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2014 | Buch

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Handbook of Functional Equations

Stability Theory

herausgegeben von: Themistocles M. Rassias

Verlag: Springer New York

Buchreihe : Springer Optimization and Its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This handbook consists of seventeen chapters written by eminent scientists from the international mathematical community, who present important research works in the field of mathematical analysis and related subjects, particularly in the Ulam stability theory of functional equations. The book provides an insight into a large domain of research with emphasis to the discussion of several theories, methods and problems in approximation theory, analytic inequalities, functional analysis, computational algebra and applications.

The notion of stability of functional equations has its origins with S. M. Ulam, who posed the fundamental problem for approximate homomorphisms in 1940 and with D. H. Hyers, Th. M. Rassias, who provided the first significant solutions for additive and linear mappings in 1941 and 1978, respectively. During the last decade the notion of stability of functional equations has evolved into a very active domain of mathematical research with several applications of interdisciplinary nature.

The chapters of this handbook focus mainly on both old and recent developments on the equation of homomorphism for square symmetric groupoids, the linear and polynomial functional equations in a single variable, the Drygas functional equation on amenable semigroups, monomial functional equation, the Cauchy–Jensen type mappings, differential equations and differential operators, operational equations and inclusions, generalized module left higher derivations, selections of set-valued mappings, D’Alembert’s functional equation, characterizations of information measures, functional equations in restricted domains, as well as generalized functional stability and fixed point theory.

Inhaltsverzeichnis

Frontmatter

On Some Functional Equations

Abstract

This chapter consists of three parts. In the first part we consider so-called Adomian’s polynomials and present the proof of the convergence of the sequence of such polynomials to the solution of the equation. The second part is devoted to present several approximation methods for finding solutions of so-called Kordylewski–Kuczma functional equation. Finally, in the last one we present a stability result in the sense of Ulam–Hyers–Rassias for generalized quadratic functional equation on topological spaces.

Marcin Adam, Stefan Czerwik, Krzysztof Król

Remarks on Stability of the Equation of Homomorphism for Square Symmetric Groupoids

Abstract

Let $(G,\star)$ and $(H,\circ)$ be square symmetric groupoids and $S\subset G$ be nonempty. We present some remarks on stability of the following conditional equation of homomorphism

$$f(x\star y)=f(x)\circ f(y) \qquad x,y\in S, x\star y\in S\;,$$

in the class of functions mapping S into H. In particular, we consider the situation where $H=\mathbb{R}$ and

$$-\nu(x,y)\le h(x\star y)-h(x)\circ h(y) \le \mu(x,y) \qquad x,y\in S, x\star y\in S\;,$$

with some functions $\mu,\nu:S^2\to [0,\infty)$.

Anna Bahyrycz, Janusz Brzdȩk

On Stability of the Linear and Polynomial Functional Equations in Single Variable

Abstract

We present a survey of selected recent results of several authors concerning stability of the following polynomial functional equation (in single variable)

$$\varphi(x)=\sum_{i=1}^m a_i(x)\varphi(\xi_i(x))^{p(i)}+F(x),$$

in the class of functions ϕ mapping a nonempty set S into a Banach algebra X over a field $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, where m is a fixed positive integer, $p(i)\in \mathbb{N}$ for $i=1,\ldots,m$, and the functions $\xi_i:S\to S$, $F:S\to X$ and $a_i:S\to X$ for $i=1,\ldots,m$, are given. A particular case of the equation, with $p(i)=1$ for $i=1,\ldots,m$, is the very well-known linear equation

$$\varphi(x)=\sum_{i=1}^m a_i(x)\varphi(\xi_i(x))+F(x).$$

Janusz Brzdȩk, Magdalena Piszczek

Selections of Set-valued Maps Satisfying Some Inclusions and the Hyers–Ulam Stability

Abstract

We present a survey of several results on selections of some set-valued functions satisfying some inclusions and also on stability of those inclusions. Moreover, we show their consequences concerning stability of the corresponding functional equations.

Janusz Brzdęk, Magdalena Piszczek

Generalized Ulam–Hyers Stability Results: A Fixed Point Approach

Abstract

We show that a recent fixed point result in (Cădariu et al., Abstr. Appl. Anal., 2012) can be used to prove some generalized Ulam–Hyers stability theorems for additive Cauchy functional equation as well as for the monomial functional equation in β-normed spaces.

Liviu Cădariu

On a Weak Version of Hyers–Ulam Stability Theorem in Restricted Domains

Abstract

In this chapter we consider a weak version of the Hyers–Ulam stability problem for the Pexider equation, Cauchy equation satisfied in restricted domains in a group when the target space of the functions is a 2-divisible commutative group. As the main result we find an approximate sequence for the unknown function satisfying the Pexider functional inequality, the limit of which is the approximate function in the Hyers–Ulam stability theorem.

Jaeyoung Chung, Jeongwook Chang

On the Stability of Drygas Functional Equation on Amenable Semigroups

Abstract

In this chapter, we will prove the Hyers–Ulam stability of Drygas functional equation

$$f(xy)+f(x\sigma(y))=2~f(x)+f(y)+f(\sigma(y)),\;x,y\in{G},$$

where G is an amenable semigroup, σ is an involution of G and $f:G\rightarrow E$ is approximatively central (i.e., $|f(xy)-f(yx)|\leq\delta$).

