scholarly journals Continuous Translation of Hölder and Lipschitz Functions

1960 ◽  
Vol 12 ◽  
pp. 674-685 ◽  
Author(s):  
H. Mirkil
Keyword(s):  
Banach Space ◽  
Real Variable ◽  
Continuous Functions ◽  
Compact Groups ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Parameter Group ◽  
Starting Point ◽  
Group Of Isometries

All functions will be complex, periodic, integrable (on [0, 2π]) functions of a real variable x. Moreover, we shall require that every function have mean zero on [0, 2π], so that in particular non-zero constants are excluded.1. Plessner's characterization of absolutely continuous functions. An old theorem of Plessner (4), generalized to arbitrary compact groups by Bochner (1), can be taken as our starting point. Consider the functions f of bounded variation on [0, 2π]. These f form a Banach space F when each f is normed by its total variation on [0, 2π]. And translations define a natural one-parameter group of isometries on F.

scholarly journals A qualitative study on generalized Caputo fractional integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed D. Kassim ◽  
Thabet Abdeljawad ◽  
Wasfi Shatanawi ◽  
Saeed M. Ali ◽  
Mohammed S. Abdo
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Continuous Functions ◽  
Fractional Operator ◽  
Stability Of Solutions ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Banach’S Fixed Point Theorem

AbstractThe aim of this article is to discuss the uniqueness and Ulam–Hyers stability of solutions for a nonlinear fractional integro-differential equation involving a generalized Caputo fractional operator. The used fractional operator is generated by iterating a local integral of the form $(I_{a}^{\rho }f)(t)=\int _{a}^{t}f(s)s^{\rho -1}\,ds$ ( I a ρ f ) ( t ) = ∫ a t f ( s ) s ρ − 1 d s . Our reported results are obtained in the Banach space of absolutely continuous functions that rely on Babenko’s technique and Banach’s fixed point theorem. Besides, our main findings are illustrated by some examples.

Uniformly bounded composition operators in the Banach space of absolutely continuous functions

2012 ◽  
Vol 75 (13) ◽  
pp. 4995-5001 ◽  
Author(s):  
D. Głazowska ◽  
J. Matkowski ◽  
N. Merentes ◽  
J.L. Sánchez Hernández
Keyword(s):  
Banach Space ◽  
Composition Operators ◽  
Continuous Functions ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Uniformly Bounded

scholarly journals On the nonlinear Hadamard-type integro-differential equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Chenkuan Li
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Continuous Functions ◽  
Contraction Principle ◽  
Uniqueness Of Solutions ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Integro Differential Equation ◽  
Banach’S Contraction Principle

AbstractThis paper studies uniqueness of solutions for a nonlinear Hadamard-type integro-differential equation in the Banach space of absolutely continuous functions based on Babenko’s approach and Banach’s contraction principle. We also include two illustrative examples to demonstrate the use of main theorems.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals The spectral theorem for well-bounded operators

1993 ◽  
Vol 54 (3) ◽  
pp. 334-351 ◽  
Author(s):  
Ian Doust ◽  
Qiu Bozhou
Keyword(s):  
Functional Calculus ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Compact Interval ◽  
Spectral Theorem ◽  
Type B ◽  
Bounded Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

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