Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis.

*(English)*Zbl 1221.39038
Springer Optimization and Its Applications 48. Berlin: Springer (ISBN 978-1-4419-9636-7/hbk; 978-1-4419-9637-4/ebook). xiii, 362 p. (2011).

One of the important questions in the theory of functional equations is “When is it true that a function satisfying a functional equation \({\mathcal E}(f)=0\) approximately must be close to an exact solution of \({\mathcal E}(f)=0\)?” If there exists an affirmative answer we say that the equation \({\mathcal E}(f)=0\) is stable. An equation \({\mathcal E}(f)=0\) is said to be superstable if any approximate solution, under certain assumptions, is a solution of the equation.

There are four methods in the study of stability of functional equations. The first method is the direct method in which one uses an iteration process producing the so-called Hyers type sequences [cf. D. H. Hyers, Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; Zbl 0061.26403)]. Another method is based on sandwich theorems which are generalizations of the Hahn-Banach separation theorems [cf. Z. Páles, Publ. Math. 58, No. 4, 651–666 (2001; Zbl 0980.39022)]. The foundation of the third method is a fixed point technique by L. Cădariu and V. Radu [J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 4, 7 p., electronic only (2003; Zbl 1043.39010)], and the fourth technique focuses on using invariant means [cf. L. Szekelyhidi, Can. Math. Bull. 25, 500–501 (1982; Zbl 0505.39002)].

This interesting book is devoted to an exposition of some new significant results of the Hyers-Ulam-Rassias stability of functional equations, difference equations and related topics in Functional Analysis. It complements the books by D. H. Hyers, G. Isac and Th. M. Rassias [Stability of functional equations in several variables. Progress in Nonlinear Differential Equations and their Applications. 34. Boston, MA: Birkhäuser (1998; Zbl 0907.39025)] and by S. Czerwik [Functional equations and inequalities in several variables. Singapore: World Scientific. (2002; Zbl 1011.39019)].

The general framework of each chapter is to introduce a functional equation and to investigate its behavior, its stability in the sense of Hyers-Ulam-Rassias, Ger or Borelli and its superstability as well as the stability of its Pexiderized version and the stability on some restricted domains by using the direct or fixed point method. Each chapter includes one of the functional equations named as Cauchy, Jensen, quadratic, isometric, trigonometric, logarithmic, multiplicative, exponential, homogeneous, gamma and Hosszú. This book is well written in a concise, clear and readable style. It is divided into 14 chapters and includes a preface, a bibliography consisting of 364 items and the subject index. The book is a good source for specialists and graduate students working in functional equations.

There are four methods in the study of stability of functional equations. The first method is the direct method in which one uses an iteration process producing the so-called Hyers type sequences [cf. D. H. Hyers, Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; Zbl 0061.26403)]. Another method is based on sandwich theorems which are generalizations of the Hahn-Banach separation theorems [cf. Z. Páles, Publ. Math. 58, No. 4, 651–666 (2001; Zbl 0980.39022)]. The foundation of the third method is a fixed point technique by L. Cădariu and V. Radu [J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 4, 7 p., electronic only (2003; Zbl 1043.39010)], and the fourth technique focuses on using invariant means [cf. L. Szekelyhidi, Can. Math. Bull. 25, 500–501 (1982; Zbl 0505.39002)].

This interesting book is devoted to an exposition of some new significant results of the Hyers-Ulam-Rassias stability of functional equations, difference equations and related topics in Functional Analysis. It complements the books by D. H. Hyers, G. Isac and Th. M. Rassias [Stability of functional equations in several variables. Progress in Nonlinear Differential Equations and their Applications. 34. Boston, MA: Birkhäuser (1998; Zbl 0907.39025)] and by S. Czerwik [Functional equations and inequalities in several variables. Singapore: World Scientific. (2002; Zbl 1011.39019)].

The general framework of each chapter is to introduce a functional equation and to investigate its behavior, its stability in the sense of Hyers-Ulam-Rassias, Ger or Borelli and its superstability as well as the stability of its Pexiderized version and the stability on some restricted domains by using the direct or fixed point method. Each chapter includes one of the functional equations named as Cauchy, Jensen, quadratic, isometric, trigonometric, logarithmic, multiplicative, exponential, homogeneous, gamma and Hosszú. This book is well written in a concise, clear and readable style. It is divided into 14 chapters and includes a preface, a bibliography consisting of 364 items and the subject index. The book is a good source for specialists and graduate students working in functional equations.

Reviewer: Mohammad Sal Moslehian (Mashhad)

##### MSC:

39B82 | Stability, separation, extension, and related topics for functional equations |

39B72 | Systems of functional equations and inequalities |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

39B52 | Functional equations for functions with more general domains and/or ranges |