Introduction to functional equations theory and problem-solving strategies for mathematical competitions and beyond
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Berkeley, Calif. [u.a.]
MSRI Mathematical Sciences Research Inst. [u.a.]
2011
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| 245 | 1 | 0 | |a Introduction to functional equations |b theory and problem-solving strategies for mathematical competitions and beyond |c Costas Efthimiou |
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Datensatz im Suchindex
| _version_ | 1804147954020253696 |
|---|---|
| adam_text | Titel: Introduction to functional equations
Autor: Sahoo, Prasanna
Jahr: 2011
Contents
Preface xiii
1 Additive Cauchy Functional Equation 1
1.1 Introduction ........................ 1
1.2 Functional Equations ................... 2
1.3 Solution of Additive Cauchy Functional Equation ... 3
1.4 Discontinuous Solution of Additive Cauchy Equation . 9
1.5 Other Criteria for Linearity................ 14
1.6 Additive Functions on the Complex Plane ....... 16
1.7 Concluding Remarks ................... 19
1.8 Exercises .......................... 21
2 Remaining Cauchy Functional Equations 25
2.1 Introduction ........................ 25
2.2 Solution of the Exponential Cauchy Equation...... 25
2.3 Solution of the Logarithmic Cauchy Equation ..... 28
2.4 Solution of the Multiplicative Cauchy Equation .... 30
2.5 Concluding Remarks ................... 34
2.6 Exercises .......................... 36
3 Cauchy Equations in Several Variables 39
3.1 Introduction ........................ 39
3.2 Additive Cauchy Equations in Several Variables .... 39
3.3 Multiplicative Cauchy Equations in Several Variables . 43
3.4 Other Two Cauchy Equations in Several Variables ... 44
3.5 Concluding Remarks ................... 45
3.6 Exercises .......................... 46
4 Extension of the Cauchy Functional Equations 49
4.1 Introduction ........................ 49
4.2 Extension of Additive Functions ............. 49
4.3 Concluding Remarks ................... 55
4.4 Exercises .......................... 58
vn
viii Contents
5 Applications of Cauchy Functional Equations 61
5.1 Introduction ........................ 61
5.2 Area of Rectangles..................... 62
5.3 Definition of Logarithm.................. 64
5.4 Simple and Compound Interest.............. 65
5.5 Radioactive Disintegration ................ 67
5.6 Characterization of Geometrie Distribution....... 68
5.7 Characterization of Discrete Normal Distribution ... 71
5.8 Characterization of Normal Distribution ........ 74
5.9 Concluding Remarks ................... 76
6 More Applications of Functional Equations 79
6.1 Introduction ........................ 79
6.2 Sum of Powers of Integers................. 79
6.2.1 Sum of the first n natural numbers........ 80
6.2.2 Sum of Square of the first n natural numbers . . 81
6.2.3 Sum of kth power of the first n natural numbers 81
6.3 Sum of Powers of Numbers on Arithmetic Progression . 84
6.4 Number of Possible Pairs Among n Things ....... 86
6.5 Cardinality of a Power Set ................ 88
6.6 Sum of Some Finite Series ................ 88
6.7 Concluding Remarks ................... 90
7 The Jensen Functional Equation 93
7.1 Introduction ........................ 93
7.2 Convex Function...................... 93
7.3 The Jensen Functional Equation............. 95
7.4 A Related Functional Equation.............. 99
7.5 Concluding Remarks ................... 101
7.6 Exercises .......................... 103
8 Pexider s Functional Equations 107
8.1 Introduction ........................ 107
8.2 Pexider s Equations .................... 107
8.3 Pexiderization of the Jensen Functional Equation ... 111
8.4 A Related Equation.................... 112
8.5 Concluding Remarks ................... 115
8.6 Exercises .......................... 117
9 Quadratic Functional Equation 119
9.1 Introduction ........................ 119
9.2 Biadditive Functions.................... 119
9.3 Continuous Solution of Quadratic Functional Equation 123
9.4 A Representation of Quadratic Functions ........ 126
Contents ix
9.5 Pexiderization of Quadratic Equation .......... 129
9.6 Concluding Remarks ................... 134
9.7 Exercises .......................... 139
10 d Alembert Functional Equation 143
10.1 Introduction ........................ 143
10.2 Continuous Solution of d Alembert Equation ...... 143
10.3 General Solution of d Alembert Equation ........ 149
