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## Über dieses Buch

No books dealing with a comprehensive illustration of the fast developing field of nonlinear analysis had been published for the mathematicians interested in this field for more than a half century until D. H. Hyers, G. Isac and Th. M. Rassias published their book, "Stability of Functional Equations in Several Variables".

This book will complement the books of Hyers, Isac and Rassias and of Czerwik (Functional Equations and Inequalities in Several Variables) by presenting mainly the results applying to the Hyers-Ulam-Rassias stability. Many mathematicians have extensively investigated the subjects on the Hyers-Ulam-Rassias stability. This book covers and offers almost all classical results on the Hyers-Ulam-Rassias stability in an integrated and self-contained fashion.

## Inhaltsverzeichnis

### Chapter 1. Introduction

In the fall of 1940, S. M. Ulam gave a wide-ranging talk before a Mathematical Colloquium at the University of Wisconsin in which he discussed a number of important unsolved problems.
Soon-Mo Jung

### Chapter 2. Additive Cauchy Equation

The functional equation $$f(x+y)=f(x)+f(y)$$ is the most famous among the functional equations. Already in 1821, A. L. Cauchy solved it in the class of continuous real-valued functions. It is often called the additive Cauchy functional equation in honor of A. L. Cauchy. The properties of this functional equation are frequently applied to the development of theories of other functional equations. Moreover, the properties of the additive Cauchy equation are powerful tools in almost every field of natural and social sciences. In Section 2.1, the behaviors of solutions of the additive functional equation are described. The Hyers–Ulam stability problem of this equation is discussed in Section 2.2, and theorems concerning the Hyers–Ulam–Rassias stability of the equation are proved in Section 2.3. The stability on a restricted domain and its applications are introduced in Section 2.4. The method of invariant means and the fixed point method will be explained briefly in Sections 2.5 and 2.6. In Section 2.7, the composite functional congruences will be surveyed. The stability results for the Pexider equation will be treated in the last section.
Soon-Mo Jung

### Chapter 3. Generalized Additive Cauchy Equations

It is very natural for one to try to transform the additive Cauchy equation into other forms. Some typically generalized additive Cauchy equations will be introduced. The functional equation $$f(ax+by)=af(x)+bf(y)$$ appears in Section 3.1. The Hyers–Ulam stability problem is discussed in connection with a question of Th. M. Rassias and J. Tabor. In Section 3.2, the functional equation (3.3) is introduced, and the Hyers–Ulam–Rassias stability for this equation is also studied. The stability result for this equation will be used to answer the question of Rassias and Tabor cited above. The last section deals with the functional equation $$f(x+y)^2=(f(x)+f(y))^2$$. The continuous solutions and the Hyers–Ulam stability for this functional equation will be investigated.
Soon-Mo Jung

### Chapter 4. Hosszú’s Functional Equation

In 1967, M. Hosszú introduced the functional equation $$f(x+y-xy)=f(x)+f(y)-f(xy)$$ in a presentation at a meeting on functional equations held in Zakopane, Poland. In honor of M. Hosszú, this equation is called Hosszú’s functional equation. As one can easily see, Hosszú’s functional equation is a kind of generalized form of the additive Cauchy functional equation. In Section 4.1, it will be proved that Hosszú’s equation is stable in the sense of C. Borelli. We discuss the Hyers–Ulam stability problem of Hosszú’s equation in Section 4.2. In Section 4.3, Hosszú’s functional equation will be generalized, and the stability (in the sense of Borelli) of the generalized equation will be proved. It is surprising that Hosszú’s functional equation is not stable on the unit interval. It will be discussed in Section 4.4. In the final section, we will survey the Hyers–Ulam stability of Hosszú’s functional equation of Pexider type.
Soon-Mo Jung

### Chapter 5. Homogeneous Functional Equation

The functional equation $$f(yx)=y^kf(x)$$ (where k is a fixed real constant) is called the homogeneous functional equation of degree k. In the case when k D 1 in the above equation, the equation is simply called the homogeneous functional equation. In Section 5.1, the Hyers–Ulam–Rassias stability of the homogeneous functional equation of degree k between real Banach algebras will be proved in the case when k is a positive integer. It will especially be proved that every “approximately” homogeneous function of degree k is a real homogeneous function of degree k. Section 5.2 deals with the superstability property of the homogeneous equation on a restricted domain and an asymptotic behavior of the homogeneous functions. The stability problem of the equation between vector spaces will be discussed in Section 5.3. In the last section, we will deal with the Hyers–Ulam–Rassias stability of the homogeneous functional equation of Pexider type.
Soon-Mo Jung

