532,879 research outputs found
Generalizations of some fixed point theorems in banach and metric spaces
A fixed point of a mapping is an element in the domain of the mapping that is mapped into itself by the mapping. The study of fixed points has been a field of interests to mathematicians since the discovery of the Banach contraction theorem, i.e. if is a complete metric space and is a contraction mapping (i.e. there exists such that for all ), then has a unique fixed point. The Banach contraction theorem has found many applications in pure and applied mathematics. Due to fixed point theory being a mixture of analysis, geometry, algebra and topology, its applications to other fields such as physics, economics, game theory, chemistry, engineering and many others has become vital. The theory is nowadays a very active field of research in which many new theorems are published, some of them applied and many others generalized. Motivated by all of this, we give an exposition of some generalizations of fixed point theorems in metric fixed point theory, which is a branch of fixed point theory about results of fixed points of mappings between metric spaces, where certain properties of the mappings involved need not be preserved under equivalent metrics. For instance, the contractive property of mappings between metric spaces need not be preserved under equivalent metrics. Since metric fixed point theory is wide, we limit ourselves to fixed point theorems for self and non-self-mappings on Banach and metric spaces. We also take a look at some open problems on this topic of study. At the end of the dissertation, we suggest our own problems for future research
Compact weighted composition operators and fixed points in convex domains
We extend a classical result of Caughran/Schwartz and another recent result
of Gunatillake by showing that if D is a bounded, convex domain in
n-dimensional complex space, m is a holomorphic function on D and bounded away
from zero toward the boundary of D, and p is a holomorphic self-map of D such
that the weighted composition operator W assigning the product of m and the
composition of f and p to a given function f is compact on a holomorphic
functional Hilbert space (containing the polynomial functions densely) on D
with reproducing kernel K blowing up along the diagonal of D toward its
boundary, then p has a unique fixed point in D. We apply this result by making
a reasonable conjecture about the spectrum of W based on previous one-variable
and multivariable results concerning compact weighted and unweighted
composition operators.Comment: 10 pages. Corrected a few typographical errors and an error in one
step of the main result's proof. This paper was presented in September 2005
at the Wabash Extramural Modern Analysis Mini-conference in Indianapoli
First steps in synthetic guarded domain theory: step-indexing in the topos of trees
We present the topos S of trees as a model of guarded recursion. We study the
internal dependently-typed higher-order logic of S and show that S models two
modal operators, on predicates and types, which serve as guards in recursive
definitions of terms, predicates, and types. In particular, we show how to
solve recursive type equations involving dependent types. We propose that the
internal logic of S provides the right setting for the synthetic construction
of abstract versions of step-indexed models of programming languages and
program logics. As an example, we show how to construct a model of a
programming language with higher-order store and recursive types entirely
inside the internal logic of S. Moreover, we give an axiomatic categorical
treatment of models of synthetic guarded domain theory and prove that, for any
complete Heyting algebra A with a well-founded basis, the topos of sheaves over
A forms a model of synthetic guarded domain theory, generalizing the results
for S
Axioms for the coincidence index of maps between manifolds of the same dimension
We study the coincidence theory of maps between two manifolds of the same
dimension from an axiomatic viewpoint. First we look at coincidences of maps
between manifolds where one of the maps is orientation true, and give a set of
axioms such that characterizes the local index (which is an integer valued
function). Then we consider coincidence theory for arbitrary pairs of maps
between two manifolds. Similarly we provide a set of axioms which characterize
the local index, which in this case is a function with values in . We also show in each setting that the group of values for the index
(either or ) is determined by the axioms.
Finally, for the general case of coincidence theory for arbitrary pairs of
maps between two manifolds we provide a set of axioms which charaterize the
local Reidemeister trace which is an element of an abelian group which depends
on the pair of functions. These results extend known results for coincidences
between orientable differentiable manifolds.Comment: 29 page
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