This  handbook Consists Of Seventeen Chapters Written By Eminent Scientists From The International Mathematical Community, Who Present Important Research Works In The Field Of Mathematical Analysis And Related Subjects, Particularly In The Ulam Stability Theory Of Functional Equations. The Book Provides An Insight Into A Large Domain Of Research With Emphasis To The Discussion Of Several Theories, Methods And Problems In Approximation Theory, Analytic Inequalities, Functional Analysis, Computational Algebra And Applications.                           The Notion Of Stability Of Functional Equations Has Its Origins With S. M. Ulam, Who Posed The Fundamental Problem For Approximate Homomorphisms In 1940 and With D. H. Hyers, Th. M. Rassias, Who Provided The First Significant Solutions For Additive And Linear Mappings In 1941 And 1978, Respectively. During The Last Decade The Notion Of Stability Of Functional Equations Has Evolved Into A Very Active Domain Of Mathematical Research With Several Applications Of Interdisciplinary Nature.                                                                                         The Chapters Of This Handbook Focus Mainly On Both Old And Recent Developments On The Equation Of Homomorphism For Square Symmetric Groupoids, The Linear And Polynomial Functional Equations In A Single Variable, The Drygas Functional Equation On Amenable Semigroups, Monomial Functional Equation,  The Cauchy–jensen Type Mappings, Differential Equations And Differential Operators, Operational Equations And Inclusions, Generalized Module Left Higher Derivations, Selections Of Set-valued Mappings, D’alembert’s Functional Equation, Characterizations Of Information Measures,  functional Equations In Restricted Domains, As Well As Generalized Functional Stability And Fixed Point Theory. On Some Functional Equations (m. Adam, S. Czerwik, K. Krol) -- Remarks On Stability Of The Equation Of Homomorphism For Square Symmetric Groupoids (a. Bahyrycz, J. Brzdek) -- On Stability Of The Linear And Polynomial Functional Equations In Single Variable (j. Brzdek, M. Piszczek) -- Selections Of Set-valued Maps Satisfying Some Inclusions And The Hyers-ulam Stability (j. Brzdek, M. Piszczek) -- Generalized Ulam-hyers Stability Results: A Fixed Point Approach (l. Caradiu) -- On A Wake Version Of Hyers-ulam Stability Theorem In Restricted Domain (j. Chung, J. Chang) -- On The Stability Of Drygas Functional Equation On Amenable Semigroups (e. Elqorachi, Y. Manar, Th.m. Rassias) -- Stability Of Quadratic And Drygas Functional Equations, With An Application For Solving An Alternative Quadratic Equation (g.l. Forti) -- A Functional Equation Having Monomials And Its Stability (m.e. Gorgji, H. Khodaei, Th.m. Rassias) -- Some Functional Equations Related To The Characterizations Of Information Measures And Their Stability (e. Gselmann, G. Maksa) -- Approximate Cauchy-jensen Type Mappings In Quasi-β Normed Spaces (h.-m. Kim, K.-w. Jun, E. Son) -- An Aqcq-functional Equation In Matrix Paranormed Spaces (j.r. Lee, C. Park, Th.m. Rassias, D.y. Shin) -- On The Generalized Hyers-ulam Stability Of The Pexider Equation On Restricted Domains (y. Manar, E. Elqorachi, Th.m. Rassias) -- Hyers-ulam Stability Of Some Differential Equations And Differential Operators (d. Popa, I. Rasa) -- Results And Problems In Ulam Stability Of Operational Equations And Inclusions (i.a. Rus) -- Superstability Of Generalized Module Left Higher Derivations On A Multi-banach Module (t.l. Shateri, Z. Afshari) -- D'alembert's Functional Equation And Superstability Problem In Hypergroups (d. Zeglami, A. Roukbi, Th.m. Rassias). Edited By Themistocles M. Rassias.

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nexusstc/Handbook of Functional Equations: Stability Theory/bbd567ed447878500a94f92ee3f55064.pdf

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Themistocles M Rassias; Springer Science+Business Media

Rassias, Themistocles M.

Springer New York : Imprint : Springer

Springer London, Limited

Springer US

Springer Series in Optimization and Its Applications, vol. 96, New York, cop. 2014

Springer optimization and its applications, volume 96, Berlin, 2014

Springer optimization and its applications, 96, New York, NY, 2014

Springer Optimization and Its Applications 96, 1, 2014

United States, United States of America

Springer Nature, Berlin, 2014

Nov 22, 2014

sm33540379

producers:
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Source title: Handbook of Functional Equations: Stability Theory (Springer Optimization and Its Applications (96))

