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Functional Equations, Inequalities and Applications provides an extensive study of several important equations and inequalities, useful in a number of problems in mathematical analysis. Subjects dealt with include the generalized Cauchy functional equation, the Ulam stability theory in the geometry of partial differential equations, stability of a quadratic functional equation in Banach modules, functional equations and mean value theorems, isometric mappings, functional inequalities of iterative type, related to a Cauchy functional equation, the median principle for inequalities and applications, Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions and approximate algebra homomorphisms. Also included are applications to some problems of pure and applied mathematics.

This book will be of particular interest to mathematicians and graduate students whose work involves functional equations, inequalities and applications.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Hyers-Ulam Stability of a Quadratic Functional Equation in Banach Modules

Abstract
We extend the Hyers-Ulam-Rassias stability of a quadratic functional equation f(x + y + z) + f(xy) + f(yz) + f(xz) = 3f(x) + 3f(z) + 3f(z) to Banach modules over a Banach algebra.
Jae-Hyeong Bae, Won-Gil Park

Chapter 2. Cauchy and Pexider Operators in Some Function Spaces

Abstract
In this paper we define Cauchy and Pexider operators by applying the well-known Cauchy and Pexider differences and then consider some properties of such operators. Other type of operators (∆-operators, quadratic operators, Jensen operators) that are also very important in the theory of functional equations are investigated. Moreover, some new functional equations closely related to this problem, are discussed as well.
Stefan Czerwik, Krzysztof Dlutek

Chapter 3. The Median Principle for Inequalities and Applications

Abstract
The “Median Principle” for different integral inequalities of Grüss and Ostrowski type is applied.
Sever S. Dragomir

Chapter 4. On the Hyers-Ulam-Rassias Stability of a Pexiderized Quadratic Equation II

Abstract
In this paper we prove a generalization of the stability of the Pexiderized quadratic equations f 1(x + y + z) + f 2(x) + f 3(y) + f 4(z) − f 5(x + y) − f 6(y + z) − f 7(x + z) = 0 and f 1(x + y + z) + f 2(x-y + z) + f 3(x + yz) + f 4(− x + y + z) − 4f 5(x) − 4f 6(y) − 4f 7(z) = 0 in the spirit of D.H. Hyers, S.M. Ulam, Th.M. Rassias and P. Gǎvruta.
2000 MSC: 39B72, 47H15(Primary) .
Kil-Woung Jun, Yang-Hi Lee

Chapter 5. On the Hyers-Ulam-Rassias Stability of a Functional Equation

Abstract
In this paper, we will introduce a new functional equation f (x 1, y 1) f (x 2, y 2) = f (x 1 x 2+ y 1 y 2, x 1 y 2y 1 x 2), which is strongly related to a well known elementary formula of number theory, and investigate the solutions of the equation. Moreover, we will also study the Hyers—Ulam—Rassias stability of that equation.
Soon-Mo Jung

Chapter 6. A Pair of Functional Inequalities of Iterative Type Related to a Cauchy Functional Equation

Abstract
It is shown that, under some general algebraic conditions on fixed real numbers a, b, α, β, every continuous at a point solution f of the system of functional inequalities f(x + a) ≤ f(x) + α, f(x + b) ≤ f(x) + β (x ∈ ℝ) must be a polynomial of order 1. Analogous results for three remaining counterparts of this simultaneous system are presented. An application to characterization of L p -norm is given.
Dorota Krassowska, Janusz Matkowski

Chapter 7. On Approximate Algebra Homomorphisms

Abstract
We are going to prove the generalized Hyers—Ulam—Rassias stability of modified Popoviciu functional equations in Banach modules over a unital C*-algebra. It is applied to show the stability of algebra homomorphisms between Banach algebras associated with modified Popoviciu functional equations in Banach al­gebras.
Chun-Gil Park

Chapter 8. Hadamard and Dragomir-Agarwal Inequalities, the Euler Formulae and Convex Functions

Abstract
The Euler formula is used with functions possessing various convexity and concavity properties to derive inequalities pertinent to numerical integration.
Josip Pečarić, Ana Vukelić

Chapter 9. On Ulam Stability in the Geometry of PDE’s

Abstract
The article is concerned with the problem of the unstability of flows corresponding to solutions of the Navier—Stokes equation in relation with the stability of a new functional equation (functional Navier—Stokes equation),that is stable as well as superstable in an extended Ulam sense. In such a framework a natural characterization of stable global laminar flows is given also.
Agostino Prástaro, Themistocles M. Rassias

Chapter 10. On Certain Functional Equations and Mean Value Theorems

Abstract
In this paper we prove certain new characterizations of mean values in the spirit of Gauss type functional equations.
Themistocles M. Rassias, Young-Ho Kim

Chapter 11. Some General Approximation Error and Convergence Rate Estimates in Statistical Learning Theory

Abstract
In statistical learning theory, reproducing kernel Hilbert spaces are used basically as the hypothese space in the approximation of the regression function. In this paper, in connection with a basic formula by S. Smale and D. X. Zhou which is fundamental in the approximation error estimates, we shall give a general formula based on the general theory of reproducing kernels combined with linear mappings in the framework of Hilbert spaces. We shall give a prototype example.
Saburou Saitoh

Chapter 12. Functional Equations on Hypergroups

Abstract
This paper presents some recent results concerning functional equations on hypergroups. The aim is to give some idea for the treatment of classical functional equation problems in the hypergroup setting. The general form of additive functions, exponentials and moment functions of second order on discrete polynomial hypergroups is given. In addition, stability problems for additive and exponential functions on hypergroups are considered.
László Székelyhidi

Chapter 13. The Generalized Cauchy Functional Equation

Abstract
The Cauchy functional equation and the Cauchy-Pexider functional equation are generalized, and their solutions are determined.
Abraham A. Ungar

Chapter 14. On the Aleksandrov-Rassias Problem for Isometric Mappings

Abstract
Let X and Y be normed real vector spaces. A mapping T: XY is called preserving the distance r if for all x,y of X with ║xy X = r then ║T(x) —T (y)║ Y = r. In this paper, we provide an overall account of the development of the Aleksandrov problem, especially the Aleksandrov—Rassias problem for mappings which preserve distances with a noninteger ratio in Hilbert spaces.
Shuhuang Xiang

Backmatter

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