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On a solution to a functional equation

Alexander E. Patkowski

Journal of Applied Analysis > 2022 > Vol. 28, nr 1 > 91--93

We offer a solution to a functional equation using properties of the Mellin transform. A new criteria for the Riemann Hypothesis is offered as an application of our main result, through a functional relationship with the Riemann xi function.

article

Stabilities and instabilities of an advanced quartic functional equation in various normed spaces

Beri V. Senthil Kumar, Hemen Dutta, Sriramulu Sabarinathan

Mathematical Methods in the Applied Sciences > 44 > 14 > 11469 - 11481

In this paper, a new and advanced quartic functional equation is introduced, and its classical stability results in various normed spaces are discussed. An appropriate illustration is also presented to invalidate the stability result for a very critical instance. At the end of this paper, we have concluded that the stability results are valid in various normed spaces.

article

A fixed point theorem for operators of Meir–Keeler type via the degree of nondensifiability and its application in dynamic programming

J. Caballero, J. Harjani, K. Sadarangani

Journal of Fixed Point Theory and Applications > 2020 > 22 > 1 > 1-14

In this paper, we prove a fixed point theorem for operators of Meir–Keeler type by using the concept of degree of nondensifiability. As an application of our result, we study the existence of solutions for a class of functional equations appearing in dynamic programming.

article

Consequences of functional equations for pairs of p-adic L-functions

Cédric Dion, Florian Sprung

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg > 2019 > 89 > 2 > 203-208

We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.

article

A note on the Levi-Civita functional equation

Keltouma Belfakih, Elhoucien Elqorachi

Aequationes mathematicae > 2019 > 93 > 6 > 1275-1291

In this paper we find the solutions of the functional equation $$\begin{aligned} f(xy)=g(x)h(y)+\sum _{j=1}^{n}g_j(x)h_j(y),\;x,y \in M, \end{aligned}$$ f ( x y ) = g ( x ) h ( y ) + ∑ j = 1 n g j ( x ) h j ( y ) , x , y ∈ M , where M is a monoid, $$n\ge 2$$ n ≥ 2 , and $$g_j$$ g j (for $$j=1,\ldots ,n$$ j = 1 , … , n ) are linear combinations...

article

Using Functional Equations to Calculate Feynman Integrals

O. V. Tarasov

Theoretical and Mathematical Physics > 2019 > 200 > 2 > 1205-1221

We propose a method for using functional equations to calculate Feynman integrals analytically. We describe the algorithm for solving the functional equations and show that a solution of a functional equation for the Feynman integral is a combination of several integrals with fewer kinematic variables. In some cases, using the functional equations, we can also reduce these integrals to integrals with...

article

Cauchy type functional equations related to some associative rational functions

Katarzyna Domańska

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica > 2019 > 18 > 145-156

L. Losonczi [4] determined local solutions of the generalized Cauchy equation f(F(x, y))= f(x) + f(y) on components of the denition of a given associative rational function F. The class of the associative rational function was described by A. Chéritat [1] and his work was followed by paper [3] of the author. The aim of the present paper is to describe local solutions of the equation considered for...

article

Riemann zeta fractional derivative—functional equation and link with primes

Emanuel Guariglia

Advances in Difference Equations > 2019 > 2019 > 1 > 1-15

This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grünwald–Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a quasisymmetric form of the aforementioned functional equation is derived (symmetric up to one complex multiplicative...

article

Identities, inequalities for Boole-type polynomials: approach to generating functions and infinite series

Yilmaz Simsek, Ji Suk So

Journal of Inequalities and Applications > 2019 > 2019 > 1 > 1-11

The main purpose and motivation of this work is to investigate and provide some new identities, inequalities and relations for combinatorial numbers and polynomials, and for Peters type polynomials with the help of their generating functions. The results of this paper involve some special numbers and polynomials such as Stirling numbers, the Apostol–Euler numbers and polynomials, Peters polynomials,...

article

Multiadditive functions satisfying certain functional equations

Masaaki Amou

Aequationes mathematicae > 2019 > 93 > 2 > 345-350

As a generalization of a result proved independently by Kurepa and Jurkat on additive functions on $${\mathbb R}$$ R satisfying certain functional equations, we determine multiadditive functions on $${\mathbb R}^n$$ R n with a positive integer n greater than 1 satisfying certain functional equations similar to those considered in the one variable case.

article

Local stationary heat fields in fibrous composites

Wojciech Baran, Krystian Kurnik, Wojciech Nawalaniec, Albashir Shareif

Silesian Journal of Pure and Applied Mathematics > 2019 > vol. 9, iss. 1 > 1--8

Consider a multiply connected domain D bounded by nonoverlapping circles. Introduce the complex potential u(z) = Re ϕ(z) in D where the function ϕ(z) is analytic in D except at infinity where ϕ(z) ∼ z. The function u(z) models the distribution of temperature in the domain D. The unknown function ϕ(z) is continuously differentiable in the closures of the considered domain. We solve approximately the...

article

Solution of a general pexiderized permanental functional equation

Vichian Laohakosol, Wuttichai Suriyacharoen

Proceedings - Mathematical Sciences > 2019 > 129 > 1 > 1-12

A general solution of the pexiderized functional equation $$\begin{aligned} f(ux+vy,uy-vx, zw)=g(x,y,z)\;h(u,v,w) \end{aligned}$$ f(ux+vy,uy-vx,zw)=g(x,y,z)h(u,v,w) is determined without any regularity assumptions. This equation arises from identities satisfied by the permanent of certain symmetric matrices. The solution so obtained are applied to deduce a number of existing related functional equations.