### Chapter 6. Linear Functional Equations

A function is called a linear function if it is homogeneous as well as additive. The homogeneity of a function, however, is a consequence of additivity if the function is assumed to be continuous. There are a number of (systems of) functional equations which include all the linear functions as their solutions. In this chapter, only a few (systems of) functional equations among them will be introduced. In Section 6.1, the superstability property of the “intuitive” system (6.1) of functional equations $$f(x+y)=f(x)+f(y)\ {\rm and}\ f(cx)=cf(x)$$ which stands for the linear functions is introduced. The stability problem for the functional equation $$f(x+cy)=f(x)+cf(y)$$ is proved in the second section and the result is applied to the proof of the Hyers–Ulam stability of the “intuitive” system (6.1). In the final section, stability problems of other systems, which describe linear functions, are discussed.
Soon-Mo Jung

### Chapter 7. Jensen’s Functional Equation

There are a number of variations of the additive Cauchy functional equation, for example, generalized additive Cauchy equations appearing in Chapter 3, Hosszú’s equation, homogeneous equation, linear functional equation, etc. However, Jensen’s functional equation is the simplest and the most important one among them. The Hyers–Ulam–Rassias stability problems of Jensen’s equation are proved in Section 7.1, and the Hyers–Ulam stability problems of that equation on restricted domains will be discussed in Section 7.2. Moreover, the stability result on a restricted domain will be applied to the study of an asymptotic property of additive functions. In Section 7.3, another approach to prove the stability will be introduced. This approach is called the fixed point method. The superstability and Ger type stability of the Loba?cevski?i functional equation will be surveyed in the last section.
Soon-Mo Jung

### Chapter 8. Quadratic Functional Equations

So far, we have discussed the stability problems of functional equations in connection with additive or linear functions. In this chapter, the Hyers–Ulam–Rassias stability of quadratic functional equations will be proved. Most mathematicians may be interested in the study of the quadratic functional equation since the quadratic functions are applied to almost every field of mathematics. In Section 8.1, the Hyers–Ulam–Rassias stability of the quadratic equation is surveyed. The stability problems for that equation on a restricted domain are discussed in Section 8.2, and the Hyers–Ulam–Rassias stability of the quadratic functional equation will be proved by using the fixed point method in Section 8.3. In Section 8.4, the Hyers–Ulam stability of an interesting quadratic functional equation different from the “original” quadratic functional equation is proved. Finally, the stability problem of the quadratic equation of Pexider type is discussed in Section 8.5.
Soon-Mo Jung

### Chapter 9. Exponential Functional Equations

The exponential function $$f(x)=e^x$$ is a powerful tool in each field of natural sciences and engineering since many natural phenomena well-known to us can be described best of all by means of it. The famous exponential functional equation $$f(x+y)=f(x)f(y)$$ simplifies the elegant property of the exponential function, for example, exCy D exey. In Section 9.1, the superstability of the exponential functional equation will be proved. Section 9.2 deals with the stability of the exponential equation in the sense of R. Ger. Stability problems of the exponential functional equation on a restricted domain and asymptotic behaviors of exponential functions are discussed in Section 9.3. Another exponential functional equation $$f(xy)=f(x)^y$$ will be introduced in Section 9.4.
Soon-Mo Jung

### Chapter 10. Multiplicative Functional Equations

The multiplicative functional equation $$f(xy)=f(x)f(y)$$ may be identified with the exponential functional equation if the domain of functions involved is a semigroup. However, if the domain space is a field or an algebra, then the former is obviously different from the latter. It is well-known that the general solution $$f\ :\ \mathbb{R}\rightarrow \mathbb{R}$$ of the multiplicative functional equation $$f(xy)=f(x)f(y)\ {\rm is}\ f(x)=0, f(x)=1, f(x)=e^{A({\rm ln}|x|)}|{\rm sign{(x)}}|,\ {\rm and} f(x)=e^{A({\rm ln}|x|)}{\rm sign(x)} \ {\rm {for\ all\ }}x \varepsilon \mathbb{R}, \ {\rm where}\ A\ :\ \mathbb{R}\rightarrow \mathbb{R}$$ is an additive function and sign $$\mathbb{R}\rightarrow \{-1, 0,1\}$$ is the sign function. If we impose the continuity on solution functions $$f:\mathbb{R}\rightarrow \mathbb{R}$$ of the multiplicative equation, then $$f(x)=0,f(x)=1,f(x)=|x|^\alpha,\ {\rm and}\ f(x)= |x|^\alpha \ {\rm sign}(x)\,{\rm {for\ all}}\ x\,\varepsilon \mathbb{R}$$, where α is a positive real constant. The first section deals with the superstability of the multiplicative Cauchy equation and a functional equation connected with the Reynolds operator. In Section 10.2, the results on i-multiplicative functionals on complex Banach algebras will be discussed in connection with the AMNM algebras which will be described in Section 10.3. Another multiplicative functional equation $$f(x^y)=f(x)^y$$ for real-valued functions defined on R will be discussed in Section 10.4. This functional equation is superstable in the sense of Ger. In the last section, we will prove that a new multiplicative functional equation $$f(x+y)=f(x)f(y)f(1/x+1/y)$$ is stable in the sense of Ger.
Soon-Mo Jung

