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2009 | Buch

Basic Equations: Cauchy and Pexider Equations Kapitel lesen Erstes Kapitel lesen

Functional Equations and Inequalities with Applications

verfasst von: Palaniappan Kannappan

Verlag: Springer US

Buchreihe : Springer Monographs in Mathematics

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

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Über dieses Buch

Functional Equations and Inequalities with Applications presents a comprehensive, nearly encyclopedic, study of the classical topic of functional equations. Nowadays, the field of functional equations is an ever-growing branch of mathematics with far-reaching applications; it is increasingly used to investigate problems in mathematical analysis, combinatorics, biology, information theory, statistics, physics, the behavioral sciences, and engineering.

This self-contained monograph explores all aspects of functional equations and their applications to related topics, such as differential equations, integral equations, the Laplace transformation, the calculus of finite differences, and many other basic tools in analysis. Each chapter examines a particular family of equations and gives an in-depth study of its applications as well as examples and exercises to support the material.

The book is intended as a reference tool for any student, professional (researcher), or mathematician studying in a field where functional equations can be applied. It can also be used as a primary text in a classroom setting or for self-study. Finally, it could be an inspiring entrée into an active area of mathematical exploration for engineers and other scientists who would benefit from this careful, rigorous exposition.

Inhaltsverzeichnis

Frontmatter
1. Basic Equations: Cauchy and Pexider Equations
Abstract
In this chapter, Cauchy’s fundamental equations are studied, some algebraic conditions and generalizations are considered, and alternate equations, conditions on restricted domains, Jensen’s equation, some special cases of Cauchy’s equations, extensions of the additive equations, Pexider equations and their extensions, some applications in economics, the allocation problem, and sums of powers of first n natural numbers are treated.
Palaniappan Kannappan
2. Matrix Equations
Abstract
This chapter is devoted to functional equations whose domain, domain and range, or range are matrices. We first consider the matrix version of the Cauchy equations (A), (E), (L), and (M). Then we treat the matrix version of cosine equation (C) and other generalizations.
Palaniappan Kannappan
3. Trigonometric Functional Equations
Abstract
In this chapter, trigonometric functional equations are studied on reals, groups, Hilbert space, etc., and cosine equations on reals, non-Abelian groups, and Hilbert space are studied. Sine equations, analytic solutions, addition and subtraction formulas, operator values, and some generalizations are studied also, and counterexamples are provided.
Palaniappan Kannappan
4. Quadratic Functional Equations
Abstract
Quadratic functional equations, bilinear forms equivalent to the quadratic equation, and some generalizations are treated in this chapter. Among the normed linear spaces (n.l.s.), inner product spaces (i.p.s.) play an important role. The interesting question when an n.l.s. is an i.p.s. led to several characterizations of i.p.s. starting with Fréchet [291], Jordan and von Neumann [398], etc. Functional equations are instrumental in many characterizations. One of the objectives of the next chapter is to bring out the involvement of functional equations in various characterizations of i.p.s.
Palaniappan Kannappan
5. Characterization of Inner Product Spaces
Abstract
The inner product space, characterization of i.p.s., parallelepiped law and generation, different characterization of i.p.s. and its generalizations, and different orthogonality are considered in this chapter.
Palaniappan Kannappan
6. Stability
Abstract
In this chapter, the stability problem is treated in general. In particular, stability of the multiplicative function, logarithmic function, trigonometric functions, vector-valued functions, alternative Cauchy equation, polynomial equation, and quadratic equation is treated. S.M. Ulam, in his famous lecture in 1940 to the Mathematics Club of the University of Wisconsin, presented a number of unsolved problems. This is the starting point of the theory of the stability of functional equations. One of the questions led to a new line of investigation, nowadays known as the stability problems.
Palaniappan Kannappan
7. Characterization of Polynomials
Abstract
In this chapter, polynomials are characterized, mostly of second and nth order. Divided difference, Rudin’s problem and generalization, Rudin’s problem on groups, Frechet’s result, and polynomials in several variables are treated.
Palaniappan Kannappan
8. Nondifferentiable Functions
Abstract
Nondifferential functions, Weierstrass functions, Vander Waerden type functions, and generalizations are considered in this chapter. In classical analysis, one of the problems that has fascinated mathematicians since the end of the nineteenth century is ‘Does there exist a continuous function that is not differentiable?’ It is an interesting question. Motivated by this exciting question, many well-known mathematicians, starting with Weierstrass (1872) [827], started to work in this area to produce such a function. It is well known that the answer is affirmative. This chapter is devoted to listing several continuous non- (nowhere) differentiable functions (c.n.d.f.s). What is of interest to us and is the primary motive of this chapter is to show that most of the well-known examples can be obtained as solutions of functional equations, highlighting the functional equation connection. Kairies’s [412] report is an excellent survey article and a main source for this chapter.
