Approximation Theorems and Fixed Point Theorems for Various Classes of 1-set-contractive Mappings in Banach Spaces

In this paper, we will prove that Ky Fan's Theorem (Math. Z. 112(1969), 234–240) is true for 1-set-contractive maps defined on a bounded closed convex subset K in a Banach space with intK≠0. This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractive maps, LANE maps and others. As applications of our theorems, some fixed point theorems of non-selfmaps are proved under various well-known boundary conditions. Our results are generalizations and improvements of the recent results obtained by many authors. Project supported by the National Natural Science Foundation of China and Natural Science Foundation of Shandong Province of China     

Variational Principles, Minimization Theorems, and Fixed-Point Theorems on Generalized Metric Spaces

In this paper, we prove a new minimization theorem by using the generalized Ekeland variational principle. We apply our minimization theorem to obtain some fixed-point theorems. Our results extend, improve, and unify many known results due to Kui, Ekeland, Takahashi, Caristi, iri, and others.     

Projection Theorems for Harmonic Measure in NTA Domains

Suppose D is an NTA domain, E D is any closed set, and P ^x ₀(E) is the projection with respect to a point x ₀D of the set E onto the boundary of D. The projection P ^x ₀ satisfies certain geometric properties so that it is a generalization of the notion of radial projection with respect to a point x ₀ onto a boundary of a domain. It is shown that the harmonic measure of E with respect to the domain DE evaluated at the point x ₀ is bounded below by a constant times the harmonic measure of the set P ^x ₀(E) with respect to the domain D evaluated at the point x ₀. The constant is independent of the set E but it may depend upon x ₀.     

Limit Theorems for Multiplicative Processes

Let W be a non-negative random variable with EW=1, and let {W _i } be a family of independent copies of W, indexed by all the finite sequences i=i ₁i _n of positive integers. For fixed r and n the random multiplicative measure ⁿ _r has, on each r-adic interval at nth level, the density with respect to the Lebesgue measure on [0,1]. If EW log Wr, the sequence { ⁿ _r } _n converges a.s. weakly to the Mandelbrot measure _r . For each fixed 1n, we study asymptotic properties for the sequence of random measures { ⁿ _r } _r as r. We prove uniform laws of large numbers, functional central limit theorems, a functional law of iterated logarithm, and large deviation principles. The function-indexed processes is a natural extension to a tree-indexed process at nth level of the usual smoothed partial-sum process corresponding to n=1. The results extend the classical ones for { ¹ _r } _r , and the recent ones for the masses of { _r } _r established in Ref. 23.     

Existence Results for Set-Valued Vector Quasiequilibrium Problems

This paper deals with the set-valued vector quasiequilibrium problem of finding a point (z ₀,x ₀) of a set E×K such that (z ₀,x ₀)∈B(z ₀,x ₀)×A(z ₀,x ₀), and, for all η∈A(z ₀,x ₀),

Euler Characteristic and Fixed-Point Theorems

We propose a geometric definition of the Euler characteristic (M) for the class of compact epi-Lipschitzian sets MRⁿ and we provide existence theorems of (generalized) equilibria for set-valued mappings F when the domain M of F is neither assumed to be convex, nor smooth but has a nonzero Euler characteristic.     

Discrete and Lexicographic Helly-Type Theorems

Helly’s theorem says that if every d+1 elements of a given finite set of convex objects in ℝ ^d have a common point, then there is a point common to all of the objects in the set. We define three new types of Helly theorems: discrete Helly theorems—where the common point should belong to an a-priori given set, lexicographic Helly theorems—where the common point should not be lexicographically greater than a given point, and lexicographic-discrete Helly theorems. We study the relations between the different types of the Helly theorems. We obtain several new discrete and lexicographic Helly numbers. An extended abstract containing parts of this work appeared in the proceedings of the Forty-Fifth Annual Symposium on Foundations of Computer Science (FOCS) 2004. This work is part of the author’s Ph.D. thesis, prepared in the school of mathematical sciences at Tel Aviv University, under the supervision of Professor Arie Tamir.     

Renewal Theorems for Singular Differential Operators

Let * be the convolution on M( ₊) associated with a second order singular differential operator L on ]0, +[. If is a probability measure on ₊ with suitable moment conditions, we study how to normalize the measures * ⁿ ; n } (resp. ) in order to get vague convergence if n+ (resp. x+). The results depend on the asymptotic drift of the operator L and on a precise study of the asymptotic behaviour of its eigenfunctions.     