article

A generalized truncated logarithm

Marina Avitabile, Sandro Mattarei

Aequationes mathematicae > 2019 > 93 > 4 > 711-734

We introduce a generalization $$G^{(\alpha )}(X)$$ G ( α ) ( X ) of the truncated logarithm $$\pounds _1(X)=\sum _{k=1}^{p-1}X^k/k$$ £ 1 ( X ) = ∑ k = 1 p - 1 X k / k in prime characteristic p, which depends on a parameter $$\alpha $$ α . The main motivation of this study is $$G^{(\alpha )}(X)$$ G ( α ) ( X ) being an inverse, in an appropriate...

article

Hyperquasipolynomials for the Theta-Function

A. A. Illarionov, M. A. Romanov

Functional Analysis and Its Applications > 2018 > 52 > 3 > 228-231

Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ℂ → ℂ satisfying f(x+y)g(x−y) = α₁(x)β₁(y)+· · ·+α_r(x)β_r(y) for some r ∈ ℕ and α_j, β_j: ℂ → ℂ are described.

article

On an Orthogonality Equation in Normed Spaces

P. Wójcik

Functional Analysis and Its Applications > 2018 > 52 > 3 > 224-227

The aim of this paper is to give an answer to the question, posed by Alsina, Sikorska, and Tomás, as to whether each norm derivative preserving mapping is necessarily linear.

article

On a functional equation characterizing linear similarities

Paweł Wójcik

Aequationes mathematicae > 2019 > 93 > 3 > 557-561

The aim of this paper is to give an answer to a question posed by Alsina, Sikorska and Tomás. Namely, we show that, under suitable assumptions, a function $$f:X\!\rightarrow \!Y$$ f : X → Y from a normed space X into a normed space Y, satisfying the functional equation $$\begin{aligned} f\left( y-\frac{\rho '_+(x,y)}{\Vert x\Vert ^2}x\right) =f(y)-\frac{\rho '_+(f(x),f(y))}{\Vert f(x)\Vert...

article

A New Invariance Identity and Means

Jimmy Devillet, Janusz Matkowski

Results in Mathematics > 2018 > 73 > 4 > 1-13

The invariance identity involving three operations $$D_{f,g}:X\times X\rightarrow X$$ Df,g:X×X→X of the form $$\begin{aligned} D_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \oplus g\left( y\right) \right) , \end{aligned}$$ Df,gx,y=f∘g-1fx⊕gy, is proposed. The connections of these operations with means is investigated. The question when the invariance equality admits...

article

A Kannan-type fixed point theorem for multivalued mappings with application

Sokol Bush Kaliaj

The Journal of Analysis > 2019 > 27 > 3 > 837-849

In this paper, we present a Kannan-type fixed point theorem for multivalued mappings defined on a complete metric space in terms of a Suzuki-type contraction. As an application, we prove an existence and uniqueness theorem for a functional equation arising in dynamic programming of continuous multistage decision processes.

article

Extensions of the Sine Addition Formula on Monoids

Bruce Ebanks, Henrik Stetkær

Results in Mathematics > 2018 > 73 > 3 > 1-13

It is known that a pair (f, g) of functions with $$f\ne 0$$ f≠0 satisfies the sine addition formula $$f(xy)=f(x)g(y)+g(x)f(y)$$ f(xy)=f(x)g(y)+g(x)f(y) on a semigroup only if $$g = (\mu _1 + \mu _2)/2$$ g=(μ1+μ2)/2 where $$\mu _1$$ μ1 and $$\mu _2$$ μ2 are multiplicative functions. Here we solve the variant $$f(xy)=g_1(x)h_1(y)+g(x)h_2(y)$$ f(xy)=g1(x)h1(y)+g(x)h2(y) for four unknown functions...

article

Stability of a mixed additive and quadratic functional equation in quasi-Banach spaces

Nguyen Dung, Vo Thi Hang

Journal of Fixed Point Theory and Applications > 2018 > 20 > 3 > 1-11

We study the stability of functional equations in quasi-Banach spaces where the quasi-norm is not assumed to be a p-norm. To overcome the modulus of concavity greater than 1 and the discontinuity of quasi-norms we use the squeeze inequality presented in an explicit revision of Aoki–Rolewicz Theorem [13, Theorem 1]. As illustrations, we prove an extension of the stability of a mixed additive and quadratic...

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Search results

On a solution to a functional equation

Stabilities and instabilities of an advanced quartic functional equation in various normed spaces

A fixed point theorem for operators of Meir–Keeler type via the degree of nondensifiability and its application in dynamic programming

Consequences of functional equations for pairs of p-adic L-functions

A note on the Levi-Civita functional equation

Using Functional Equations to Calculate Feynman Integrals

Cauchy type functional equations related to some associative rational functions

Riemann zeta fractional derivative—functional equation and link with primes

Identities, inequalities for Boole-type polynomials: approach to generating functions and infinite series

Multiadditive functions satisfying certain functional equations

Local stationary heat fields in fibrous composites

Solution of a general pexiderized permanental functional equation

A generalized truncated logarithm

Hyperquasipolynomials for the Theta-Function

On an Orthogonality Equation in Normed Spaces

On a functional equation characterizing linear similarities

A New Invariance Identity and Means

A Kannan-type fixed point theorem for multivalued mappings with application

Extensions of the Sine Addition Formula on Monoids

Stability of a mixed additive and quadratic functional equation in quasi-Banach spaces

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