scholarly journals Stability of Pexider Equations on Semigroup with No Neutral Element

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jaeyoung Chung
Keyword(s):  
Banach Space ◽  
Closed Form ◽  
Direct Consequence ◽  
Neutral Element ◽  
Complex Numbers ◽  
Ulam Stability ◽  
Exponential Equation ◽  
Pexider Equation ◽  
Pexider Equations ◽  
Jung’S Theorem

LetSbe a commutative semigroup with no neutral element,Ya Banach space, andℂthe set of complex numbers. In this paper we prove the Hyers-Ulam stability for Pexider equationfx+y-gx-h(y)≤ϵfor allx,y∈S, wheref,g,h:S→Y. Using Jung’s theorem we obtain a better bound than that usually obtained. Also, generalizing the result of Baker (1980) we prove the superstability for Pexider-exponential equationft+s-gth(s)≤ϵfor allt,s∈S, wheref,g,h:S→ℂ. As a direct consequence of the result we also obtain the general solutions of the Pexider-exponential equationft+s=gth(s)for allt,s∈S, a closed form of which is not yet known.

scholarly journals Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1425
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Function ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Hyperbolic Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Arbitrary Complex ◽  
Famous Book

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

scholarly journals Note on a Stieltjes Transform in terms of the Lerch Function

2021 ◽  
Vol 14 (3) ◽  
pp. 723-736
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Special Functions ◽  
Complex Numbers ◽  
Definite Integral ◽  
Fundamental Constants ◽  
Logarithmic Function ◽  
Stieltjes Transform ◽  
Closed Form Solutions ◽  
General Complex

In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by\[\int_{0}^{\infty} \frac{(1-b x)^m \log ^k(c (1-b x))+(b x+1)^m \log ^k(c (b x+1))}{a+x^2}dx\]where $a,b,c,m$ and $k$ are general complex numbers subject to the restrictions given in connection with the formulas.

scholarly journals On some properties of determinants with complex elements and their practical application

2021 ◽  
pp. 67-77
Author(s):  
Mykhailo Fys ◽  
Roman Kvit ◽  
Tetyana Salo
Keyword(s):  
Closed Form ◽  
Imaginary Part ◽  
Complex Numbers ◽  
System Of Equations ◽  
Order Complex ◽  
Symmetric Polynomials ◽  
Practical Application ◽  
The Real ◽  
The Given ◽  
Selection Of

The formulas presented in this paper make it possible to select the real and imaginary part of the determinant value of the n -th order complex quantity, greatly simplifying the process of its deployment. Moreover, its module is given by the determinant of the 2n -th order, the elements of which are the real and imaginary parts of complex numbers. This makes it possible to analyze analytically the process described using determinants with complex numbers. The real and imaginary parts are also determined by the sum of determinants already with n rows and columns, the elements of which make up complex elements. The terms of this sum are solutions of a system of equations represented in closed form using symmetric polynomials, the arguments of which are its coefficients. Part of this combination is expressed by two determinants of the n -th order, the elements of which are the sum and difference of the real and imaginary parts of the elements. This significantly reduces the number of arithmetic operations during the deployment of a complex determinant and the selection of its real and imaginary parts. The given numerical example confirms the feasibility of this approach.

scholarly journals Szemerédi's theorem, frequent hypercyclicity and multiple recurrence

2012 ◽  
Vol 110 (2) ◽  
pp. 251 ◽  
Author(s):  
George Costakis ◽  
Ioannis Parissis
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Multiplication Operator ◽  
Complex Banach Space ◽  
Multiplication Operators ◽  
Complex Numbers ◽  
Multiple Recurrence ◽  
Weighted Shifts ◽  
Bounded Linear ◽  
Previous Assumption