Elhoucien Elqorachi, Youssef Manar, Themistocles M. Rassias

Stability of Quadratic and Drygas Functional Equations, with an Application for Solving an Alternative Quadratic Equation

Abstract

The aim of this survey is to present stability results obtained in the last years (roughly after 1995) for the quadratic equation and its various generalizations, and the Drygas equation. The number of papers on this subject is very high, hence, the author of the present chapter made a (quite arbitrary) choice of some of them to be shown in detail. The last section is devoted to an application of stability for solving an alternative form of the quadratic equation.

Gian Luigi Forti

A Functional Equation Having Monomials and Its Stability

Abstract

We use some results about the Fréchet functional equation to consider the following functional equation:

$$\begin{aligned} f\left(\left(\sum_{i=1}^{m}a_ix_i^p\right)^\frac{1}{p}\right)=\sum_{i=1}^{m}a_if(x_i).\end{aligned}$$

We also apply a fixed point method and homogeneous functions of degree α to investigate some stability results for this functional equation in β-Banach spaces.

M. E. Gordji, H. Khodaei, Themistocles M. Rassias

Some Functional Equations Related to the Characterizations of Information Measures and Their Stability

Abstract

The main purpose of this chapter is to investigate the stability problem of some functional equations that appear in the characterization problem of information measures.

Eszter Gselmann, Gyula Maksa

Approximate Cauchy–Jensen Type Mappings in Quasi-β-Normed Spaces

Abstract

In this chapter, we find the general solution of the following Cauchy–Jensen type functional equation

$$f(\frac{x+y}{n}+z)+f(\frac{y+z}{n}+x)+f(\frac{z+x}{n}+y)=\frac{n+2}{n}[f(x)+f(y)+f(z)],$$

and then investigate the generalized Hyers–Ulam stability of the equation in quasi-β-normed spaces for any fixed nonzero integer n.

Hark-Mahn Kim, Kil-Woung Jun, Eunyoung Son

An AQCQ-Functional Equation in Matrix Paranormed Spaces

Abstract

In this chapter, we prove the Hyers–Ulam stability of an additive-quadratic-cubic-quartic functional equation in matrix paranormed spaces. Moreover, we prove the Hyers–Ulam stability of an additive-quadratic-cubic-quartic functional equation in matrix β-homogeneous F-spaces.

Jung Rye Lee, Choonkil Park, Themistocles M. Rassias, Dong Yun Shin

On the Generalized Hyers–Ulam Stability of the Pexider Equation on Restricted Domains

Abstract

Let σ: $E\longrightarrow E$ be an involution of the normed space E and let p, M, d be nonnegative real numbers, such that $0<p<1$. In this chapter, we investigate the Hyers–Ulam–Rassias stability of the Pexider functional equations

$$\begin{aligned} f(x+y) = &\, g(x)+h(y),\ f(x+y)+g(x-y)=h(x)+k(y),\\ & f(x+y)+g(x+\sigma(y))=h(x)+k(y), x,y\in{E}\end{aligned}$$

on restricted domains $\mathcal{B}=\{(x,y)\in{E^{2}}: \|x\|^{p}+\|y\|^{p}\geq M^{p}\}$ and $\mathcal{C}=\{(x,y)\in{E}^{2}:\|x\|\geq d\; or\; \|y\|\geq d\}.$

Youssef Manar, Elhoucien Elqorachi, Themistocles M. Rassias

Hyers-Ulam Stability of Some Differential Equations and Differential Operators

Abstract

This chapter contains results on generalized Hyers–Ulam stability, obtained by the authors, for linear differential equations, linear differential operators and partial differential equations in Banach spaces. As a consequence we improve some known estimates of the difference between the perturbed and the exact solution.

Dorian Popa, Ioan Raşa

Results and Problems in Ulam Stability of Operatorial Equations and Inclusions

Abstract

In this chapter we survey some results and problems in Ulam stability of fixed point equations, coincidence point equations, operatorial inclusions, integral equations, ordinary differential equations, partial differential equations and functional inclusions. Some new results and problems are also presented.

Ioan A. Rus

Superstability of Generalized Module Left Higher Derivations on a Multi-Banach Module

Abstract

The problem of stability of functional equations was originally raised by Ulam in 1940. During the last decades, several stability problems for various functional equations have been investigated by several authors. In this chapter, by defining a multi-Banach space, we introduce a multi-Banach module. Also, we define the notion of generalized module left higher derivations and approximate generalized module left higher derivations. Then, we discuss the superstability of an approximate generalized module left higher derivation on a multi-Banach module. In fact, we show that an approximate generalized module left higher derivation on a multi-Banach module is a generalized module left higher derivation. Finally, we get the similar result for a linear generalized module left higher derivation.

T. L. Shateri, Z. Afshari

D’Alembert’s Functional Equation and Superstability Problem in Hypergroups

Abstract

Our main goal is to determine the continuous and bounded complex valued solutions of the functional equation

$$ \langle \delta_{x}\ast \delta_{y},g\rangle +\langle \delta_{x}\ast \delta_{\check{y}},g\rangle =2~g(x)g(y),\;x,y\in X,$$

where X is a hypergroup. The solutions are expressed in terms of 2 -dimensional representations of X. The papers of Davison [10] and Stetkaer [25, 26] are the essential motivation for this first part of the present work and the methods used here are closely related to and inspired by those in [10, 25, 26]. In addition, superstability problem for this functional equation on any hypergroup and without any condition on f is considered.

D. Zeglami, A. Roukbi, Themistocles M. Rassias

Titel: Handbook of Functional Equations
herausgegeben von: Themistocles M. Rassias
Verlag: Springer New York
Electronic ISBN: 978-1-4939-1286-5
Print ISBN: 978-1-4939-1285-8
DOI: https://doi.org/10.1007/978-1-4939-1286-5