Introduction to functional equations

Contents......Page 8
Preface......Page 14
1.1 Introduction......Page 20
1.2 Functional Equations......Page 21
1.3 Solution of Additive Cauchy Functional Equation......Page 22
1.4 Discontinuous Solution of Additive Cauchy Equation......Page 28
1.5 Other Criteria for Linearity......Page 33
1.6 Additive Functions on the Complex Plane......Page 35
1.7 Concluding Remarks......Page 38
1.8 Exercises......Page 40
2.2 Solution of the Exponential Cauchy Equation......Page 44
2.3 Solution of the Logarithmic Cauchy Equation......Page 47
2.4 Solution of the Multiplicative Cauchy Equation......Page 49
2.5 Concluding Remarks......Page 53
2.6 Exercises......Page 55
3.2 Additive Cauchy Equations in Several Variables......Page 58
3.3 Multiplicative Cauchy Equations in Several Variables......Page 62
3.4 Other Two Cauchy Equations in Several Variables......Page 63
3.5 Concluding Remarks......Page 64
3.6 Exercises......Page 65
4.2 Extension of Additive Functions......Page 68
4.3 Concluding Remarks......Page 74
4.4 Exercises......Page 77
5.1 Introduction......Page 80
5.2 Area of Rectangles......Page 81
5.3 Definition of Logarithm......Page 83
5.4 Simple and Compound Interest......Page 84
5.5 Radioactive Disintegration......Page 86
5.6 Characterization of Geometric Distribution......Page 87
5.7 Characterization of Discrete Normal Distribution......Page 90
5.8 Characterization of Normal Distribution......Page 93
5.9 Concluding Remarks......Page 95
6.2 Sum of Powers of Integers......Page 98
6.2.1 Sum of the first n natural numbers......Page 99
6.2.3 Sum of kth power of the first n natural numbers......Page 100
6.3 Sum of Powers of Numbers on Arithmetic Progression......Page 103
6.4 Number of Possible Pairs Among n Things......Page 105
6.6 Sum of Some Finite Series......Page 107
6.7 Concluding Remarks......Page 109
7.2 Convex Function......Page 112
7.3 The Jensen Functional Equation......Page 114
7.4 A Related Functional Equation......Page 118
7.5 Concluding Remarks......Page 120
7.6 Exercises......Page 122
8.2 Pexider's Equations......Page 126
8.3 Pexiderization of the Jensen Functional Equation......Page 130
8.4 A Related Equation......Page 131
8.5 Concluding Remarks......Page 134
8.6 Exercises......Page 136
9.2 Biadditive Functions......Page 138
9.3 Continuous Solution of Quadratic Functional Equation......Page 142
9.4 A Representation of Quadratic Functions......Page 145
9.5 Pexiderization of Quadratic Equation......Page 148
9.6 Concluding Remarks......Page 153
9.7 Exercises......Page 158
10.2 Continuous Solution of d'Alembert Equation......Page 162
10.3 General Solution of d'Alembert Equation......Page 168
10.4 A Charcterization of Cosine Functions......Page 176
10.5 Concluding Remarks......Page 178
10.6 Exercises......Page 182
11.1 Introduction......Page 184
11.2 Solution of a Cosine-Sine Functional Equation......Page 185
11.3 Solution of a Sine-Cosine Functional Equation......Page 189
11.4 Solution of a Sine Functional Equation......Page 192
11.5 Solution of a Sine Functional Inequality......Page 202
11.6 An Elementary Functional Equation......Page 203
11.7 Concluding Remarks......Page 206
11.8 Exercises......Page 213
12.2 Solution of the Pompeiu Functional Equation......Page 216
12.3 A Generalized Pompeiu Functional Equation......Page 218
12.4 Pexiderized Pompeiu Functional Equation......Page 221
12.5 Concluding Remarks......Page 227
12.6 Exercises......Page 228
13.2 Hosszú Functional Equation......Page 230
13.3 A Generalization of Hosszú Equation......Page 233
13.4 Concluding Remarks......Page 241
13.5 Exercises......Page 243
14.2 Continuous Solution of Davison Functional Equation......Page 246
14.3 General Solution of Davison Functional Equation......Page 249
14.4 Concluding Remarks......Page 250
14.5 Exercises......Page 251
15.1 Introduction......Page 254
15.2 General Solution of the Abel Functional Equation......Page 255
15.3 Concluding Remarks......Page 258
15.4 Exercises......Page 259
16.2 The Mean Value Theorem......Page 262
16.3 A Mean Value Type Functional Equation......Page 264
16.4 Generalizations of Mean Value Type Equation......Page 266
16.5 Concluding Remarks......Page 280
16.6 Exercises......Page 285
17.1 Introduction......Page 288
17.2 Solution of Two Functional Equations......Page 292
17.3 Some Auxiliary Results......Page 297
17.4 Solution of a Generalized Functional Equation......Page 305
17.5 Concluding Remarks......Page 306
17.6 Exercises......Page 309
18.1 Introduction......Page 312
18.2 Cauchy Sequence and Geometric Series......Page 313
18.3 Hyers' Theorem......Page 314
18.4 Generalizations of Hyers' Theorem......Page 319
18.5 Concluding Remarks......Page 324
18.6 Exercises......Page 328
19.2 Stability of Exponential Equation......Page 332
19.3 Ger Type Stability of Exponential Equation......Page 339
19.4 Concluding Remarks......Page 341
19.5 Exercises......Page 345
20.2 Stability of d'Alembert Equation......Page 348
20.3 Stability of Sine Equation......Page 355
20.4 Concluding Remarks......Page 359
20.5 Exercises......Page 364
21.2 Stability of the Quadratic Equation......Page 368
21.3 Stability of Generalized Quadratic Equation......Page 372
21.4 Stability of a Functional Equation of Drygas......Page 379
21.5 Concluding Remarks......Page 388
21.6 Exercises......Page 395
22.2 Stability of Davison Functional Equation......Page 400
22.3 Generalized Stability of Davison Equation......Page 403
22.4 Concluding Remarks......Page 406
22.5 Exercises......Page 409
23.1 Introduction......Page 410
23.2 Stability of Hosszú Functional Equation......Page 411
23.3 Stability of Pexiderized Hosszú Functional Equation......Page 413
23.4 Concluding Remarks......Page 420
23.5 Exercises......Page 422
24.2 Stability Theorem......Page 424
24.3 Concluding Remarks......Page 428
24.4 Exercises......Page 434
Bibliography......Page 436