10.4 A Charcterization of Cosine Functions.......... 157
10.5 Concluding Remarks ................... 159
10.6 Exercises .......................... 163
11 Trigonometrie Functional Equations 165
11.1 Introduction ........................ 165
11.2 Solution of a Cosine-Sine Functional Equation ..... 166
11.3 Solution of a Sine-Cosine Functional Equation ..... 170
11.4 Solution of a Sine Functional Equation ......... 173
11.5 Solution of a Sine Functional Inequality......... 183
11.6 An Elementary Functional Equation........... 184
11.7 Concluding Remarks ................... 187
11.8 Exercises .......................... 194
12 Pompeiu Functional Equation 197
12.1 Introduction ........................ 197
12.2 Solution of the Pompeiu Functional Equation...... 197
12.3 A Generalized Pompeiu Functional Equation...... 199
12.4 Pexiderized Pompeiu Functional Equation ....... 202
12.5 Concluding Remarks ................... 208
12.6 Exercises .......................... 209
13 Hosszü Functional Equation 211
13.1 Introduction ........................ 211
13.2 Hosszü Functional Equation ............... 211
13.3 A Generalization of Hosszü Equation .......... 214
13.4 Concluding Remarks ................... 222
13.5 Exercises .......................... 224
14 Davison Functional Equation 227
14.1 Introduction ........................ 227
14.2 Continuous Solution of Davison Functional Equation . 227
14.3 General Solution of Davison Functional Equation . . . 230
14.4 Concluding Remarks ................... 231
14.5 Exercises .......................... 232
x Contents
15 Abel Functional Equation 235
15.1 Introduction ........................ 235
15.2 General Solution of the Abel Functional Equation . . . 236
15.3 Concluding Remarks ................... 239
15.4 Exercises .......................... 240
16 Mean Value Type Functional Equations 243
16.1 Introduction ........................ 243
16.2 The Mean Value Theorem ................ 243
16.3 A Mean Value Type Functional Equation........ 245
16.4 Generalizations of Mean Value Type Equation ..... 247
16.5 Concluding Remarks ................... 261
16.6 Exercises .......................... 266
17 Functional Equations for Distance Measures 269
17.1 Introduction ........................ 269
17.2 Solution of Two Functional Equations.......... 273
17.3 Some Auxiliary Results .................. 278
17.4 Solution of a Generalized Functional Equation ..... 286
17.5 Concluding Remarks ................... 287
17.6 Exercises .......................... 290
18 Stability of Additive Cauchy Equation 293
18.1 Introduction ........................ 293
18.2 Cauchy Sequence and Geometrie Series ......... 294
18.3 Hyers Theorem ...................... 295
18.4 Generalizations of Hyers Theorem............ 300
18.5 Concluding Remarks ................... 305
18.6 Exercises .......................... 309
19 Stability of Exponential Cauchy Equations 313
19.1 Introduction ........................ 313
19.2 Stability of Exponential Equation ............ 313
19.3 Ger Type Stability of Exponential Equation ...... 320
19.4 Concluding Remarks ................... 322
19.5 Exercises .......................... 326
20 Stability of d Alembert and Sine Equations 329
20.1 Introduction ........................ 329
20.2 Stability of d Alembert Equation............. 329
20.3 Stability of Sine Equation................. 336
20.4 Concluding Remarks ................... 340
20.5 Exercises .......................... 345
Contents xi
21 Stability of Quadratic Functional Equations 349
21.1 Introduction ........................ 349
21.2 Stability of the Quadratic Equation ........... 349
21.3 Stability of Generalized Quadratic Equation ...... 353
21.4 Stability of a Functional Equation of Drygas ...... 360
21.5 Concluding Remarks ................... 369
21.6 Exercises .......................... 376
22 Stability of Davison Functional Equation 381
22.1 Introduction ........................ 381
22.2 Stability of Davison Functional Equation ........ 381
22.3 Generalized Stability of Davison Equation ....... 384
22.4 Concluding Remarks ................... 387
22.5 Exercises .......................... 390
23 Stability of Hosszü Functional Equation 391
23.1 Introduction ........................ 391
23.2 Stability of Hosszü Functional Equation......... 392
23.3 Stability of Pexiderized Hosszü Functional Equation . . 394
23.4 Concluding Remarks ................... 401
23.5 Exercises .......................... 403
