Functional Equations and Inequalities in Several Variables

Stefan Czerwik9789810248376, 981-02-4837-7

This book outlines the modern theory of functional equations and inequalities in several variables. It consists of three parts. The first is devoted to additive and convex functions defined on linear spaces with semilinear topologies. In the second part, the problems of stability of functional equations in the sense of Ulam-Hyers-Rassias and in some function spaces are considered. In the last part, the functional equations in set-valued functions are dealt with – for the first time in the mathematical literature. The book contains many fresh results concerning those problems.

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Table of contents :
Contents……Page 8
Preface……Page 6
PART I Functional Equations and Inequalities in Linear Spaces……Page 11
1 Linear spaces and semilinear topology……Page 13
2 Convex functions……Page 19
3 Lower hull of a convex function……Page 29
4 Theorems of Bernstein-Doetsch Piccard and Mehdi……Page 33
5 Some set classes of continuous and J-convex functions……Page 39
6 Cauchy’s exponential equation……Page 45
7 D’Alembert’s equation on abelian groups……Page 53
8 D’Alembert’s equation on topological groups……Page 59
9 Polynomial functions and their extensions……Page 75
10 Quadratic mappings……Page 99
11 Quadratic equation on an interval……Page 123
12 Functional equations for quadratic differences……Page 131
PART II Ulam-Hyers-Rassias Stability of Functional Equations……Page 137
13 Additive Cauchy equation……Page 139
14 Multiplicative Cauchy equation……Page 151
15 Jensen’s functional equation……Page 157
16 Pexider’s functional equation……Page 163
17 Gamma functional equation……Page 167
18 D’Alembert’s and Lobaczevski’s functional equations……Page 171
19 Stability of homogeneous mappings……Page 179
20 Quadratic functional equation……Page 195
21 Stability of functional equations in function spaces……Page 213
22 Cauchy difference operator in Lp spaces……Page 223
23 Pexider difference operator in LP spaces……Page 231
24 Cauchy and Pexider operators in X/ spaces……Page 239
25 Stability in the Lipschitz norms……Page 245
26 Round-off stability of iterations……Page 255
27 Quadratic difference operator in LP spaces……Page 261
PART III Functional Equations in Set-Valued Functions……Page 271
28 Cauchy’s set-valued functional equation……Page 273
29 Jensen’s functional equation……Page 289
30 Pexider’s functional equation……Page 297
31 Quadratic set-valued functions……Page 303
32 Subadditive set-valued functions……Page 311
33 Superadditive set-valued functions and generalization of Banach-Steinhaus theorem……Page 333
34 Hahn-Banach type theorem and applications……Page 341
35 Subquadratic set-valued functions……Page 355
36 K-convex and K-concave set-valued functions……Page 365
37 Iteration semigroups of set-valued functions……Page 389
References……Page 397
Index……Page 417