Let $T$ be a bounded linear operator acting on a complex Banach space $X$ and $(\lambda_n)_{n\in\mathsf{N}}$ a sequence of complex numbers. Our main result is that if $|\lambda_n|/|\lambda_{n+1}|\to 1$ and the sequence $(\lambda_n T^n)_{n\in\mathsf{N}}$ is frequently universal then $T$ is topologically multiply recurrent. To achieve such a result one has to carefully apply Szemerédi's theorem in arithmetic progressions. We show that the previous assumption on the sequence $( \lambda_n)_{n\in\mathsf{N}}$ is optimal among sequences such that $|\lambda_{n}|/|\lambda_{n+1}|$ converges in $[0,\infty]$. In the case of bilateral weighted shifts and adjoints of multiplication operators we provide characterizations of topological multiple recurrence in terms of the weight sequence and the symbol of the multiplication operator respectively.

scholarly journals BIDUAL OCTAHEDRAL RENORMINGS AND STRONG REGULARITY IN BANACH SPACES

2019 ◽  
pp. 1-17 ◽  
Author(s):  
Johann Langemets ◽  
Ginés López-Pérez
Keyword(s):  
Banach Space ◽  
Studia Math ◽  
Direct Consequence ◽  
Baire Class ◽  
Dual Norm ◽  
Separable Predual ◽  
Separable Case ◽  
Strongly Regular ◽  
First Baire Class

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

Representations and Divisibility of Operator Polynomials

1978 ◽  
Vol 30 (5) ◽  
pp. 1045-1069 ◽  
Author(s):  
I. Gohberg ◽  
P. Lancaster ◽  
L. Rodman
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

Let be a complex Banach space and the algebra of bounded linear operators on . In this paper we study functions from the complex numbers to of the form

scholarly journals Ulam Stability of a Second Linear Differential Operator with Nonconstant Coefficients

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1451
Author(s):  
Liviu Cădariu ◽  
Dorian Popa ◽  
Ioan Raşa
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Operator ◽  
Global Solution ◽  
Linear Differential Operator ◽  
Second Order ◽  
Order Differential Operator ◽  
Ulam Stability ◽  
Riccati Differential Equation ◽  
Linear Differential

In this paper, we obtain a result on Ulam stability for a second order differential operator acting on a Banach space. The result is connected to the existence of a global solution for a Riccati differential equation and some appropriate conditions on the coefficients of the operator.

scholarly journals On Generalized Hyers-Ulam Stability of Admissible Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Banach Spaces ◽  
Unit Disk ◽  
Fractional Differential Equations ◽  
Complex Banach Space ◽  
Ulam Stability ◽  
Fractional Operators ◽  
Fractional Differential ◽  
Admissible Functions

We consider the Hyers-Ulam stability for the following fractional differential equations in sense of Srivastava-Owa fractional operators (derivative and integral) defined in the unit disk:Dzβf(z)=G(f(z),Dzαf(z),zf'(z);z),0<α<1<β≤2, in a complex Banach space. Furthermore, a generalization of the admissible functions in complex Banach spaces is imposed, and applications are illustrated.

Solution and stability of an n-dimensional functional equation

Analysis ◽  
2019 ◽  
Vol 39 (3) ◽  
pp. 107-115 ◽  
Author(s):  
Sandra Pinelas ◽  
V. Govindan ◽  
K. Tamilvanan
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Fixed Point ◽  
Positive Integer ◽  
General Solution ◽  
Functional Equations ◽  
Ulam Stability ◽  
Fixed Point Methods ◽  
Fixed Positive Integer

AbstractIn this paper, we prove the general solution and generalized Hyers–Ulam stability of n-dimensional functional equations of the form\sum_{\begin{subarray}{c}i=1\\ i\neq j\neq k\end{subarray}}^{n}f\biggl{(}-x_{i}-x_{j}-x_{k}+\sum_{% \begin{subarray}{c}l=1\\ l\neq i\neq j\neq k\end{subarray}}^{n}x_{l}\biggr{)}=\biggl{(}\frac{n^{3}-9n^{% 2}+20n-12}{6}\biggr{)}\sum_{i=1}^{n}f(x_{i}),where n is a fixed positive integer with \mathbb{N}-\{0,1,2,3,4\}, in a Banach space via direct and fixed point methods.

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