24 Stability of Abel Functional Equation 405
24.1 Introduction ........................ 405
24.2 Stability Theorem ..................... 405
24.3 Concluding Remarks ................... 409
24.4 Exercises .......................... 415
Bibliography 417
Index 441
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| spelling | Efthimiou, Costas Verfasser aut Introduction to functional equations theory and problem-solving strategies for mathematical competitions and beyond Costas Efthimiou Berkeley, Calif. [u.a.] MSRI Mathematical Sciences Research Inst. [u.a.] 2011 XIV, 363 S. txt rdacontent n rdamedia nc rdacarrier MSRI mathematical circles library 6 Functional equations / Problems, exercises, etc General / Instructional exposition (textbooks, tutorial papers, etc.) msc General / General and miscellaneous specific topics / Problem books msc Real functions / Instructional exposition (textbooks, tutorial papers, etc.) msc Real functions / Functions of one variable / One-variable calculus msc Real functions / Functions of one variable / Iteration msc Real functions / Functions of several variables / Continuity and differentiation questions msc Funktionalgleichung (DE-588)4018923-5 gnd rswk-swf Funktionalgleichung (DE-588)4018923-5 s DE-604 MSRI mathematical circles library 6 (DE-604)BV024626790 6 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024141056&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Efthimiou, Costas Introduction to functional equations theory and problem-solving strategies for mathematical competitions and beyond MSRI mathematical circles library Functional equations / Problems, exercises, etc General / Instructional exposition (textbooks, tutorial papers, etc.) msc General / General and miscellaneous specific topics / Problem books msc Real functions / Instructional exposition (textbooks, tutorial papers, etc.) msc Real functions / Functions of one variable / One-variable calculus msc Real functions / Functions of one variable / Iteration msc Real functions / Functions of several variables / Continuity and differentiation questions msc Funktionalgleichung (DE-588)4018923-5 gnd |
| subject_GND | (DE-588)4018923-5 |
| title | Introduction to functional equations theory and problem-solving strategies for mathematical competitions and beyond |
| title_auth | Introduction to functional equations theory and problem-solving strategies for mathematical competitions and beyond |
| title_exact_search | Introduction to functional equations theory and problem-solving strategies for mathematical competitions and beyond |
| title_full | Introduction to functional equations theory and problem-solving strategies for mathematical competitions and beyond Costas Efthimiou |
| title_fullStr | Introduction to functional equations theory and problem-solving strategies for mathematical competitions and beyond Costas Efthimiou |
| title_full_unstemmed | Introduction to functional equations theory and problem-solving strategies for mathematical competitions and beyond Costas Efthimiou |
| title_short | Introduction to functional equations |
| title_sort | introduction to functional equations theory and problem solving strategies for mathematical competitions and beyond |
| title_sub | theory and problem-solving strategies for mathematical competitions and beyond |
| topic | Functional equations / Problems, exercises, etc General / Instructional exposition (textbooks, tutorial papers, etc.) msc General / General and miscellaneous specific topics / Problem books msc Real functions / Instructional exposition (textbooks, tutorial papers, etc.) msc Real functions / Functions of one variable / One-variable calculus msc Real functions / Functions of one variable / Iteration msc Real functions / Functions of several variables / Continuity and differentiation questions msc Funktionalgleichung (DE-588)4018923-5 gnd |
| topic_facet | Functional equations / Problems, exercises, etc General / Instructional exposition (textbooks, tutorial papers, etc.) General / General and miscellaneous specific topics / Problem books Real functions / Instructional exposition (textbooks, tutorial papers, etc.) Real functions / Functions of one variable / One-variable calculus Real functions / Functions of one variable / Iteration Real functions / Functions of several variables / Continuity and differentiation questions Funktionalgleichung |
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