Palaniappan Kannappan
9. Characterization of Groups, Loops, and Closure Conditions
Abstract
Groupoids, quasigroups, loops, semigroups, and groups are considered in this chapter. Closure conditions, isotopy, and groups are characterized by various identities. Functional equations arising out of Bol, Moufang, and extra loops are considered. Mediality, the left inverse property, Steiner loops, and generalized bisymmetry are treated. This chapter is devoted to algebraic identities (and their generalizations leading to the involvement of functional equations) connected to groups and well-known special loops such as Bol, Moufang, left inverse property (l.i.p.), and weak inverse property (w.i.p.). An identity in a binary system (such as transitivity, bisymmetry, distributivity, Bol, Moufang, etc.) induces a generalized identity—a functional equation—in a class of quasigroups. We treat several of them in this chapter. First we give some notation and definitions.
Palaniappan Kannappan
10. Functional Equations from Information Theory
Abstract
In this chapter, information theory using desirable properties and representation, Shannon’s entropy, directed divergence, generalized directed divergence, entropy of order α and degree β, and weighted entropy are treated. The Fundamental equations of entropy and their generalizations and sum form equations and their generalizations are treated. Distance measures and inset measures are performed, and applications are treated.
Palaniappan Kannappan
11. Abel Equations and Generalizations
Abstract
In this chapter, Abel’s equation, exponential iteration, associative and commutative equations, trigonometric equations, and systems of equations are treated. Generalizations and connections to information measures are treated. Hilbert, in his famous address to the International Congress of Mathematicians held in Paris in 1900 [372], posed many unsolved problems. The second part of his fifth problem is devoted in general to functional equations and in particular to functional equations treated by Abel, who was the first to treat functional equations systematically. He presented a general method of solving equations of quite general forms. For the most part, he solved functional equations by reducing them to differential equations.
Palaniappan Kannappan
12. Regularity Conditions—Christensen Measurability
Abstract
Regularity conditions and Christensen measurability and its applications are treated in this chapter. Conditions such as boundedness, monotonicity, measurability, continuity at a point, continuity, the Baire property, integrability, differentiability, and analyticity, for example, are called regularity conditions. To solve functional equations, it was customary to assume a rich regularity property like differentiability and reduce a functional equation to a differential equation and solve it. The trend for quite some time has been to solve functional equations under weaker regularity conditions like integrability or measurability or no regularity condition at all (solve algebraically).
Palaniappan Kannappan
13. Difference Equations
Abstract
In this chapter, we deal with several difference equations arising from the wellknown Cauchy, Pexider, quadratic, cosine, and other functional equations. Cauchy difference, difference that depends on the product, the Pompeiu functional equation and its generalizations, quadratic difference, and Pexider difference are treated.
Palaniappan Kannappan
14. Characterization of Special Functions
Abstract
This chapter is devoted to functional equations connected to a few special functions such as the gamma function, beta function, Riemann’s zeta function, Riemann’s function, Cantor-Lebesgue singular functions,Minkowski’s function, and Dirichlet’s distribution.
Palaniappan Kannappan
15. Miscellaneous Equations
Abstract
This chapter is devoted to solutions by the methods of determinants and means and revisits logarithmic, trigonometric, and polynomial (mean value theorem) functions and inner product spaces. We start off with a method known as the method of determinants, due to E. Vincze, and solve many of the equations solved before.
Palaniappan Kannappan
16. General Inequalities
Abstract
This chapter is devoted to functional inequalities. Functional inequalities occur in several fields, such as information theory, inner product spaces, geometry, complex analysis, trigonometry, and Cauchy, gamma, and beta equations. Classical A.M. = G.M., logarithmic inequality, multiplicative inequality, trigonometric functional inequality, parallelogram identity, quadratic inequality, inequalities for the gamma and beta functions, Simpson’s inequality, inequalities from information theory and mixed theory, and reproducing scoring systems are treated.
Palaniappan Kannappan
17. Applications
Abstract
Applications in binomial expansion, scalar products, economics, the Cobb-Douglas production function and quasilinearity, interest formulas, physics, Gaussian functions, Chebyshev polynomials, determinants, geometry, statistics, information theory, and sums of powers are considered in this chapter. Functional equations is a growing and fascinating part of mathematics because of its intrinsic mathematical beauty as well as its applications. Applications of functional equations abound in mathematics and elsewhere in diverse fields. This chapter is devoted to applications of functional equations in various fields. Applications have contributed to the growth and development of functional equations.
Palaniappan Kannappan
Backmatter
Metadaten
Titel
Functional Equations and Inequalities with Applications
verfasst von
Palaniappan Kannappan
Copyright-Jahr
2009
Verlag
Springer US
Electronic ISBN
978-0-387-89492-8
Print ISBN
978-0-387-89491-1
DOI
https://doi.org/10.1007/978-0-387-89492-8

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