### Chapter 11. Logarithmic Functional Equations

It is not difficult to demonstrate the Hyers–Ulam stability of the logarithmic functional equation $$f(xy)=f(x)+f(y)\ {\rm for\ functions}\ f:(0,\infty)\rightarrow E$$, where E is a Banach space. More precisely, if a function $$f:(0,\infty)\rightarrow E$$ satisfies the functional inequality $$\| f(xy)-f(x)-f(y)\| \leq \delta\ {\rm for\ some}\ \delta > 0\ {\rm and\ for\ all}\ x,y > 0$$, then there exists a unique logarithmic function $$L:(0,\infty)\rightarrow E$$ (this means that $$L(xy)=L(x)+L(y)\ {\rm for\ all}\ x,y >0$$) such that $$\| f(x)-L(x)\| \leq \delta\ {\rm for\ any}\ x >0$$. In this chapter, we will introduce a new functional equation $$f(x^y)=yf(x)$$ which has the logarithmic property in the sense that the logarithmic function $$f(x)={\rm ln}x (x > 0)$$ is a solution of the equation. Moreover, the functional equation of Heuvers $$f(x+y)=f(x)+f(y)+f(1/x+1/y)$$ will be discussed.
Soon-Mo Jung

### Chapter 12. Trigonometric Functional Equations

The famous addition and subtraction rules for trigonometric functions can be represented by using functional equations. Some of these equations will be introduced and the stability problems for them will be surveyed. Section 12.1 deals with the superstability phenomenon of the cosine functional equation (12.1) which stands for an addition theorem of cosine function. Similarly, the superstability of the sine functional equation (12.3) is proved in Section 12.2. In Section 12.3, trigonometric functional equations (12.8) and (12.9) with two unknown functions will be discussed. It is very interesting that these functional equations for complex-valued functions defined on an amenable group are not superstable, but they are stable in the sense of Hyers and Ulam, whereas the equations (12.1) and (12.3) are superstable. In Section 12.4, we will deal with the Hyers–Ulam stability of the Butler–Rassias functional equation.
Soon-Mo Jung

### Chapter 13. Isometric Functional Equation

An isometry is a distance-preserving map between metric spaces. For normed spaces E1 and E2, a function $$f:\ E_1 \rightarrow E_2$$ is called an isometry if f satisfies the isometric functional equation $$\| f(x)-f(y)\| = \|x-y\|\ {\rm for\ all}\ x,y \varepsilon E_1$$. The historical background for Hyers–Ulam stability of isometries will be introduced in Section 13.1. The Hyers–Ulam–Rassias stability of isometries on a restricted domain will be surveyed in Section 13.2. Section 13.3 will be devoted to the fixed point method for studying the stability problem of isometries. In the final section, the Hyers–Ulam–Rassias stability of Wigner equation $$|\langle f(x), f(y)\rangle|= |\langle x, y \rangle|$$ on a restricted domain will be discussed.
Soon-Mo Jung

### Chapter 14. Miscellaneous

One of the simplest functional equations is the associativity equation. This functional equation represents the famous associativity axiom $$x.(y.z)=(x.y).z$$. Section 14.1 deals with the superstability of the associativity equation. In Section 14.2, an important functional equation defining multiplicative derivations in algebras will be introduced, and the Hyers–Ulam stability of the equation for functions on (0, 1] will be proved. The gamma function Г is very useful to develop other functions which have physical applications. In Section 14.3, the Hyers–Ulam–Rassias stability of the gamma functional equation and a generalized beta functional equation will be proved. The Hyers–Ulam stability of the Fibonacci functional equation will be proved in the last section.
Soon-Mo Jung

### Backmatter

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