scholarly journals On dynamics generated by a uniformly convergent sequence of maps

2018 ◽  
Vol 247 ◽  
pp. 81-90 ◽  
Author(s):  
Puneet Sharma ◽  
Manish Raghav
Keyword(s):  
Convergent Sequence ◽  
Uniformly Convergent

scholarly journals A cauchy criterion and a convergence theorem for Riemann-complete integral

1972 ◽  
Vol 13 (1) ◽  
pp. 21-24
Author(s):  
H. W. Pu
Keyword(s):  
Convergence Theorem ◽  
Convergent Sequence ◽  
Monotone Convergence ◽  
Lebesgue Integral ◽  
Necessary And Sufficient Condition ◽  
Integrable Functions ◽  
Type Integral ◽  
Uniformly Convergent ◽  
Necessary And Sufficient ◽  
Generalized Type

In 1957 Kurzweil [1] proved some theorems concerning a generalized type of differential equations by defining a Riemann-type integral, but he did not study its properties beyond the needs of that research. This was done by R. Henstock [2, 3], who named it a Riemann-complete integral. He showed that the Riemann-complete integral includes the Lebesgue integral and that the Levi monotone convergence theorem holds. The purpose of the present paper is to give a necessary and sufficient condition for a function to be Riemann-complete integrable and to establish a termwise integration theorem for a uniformly convergent sequence of Riemann-complete integrable functions.

A generalized Bernstein approximation theorem

1988 ◽  
Vol 104 (2) ◽  
pp. 317-330 ◽  
Author(s):  
João B. Prolla
Keyword(s):  
Convex Cone ◽  
Hausdorff Space ◽  
Bernstein Polynomials ◽  
Convergent Sequence ◽  
Vector Subspace ◽  
Weierstrass Theorem ◽  
Classic Result ◽  
Bernstein Approximation ◽  
Uniformly Convergent ◽  
Image Position

A celebrated theorem of Weierstrass states that any continuous real-valued function f defined on the closed interval [0, 1] ⊂ ℝ is the limit of a uniformly convergent sequence of polynomials. One of the most elegant and elementary proofs of this classic result is that which uses the Bernstein polynomials of fone for each integer n ≥ 1. Bernstein's Theorem states that Bn(f) → f uniformly on [0, 1] and, since each Bn(f) is a polynomial of degree at most n, we have as a consequence Weierstrass' theorem. See for example Lorentz [9]. The operator Bn, defined on the space C([0, 1]; ℝ) with values in the vector subspace of all polynomials of degree at most n has the property that Bn(f) ≥ 0 whenever f ≥ 0. Thus Bernstein's Theorem also establishes the fact that each positive continuous real-valued function on [0, 1] is the limit of a uniformly convergent sequence of positive polynomials. This raises the following natural question: consider a compact Hausdorff space X and the convex cone C+(X):= {f ∈ C(X; ℝ); f ≥ 0}. Now the analogue of Bernstein's Theorem would be a theorem stating when a convex cone contained in C+(X) is dense in it. More generally, one raises the question of describing the closure of a convex cone contained in C(X; ℝ), and, in particular, the closure of A+:= {f ∈ A; f ≥ 0}, where A is a subalgebra of C(X; ℝ).

Sequentially Relatively Uniformly Complete Riesz Spaces and Vulikh Algebras

1972 ◽  
Vol 24 (6) ◽  
pp. 1110-1113 ◽  
Author(s):  
C. T. Tucker
Keyword(s):  
Positive Integer ◽  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Zero Element ◽  
Riesz Space ◽  
Riesz Spaces ◽  
Weak Unit ◽  
Uniformly Convergent ◽  
Positive Integers ◽  
Unique Limit

Throughout this paper V will denote an Archimedean Riesz space with a weak unit e and a zero element θ. A sequence f1,f2,f3, … of points of V is said to converge relatively uniformly to a point f (with regulator the point g of V) if, for each ∈ > 0, there is a number N such that, if n is a positive integer and n > N, then |f — fn| < ∈g. In an Archimedean Riesz space a relatively uniformly convergent sequence has a unique limit. The sequence f1, f2, f3, … is called a relatively uniform Cauchy sequence (with regulator g) if, for each ∈ > 0, there is a number N such that if n and m are positive integers and n, m > N, then |fn — fm| < eg. A subset M of V is said to be sequentially relatively uniformly complete, written s.r.u.-complete, whenever every relatively uniform Cauchy sequence of points of M (with regulator in V) converges to a point of M. This property was defined by Luxemburg and Moore in [4] and some related conditions were derived.

scholarly journals Devaney Chaos in Nonautonomous Discrete Systems

2016 ◽  
Vol 26 (11) ◽  
pp. 1650190 ◽  
Author(s):  
Hao Zhu ◽  
Yuming Shi ◽  
Hua Shao
Keyword(s):  
Metric Space ◽  
Chaotic Behavior ◽  
Convergent Sequence ◽  
Discrete Systems ◽  
Periodic Points ◽  
The Other ◽  
Compact Metric Space ◽  
Devaney Chaos ◽  
Continuous Maps ◽  
Uniformly Convergent

This paper is concerned with Devaney chaos in nonautonomous discrete systems. It is shown that in its definition, the two former conditions, i.e. transitivity and density of periodic points, in a set imply the last one, i.e. sensitivity, in the case that the set is unbounded, while a similar result holds under two additional conditions in the other case that the set is bounded. Some chaotic behavior is studied for a class of nonautonomous discrete systems, each of which is governed by a convergent sequence of continuous maps. In addition, the concepts of some pseudo-orbits and shadowing properties are introduced for nonautonomous discrete systems, and it is shown that some shadowing properties of the system and density of periodic points imply that the system is Devaney chaotic under the condition that the sequence of continuous maps is uniformly convergent in a compact metric space.

scholarly journals Hyers-Ulam Stability of Iterative Equation in the Class of Lipschitz Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Chao Xia ◽  
Wei Song
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Functional Equations ◽  
Convergent Sequence ◽  
Lipschitz Functions ◽  
Ulam Stability ◽  
Iterative Equation ◽  
Iterative Equations ◽  
Uniformly Convergent ◽  
Sequence Of Functions

Hyers-Ulam stability is a basic sense of stability for functional equations. In the present paper we discuss the Hyers-Ulam stability of a kind of iterative equations in the class of Lipschitz functions. By the construction of a uniformly convergent sequence of functions we prove that, for every approximate solution of such an equation, there exists an exact solution near it.

Synthesis of B-ring-fluorinated (−)-epicatechin gallate derivatives

2020 ◽  
Vol 18 (21) ◽  
pp. 4024-4028
Author(s):  
David D. S. Thieltges ◽  
Kai D. Baumgarten ◽  
Carina S. Michaelis ◽  
Constantin Czekelius
Keyword(s):  
Kinetic Resolution ◽  
Convergent Sequence ◽  
Epicatechin Gallate

Electronically modified, fluorinated catechins and epicatechins are enantioselectively synthesized in a short, convergent sequence via kinetic resolution.

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