Banach空间中多类1-集-压缩映象的逼近定理和不动点定理

本文证明了 Ｋｙ Ｆａｎ定理［１］对定义在 Ｂａｎａｃｈ空间中含有内点的有界闭凸子集上的１－集－压缩映象正确．１－集－压缩映象类包含凝聚映象、非扩张映象、半压缩映象、局部几乎非扩张映象和其它映象．作为定理的应用,得到了一些非自映象在许多众知边界条件下的不动点定理,改进和推广了许多作者的最近结果．     

Existence of solutions for a class of nonlinear control problems

We consider problems of control and problems of optimal control, monitored by an abstract equation of the formEx=N _u x in a finite interval [0,T]; here,x is the state variable with values in a reflexive Banach space;u is the control variable with values in a metric space;E is linear and monotone; andN _u is nonlinear of the Nemitsky type. Thus, by well-known devices, the results apply also to parabolic partial differential equations in a cylinder [0,T]×G,G ⁿ , with Cauchy data fort=0 and Dirichlet or Neumann conditions on the lateral surface of the cylinder. We prove existence theorems for solutions and existence theorems for optimal solutions, by reduction to a theorem of Kemochi for reflexive Banach spaces.     

Existence Theorems for Certain Elliptic and Parabolic Semilinear Equations

Existence theorems for the nonlinear parabolic differential equation −∂u/∂t + Δu + |u|^p + f(x, t) = 0 in ⁿ × [0, ∞) with zero initial value are established given explicit conditions on the nonhomogeneous termf(x, t). An existence theorem is also demonstrated for the corresponding elliptic equation.     

The Asymptotic Behavior of a Family of Sequences via Tauberian Theorems

We study the asymptotic behavior of a family of sequences defined by the following nonlinear induction relation c₀ = 1 and c_n ∑^k_{j = 1} r_jc_[n/mj] + ∑^k_{j = k + 1} r_jc_{[(n + 1)^1/mj] − 1} for n ≥ 1, where the r_j are real positive numbers and m_j are integers greater than or equal to 2. Depending on the fact that ∑^k_{j = 1} r_j is greater or lower than 1, we prove that c_n/n^α or c_n/(ln n)^α goes to some finite limit for some explicit α. Our study is based on Tauberian theorems and extends a result of Erdös et al.     

Absolute Summability Factors and Absolute Tauberian Theorems for Double Series and Sequences

Let A and B be the linear methods of the summability of double series with fields of bounded summability MA _b ^' and B _b ^' , respectively. Let T be certain set of double series. The condition x T is called B _b-Tauberian for A if A _b ^' B _b ^' .Some theorems about summability factors enable one to find new B _b-Tauberian conditions for A from the already known B _b-Tauberian conditions for A.     

Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms

We consider quadratic forms of the type

Surface integral and Gauss-Ostrogradskij theorem from the viewpoint of applications

Making use of a surface integral defined without use of the partition of unity, trace theorems and the Gauss-Ostrogradskij theorem are proved in the case of three-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces H ^1,p () (1 p < ). The paper is a generalization of the previous author's paper which is devoted to the line integral.     

Extended iterative methods for the solution of operator equations

Summary Given an iterative methodM ₀, characterized byx ^(k+1=G ₀(x( ^k )) (k0) (x(⁰) prescribed) for the solution of the operator equationF(x)=0, whereF:XX is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM _p characterized byx ^(k+1=G _p (x( ^k )) (k0) (x(⁰) prescribed) with order of convergence higher than that ofM _o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM ₀ is Newton's method.Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM _p, which are referred to as extensions ofM ₀. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved.Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.     

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.     

Tauberian and Abelian theorems for rapidly decaying distributions and their applications to stable laws

We establish some assertions of Tauberian and Abelian types which enable us to find connections between the asymptotic properties of the Laplace transform at infinity and the asymptotics of the corresponding densities of rapidly decaying distributions (at infinity or in some neighborhood of zero). As applications of our Tauberian type theorems we present asymptotics for the density f ^(α,ρ)(x) of “extreme” stable laws with parameters (α, ρ) for ρ = ±1 and x lying in the domain of rapid decay of f ^(α,ρ)(x). This asymptotics had been found in [1–5] by a more complicated method.     

The order of the giant component of random hypergraphs

We establish central and local limit theorems for the number of vertices in the largest component of a random d‐uniform hypergraph H_d(n,p) with edge probability p = c/ , where c > (d ‐ 1)^‐1 is a constant. The proof relies on a new, purely probabilistic approach. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Coherence and Uniqueness Theorems for Averaging Processes in Statistical Mechanics

Let S be the set of scalings {n ^–1:n=1,2,3,...} and let L _z =z Z ², zS, be the corresponding set of scaled lattices in R ². In this paper averaging operators are defined for plaquette functions on L _z to plaquette functions on L _z for all z, zS, z=dz, d{2,3,4,...}, and their coherence is proved. This generalizes the averaging operators introduced by Balaban and Federbush. There are such coherent families of averaging operators for any dimension D=1,2,3,... and not only for D=2. Finally there are uniqueness theorems saying that in a sense, besides a form of straightforward averaging, the weights used are the only ones that give coherent families of averaging operators.