, 2001     

The limit distribution of the number of nodes in low strata of a random mapping

We consider random single-valued mappings of an n-element set into itself. Using simple probabilistic facts, we show that the number of nodes in low strata and the number of cyclic nodes of the graphs of such mappings are identically distributed as n → ∞.     

BASIC ASPECTS OF UNIRATIONAL HOMOTOPY THEORY

Abstract

In this note clopen-to-closed mappings are used to characterize strongly zero-dimensional spaces and ultranormal spaces.     

Univalence Criteria Starting from the Method of Loewner Chains

The paper continues the work of Royster (Duke Math J 19:447–457, 1952), Mocanu [Mathematica (Cluj) 22(1):77–83, 1980; Mathematica (Cluj) 29:49–55, 1987], Cristea [Mathematica (Cluj) 36(2):137–144, 1994; Complex Var 42:333–345, 2000; Mathematica (Cluj) 43(1):23–34, 2001; Mathematica (Cluj), 2010, to appear; Teoria Topologica a Functiilor Analitice, Editura Universitatii Bucuresti, Romania, 1999] of extending univalence criteria for complex mappings to C ¹ mappings. We improve now the method of Loewner chains which is usually used in complex univalence theory for proving univalence criteria or for proving quasiconformal extensions of holomorphic mappings f : B → C ⁿ to C ⁿ . The results are surprisingly strong. We show that the usual results from the theory, like Becker’s univalence criteria remain true for C ¹ mappings and since we use a stronger form of Loewner’s theory, we obtain results which are stronger even for holomorphic mappings f : B → C ⁿ . In our main result (Theorem 4.1) we end the researches dedicated to quasiconformal extensions of K-quasiregular and holomorphic mappings f : B → C ⁿ to C ⁿ . We show that a C ¹ quasiconformal map f : B → C ⁿ can be extended to a quasiconformal map F : C ⁿ → C ⁿ , without any metric condition imposed to the map f.     

Growth and Distortion Results for Convex Mappings in Infinite Dimensional Spaces

In this paper, we obtain the growth result for normalized convex mappings on the unit ball of a complex Banach space. Also we give some bounds of coefficients and a distortion result for convex mappings on the unit ball of a complex Hilbert space.     

Coincidences of Nonself Operators by a Simpler Algorithm

Abstract

In this article, we introduce a new class of contractive mappings and study analytical and computational aspects of a special case of Jungck-Khan iterative algorithm generated by this class of mappings. In particular, we improve upon strong convergence, rate of convergence and data dependence results existing in the current literature. Analytical as well numerical illustrative examples are given to support the new results.     

Convergence theorems for semigroup-type families of non-self mappings

Motivated by a paper Chidume and Zegeye [Strong convergence theorems for common fixed points of uniformly L-Lipschitzian pseudocontractive semi-groups, Applicable Analysis, 86 (2007), 353–366], we prove several strong convergence theorems for a family (not necessarily a semigroup) ℱ = {T(t): t ∈ G} of nonexpansive or pseudocontractive non-self mappings in a reflexive strictly convex Banach space with a uniformly Gateaux differentiable norm, where G is an unbounded subset of ℝ⁺. Our results extend and improve the corresponding ones byMatsushita and Takahashi [Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,Nonlinear Analysis, 68 (2008), 412–419],Morales and Jung [Convergence of paths for pseudo-contractive mappings in Banach spaces, Proceedings of American Mathematical Society, 128 (2000), 3411–3419], Song [Iterative approximation to common fixed points of a countable family of nonexpansive mappings, Applicable Analysis, 86 (2007), 1329–1337], Song and Xu [Strong convergence theorems for nonexpansive semigroup in Banach spaces, Journal of Mathematical Analysis and Applications, 338 (2008), 152–161], Wong, Sahu, and Yao [Solving variational inequalities involving nonexpansive type mappings, Nonlinear Analysis, (2007) doi:10.1016/j.na. 2007.11.025] in the context of a non-semigroup family of non-self mappings.      

Weak Subdifferential of Set-Valued Mappings

In this note we introduce a notion of the weak contingent generalized gradient for set-valued mappings associated with the contingent epiderivative of set-valued mappings introduced in "E. Bednarczuk and W. Song (1998). Contingent epiderivative and its applications to set-valued optimization. Control and Cybernetics, 27, 376-386; G.Y. Chen and J. Jahn (1998). Optimally conditions for set-valued optimization problems. Mathematical Methods of Operations Research, 48, 187-200." and prove that, under some additional condition, it coincides with the weak subdifferential introduced in "T. Tanino (1992). Conjugate duality in vector optimization. Journal of Mathematical Analysis and Applications, 167, 84-97." when the set-valued map is cone-convex. We also study the weak contingent generalized gradient of a sum of two set-valued mappings and optimality conditions for a set-valued vector optimization problem.     

Stochastic Integrals with Respect to Consistent Random Measures

We consider integrals of random mappings with respect to consistent random measures in C([0; 1]).     

Coupled Random Fixed Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces

Abstract

Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete separable metric space and (Ω, Σ) be a measurable space. In this article a pair of random mappings F: Ω × (X × X) → X and g: Ω × X → X, where F has a mixed g-monotone property on X, and F and g satisfy the non-linear contractive condition (5) below, are introduced and investigated. Two coupled random coincidence and coupled random fixed point theorems are proved. These results are random versions and extensions of recent results of Lakshmikantham and ?iri? [V. Lakshmikantham and Lj. ?iri?, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.—Theor. 70(12) (2009): 4341–4349] and include several recent developments.     

The Markov chain asymptotics of random mapping graphs

In this paper the limit behavior of random mappings with n vertices is investigated. We first compute the asymptotic probability that a fixed class of finite non-intersected subsets of vertices are located in different components and use this result to construct a scheme of allocating particles with a related Markov chain. We then prove that the limit behavior of random mappings is actually embedded in such a scheme in a certain way. As an application, we shall give the asymptotic moments of the size of the largest component.     

A classification of proper holomorphic mappings between generalized pseudoellipsoids of different dimensions

ABSTRACT

We classify proper holomorphic mappings between generalized pseudoellipsoids of different dimensions. Those domains are parametrized by the exponents. The relations among them are also obtained. Main tool is the orthogonal decomposition of a CR bundle. Such a decomposition derives the ‘variable-splitting’ of the mapping.     

Invariant Curves of Reversible Mappings with Small Twist

In this paper,we are concerned with the existence of invariant curves of reversible mappings.A variant of the classical small twist theorem is given.     

On the solution existence of implicit quasivariational inequalities with discontinuous multifunctions

In this article, we established some solution existence theorems for implicit quasivariational inequalities. We first established some results in finite-dimensional spaces and then a solution existence result in infinite-dimensional spaces was derived. Our theorems are proved for discontinuous mappings and sets which may be unbounded. The results presented in this article are improvements of results in Cubiotti, and Yao (Cubiotti, P. and Yao, J.C., 1997, Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings. Mathematical Methods of Operations Research, 46, 213–328; Cubiotti, P. and Yao, J.C., 2007, Discontinuous implicit generalized quasi-variational inequalities in Banach spaces. Journal of Global Optimization (To appear))     

Viscosity Approximation Methods for Generalized Multi-Valued Nonexpansive Mappings with Applications

Abstract

In this article, we study viscosity approximation methods for generalized multi-valued nonexpansive mappings and we present some new results related to strong convergence, variational inequality, convex optimization, split and common split feasibility problems (SFPs). Some numerical computations are also presented to illustrate our results.