Preface 5
Contents 6
Contributors 8
On Some Functional Equations 10
1 On the Convergence of Adomian's Method 45
1.1 Introduction 10
1.2 The Decomposition Method 11
1.3 Convergence Result 13
2 Approximation Methods for Solving Functional Equations 48
2.1 The Collocation Method 15
2.2 The Method of Moments 19
2.3 The Least Squares Method 25
2.4 The Adomian Decomposition Method 27
3 Stability of the Generalized Quadratic Functional Equation on Topological Spaces 50
3.1 Introduction 30
3.2 Stability 33
References 43
Remarks on Stability of the Equation of Homomorphism for Square Symmetric Groupoids 45
1 Introduction 45
2 An Auxiliary Result 48
3 Modified Stability on Square Symmetric Groupoids 50
4 Some Complementary Results 56
References 63
On Stability of the Linear and Polynomial Functional Equations in Single Variable 66
1 Introduction 66
2 Stability of Zeros of Polynomials 68
3 Stability of the Linear Equation: The General Case 70
4 Stability of the Linear Equation: Iterative Case 75
5 Set-Valued Case 82
6 Stability of the Polynomial Equation 84
References 85
Selections of Set-valued Maps Satisfying Some Inclusions and the Hyers–Ulam Stability 89
1 Introduction 89
2 Linear Inclusions 90
3 Inclusions in a Single Variable 96
4 Applications 102
References 105
Generalized Ulam–Hyers Stability Results: A Fixed Point Approach 107
1 Preliminaries 107
2 Results 110
References 116
On a Weak Version of Hyers–Ulam Stability Theorem in Restricted Domains 118
1 Introduction 118
2 A Weak Stability of Pexider Equation 120
3 Weak Stability of Pexider Equation in Restricted Domains 125
References 136
On the Stability of Drygas Functional Equation on Amenable Semigroups 139
1 Introduction 139
2 Hyers–Ulam Stability of the Drygas Functional Equation in Amenable Semigroups 141
References 157
Stability of Quadratic and Drygas Functional Equations, with an Application for Solving an Alternative Quadratic Equation 159
1 Introduction 159
2 Stability of the Quadratic Equation 160
3 Stability of the Drygas Equation 171
4 Alternative Quadratic Equation 176
References 178
A Functional Equation Having Monomials and Its Stability 184
1 Introduction 184
2 Preliminaries 185
3 Multi-additive and Monomial Mappings 188
4 Fixed Points and Stability of Monomial Functional Equations 196
References 199
Some Functional Equations Related to the Characterizations of Information Measures and Their Stability 201
1 Introduction and Preliminaries 201
1.1 Information Measures 202
1.2 The Characterization Problem and Functional Equations 204
1.3 Prerequisites from the Theory of Functional Equations 205
2 Results on the Fundamental Equation of Information and on the Sum Form Equations 207
2.1 Information Functions 207
2.2 Sum Form Equations 209
3 Stability Problems 212
3.1 The Cases α = 0 and 0 < α = 1 214
3.2 The Case α < 0 225
3.3 Related Equations 231
3.3.1 Stability of the Entropy Equation 231
3.3.2 Stability of the Modified Entropy Equation 235
3.4 Stability of Sum Form Equations 238
References 241
Approximate Cauchy–Jensen Type Mappings in Quasi-β-Normed Spaces 244
1 Introduction 244
2 Generalized Hyers–Ulam Stability of (2) 247
3 Alternative Generalized Hyers–Ulam Stability of (2) 252
References 254
An AQCQ-Functional Equation in Matrix Paranormed Spaces 256
1 Introduction and Preliminaries 256
2 Hyers–Ulam Stability of the AQCQ-Functional Equation (3) in Matrix Paranormed Spaces: Odd Mapping Case 259
3 Hyers–Ulam Stability of the AQCQ-Functional Equation (3) in Matrix Paranormed Spaces: Even Mapping Case 265
4 Hyers–Ulam Stability of the AQCQ-Functional Equation (3) in Matrix β-Homogeneous F∗-Spaces: Odd Mapping Cas 268
5 Hyers–Ulam Stability of the AQCQ-Functional Equation (3) in Matrix β-Homogeneous F∗-Spaces: Even Mapping Case 273
6 Conclusions 275
References 277
On the Generalized Hyers–Ulam Stability of the Pexider Equation on Restricted Domains 279
1 Introduction 279
2 Stability of Eq. (7) on Restricted Domains 282
3 Stability of Eq. (8) on Restricted Domains 287
4 Stability of Eq. (9) on Restricted Domain 292
References 296
Hyers-Ulam Stability of Some Differential Equations and Differential Operators 300
1 Introduction 300
2 Stability of the Linear Differential Equation of Order One 301
3 Stability of the Linear Differential Equation of Higher Order with Constant Coefficients 305
4 Stability of First Order Linear Differential Operator 308
5 Stability of Higher Order Linear Differential Operator 313
6 Stability of Partial Differential Equations 314
References 320
Results and Problems in Ulam Stability of Operatorial Equations and Inclusions 322
1 Introduction 322
2 Operatorial Equations and Inclusions 323
3 From the Ulam Problem to the Notion of Ulam-Hyers Stability of an Operatorial Equation 325
4 ψ-Weakly Picard Operators and Ulam-Hyers Stability of a Fixed Point Equation 327
5 Ulam-Hyers Stability of a Coincidence Equation 334
6 The Case of Spaces of Functions: Ulam-Hyers and Ulam-Hyers-Rassias Stability 335
7 Equations with Multivalued Operators 338
8 Other Problems 340
8.1 Ulam Stability in the Case of a Generalized Metric Space (d(x, y) ∈ R+) 340
8.2 Ulam Stability in the Case of a Generalized Metric Space (d(x, y) ∈ E+) 342
8.3 Ulam Stability in the Case of Equations with Set-To-Point Operators 344
8.4 Difference Equations as Operatorial Equations 344
8.5 Ulam Stability of Fractal Equations 345
References 346
Superstability of Generalized Module Left Higher Derivations on a Multi-Banach Module 352
1 Introduction and Preliminaries 352
2 Main Results 355
References 363
D’Alembert’s Functional Equation and Superstability Problem in Hypergroups 365
1 Introduction 365
2 Preliminaries and Notations 367
2.1 Hypergroups 367
2.2 D’Alembert Function on Hypergroups 369
3 Properties of D’Alembert Functions 369
4 The Space W(g) of Wilson Functions 377
5 The General Case 387
6 Superstability of the D’Alembert Equation (4) 388
References 392

