Some Structure Theorems for Inverse Limits with Set-valued Functions
نویسنده
چکیده
We investigate inverse limits with set-valued bonding functions. We generalize theorems of W. T. Ingram and William S. Mahavier, and of Van Nall, on the connectedness of the inverse limit space. We establish a fixed point theorem and show that under certain conditions, inverse limits with set-valued bonding functions can be realized as ordinary inverse limits. We also obtain some results that are useful in determining the existence of certain subcompacta of the inverse limit on a single space with a single set-valued bonding function. All spaces considered in this paper will be metric. A continuum is a compact, connected metric space. A continuous function f : X → Y will be referred to as a mapping. We wish to consider inverse limits on inverse sequences X1, X2, . . . of compacta with upper semi-continuous bonding functions G n : Xn+1 → 2n . These inverse limits have been called generalized inverse limits and inverse limits with set-valued bonding functions. The literature on and interest in these inverse limits is growing fairly rapidly (see [3], [4], [5], [6], [7], [8], [10], [11], [12]). Perusing these papers, one notices that there are some commonly-used notations and terminology for important concepts related to a function G : X → 2 that are defined, in a natural way, relative to X, Y , and X × Y . Ordinarily, the graph of G would lie in X × 2 with the product topology induced from the topology of X and the topology of 2 . However, we wish to view the graph of G as a subset of X × Y , and for x ∈ X, we view G(x) as a subset of Y rather than a point in 2 . For essentially all of the properties we are interested in, the topologies of X, Y , and X × Y will influence 2010 Mathematics Subject Classification. Primary 54C60, 54D80; Secondary 54B10, 54C15, 54F15, 54H25.
منابع مشابه
Inverse Limits of Families of Set-valued Functions
In this paper we investigate inverse limits of two related parameterized families of upper semi-continuous set-valued functions. We include a theorem one consequence of which is that certain inverse limits with a single bonding function from one of these families are the closure of a topological ray (usually with indecomposable remainder). Also included is a theorem giving a new sufficient cond...
متن کاملFuzzy number-valued fuzzy relation
It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from domain to range, binary relations may assign to each element of domain two or more elements of range. Some basic operations on functions such as the inverse and composition are applicable to binary relations as well. Depending on the domain or range or both are fuzzy value fuzzy set, i...
متن کاملTopology Proceedings 36 (2010) pp. 353-373: Inverse Limits with Upper Semi-Continuous Bonding Functions: Problems and Some Partial Solutions
By means of numerous examples we call attention to several problems in the theory of inverse limits with upper semi-continuous bonding functions. Along with the problems we present a few partial solutions. Most of the problems we discuss arise from the failure of certain theorems from the theory of inverse limits with mappings to carry over to the setting of inverse limits with set-valued funct...
متن کاملSome local fixed point results under $C$-class functions with applications to coupled elliptic systems
The main objective of the paper is to state newly fixed point theorems for set-valued mappings in the framework of 0-complete partial metric spaces which speak about a location of a fixed point with respect to an initial value of the set-valued mapping by using some $C$-class functions. The results proved herein generalize, modify and unify some recent results of the existing literature. As an ...
متن کاملSome Fixed Point Theorems in Generalized Metric Spaces Endowed with Vector-valued Metrics and Application in Linear and Nonlinear Matrix Equations
Let $mathcal{X}$ be a partially ordered set and $d$ be a generalized metric on $mathcal{X}$. We obtain some results in coupled and coupled coincidence of $g$-monotone functions on $mathcal{X}$, where $g$ is a function from $mathcal{X}$ into itself. Moreover, we show that a nonexpansive mapping on a partially ordered Hilbert space has a fixed point lying in the unit ball of the Hilbert space. ...
متن کامل