This {OCLCbr#A0}handbook consists of seventeen chapters written by eminent scientists from the international mathematical community, who present important research works in the field of mathematical analysis and related subjects, particularly in the Ulam stability theory of functional equations. The book provides an insight into a large domain of research with emphasis to the discussion of several theories, methods and problems in approximation theory, analytic inequalities, functional analysis, computational algebra and applications.{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0} The notion of stability of functional equations has its origins with S. M. Ulam, who posed the fundamental problem for approximate homomorphisms in 1940{OCLCbr#A0}and with D. H. Hyers, Th. M. Rassias, who provided the first significant solutions for additive and linear mappings in 1941 and 1978, respectively. During the last decade the notion of stability of functional equations has evolved into a very active domain of mathematical research with several applications of interdisciplinary nature.{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0}{OCLCbr#A0} The chapters of this handbook focus mainly on both old and recent developments on the equation of homomorphism for square symmetric groupoids, the linear and polynomial functional equations in a single variable, the Drygas functional equation on amenable semigroups, monomial functional equation,{OCLCbr#A0} the Cauchy-Jensen type mappings, differential equations and differential operators, operational equations and inclusions, generalized module left higher derivations, selections of set-valued mappings, D'Alembert's functional equation, characterizations of information measures, {OCLCbr#A0}functional equations in restricted domains, as well as generalized functional stability and fixed point theory

"This handbook consists of seventeen chapters written by eminent scientists from the international mathematical community, who present important research works in the field of mathematical analysis and related subjects, particularly in the Ulam stability theory of functional equations. The book provides an insight into a large domain of research with emphasis to the discussion of several theories, methods and problems in approximation theory, analytic inequalities, functional analysis, computational algebra and applications. The notion of stability of functional equations has its origins with S.M. Ulam, who posed the fundamental problem for approximate homomorphisms in 1940 and with D.H. Hyers, Th. M. Rassias, who provided the first significant solutions for additive and linear mappings in 1941 and 1978, respectively. During the last decade the notion of stability of functional equations has evolved into a very active domain of mathematical research with several applications of interdisciplinary nature. The chapters of this handbook focus mainly on both old and recent developments on the equation of homomorphism for square symmetric groupoids, the linear and polynomial functional equations in a single variable, the Drygas functional equation on amenable semigroups, monomial functional equation, the Cauchy-Jensen type mappings, differential equations and differential operators, operational equations and inclusions, generalized module left higher derivations, selections of set-valued mappings, D'Alembert's functional equation, characterizations of information measures, functional equations in restricted domains, as well as generalized functional stability and fixed point theory."--Page 4 of cover

Front Matter....Pages i-x
On Some Functional Equations....Pages 1-35
Remarks on Stability of the Equation of Homomorphism for Square Symmetric Groupoids....Pages 37-57
On Stability of the Linear and Polynomial Functional Equations in Single Variable....Pages 59-81
Selections of Set-valued Maps Satisfying Some Inclusions and the Hyers–Ulam Stability....Pages 83-100
Generalized Ulam–Hyers Stability Results: A Fixed Point Approach....Pages 101-111
On a Weak Version of Hyers–Ulam Stability Theorem in Restricted Domains....Pages 113-133
On the Stability of Drygas Functional Equation on Amenable Semigroups....Pages 135-154
Stability of Quadratic and Drygas Functional Equations, with an Application for Solving an Alternative Quadratic Equation....Pages 155-179
A Functional Equation Having Monomials and Its Stability....Pages 181-197
Some Functional Equations Related to the Characterizations of Information Measures and Their Stability....Pages 199-241
Approximate Cauchy–Jensen Type Mappings in Quasi- β -Normed Spaces....Pages 243-254
An AQCQ-Functional Equation in Matrix Paranormed Spaces....Pages 255-277
On the Generalized Hyers–Ulam Stability of the Pexider Equation on Restricted Domains....Pages 279-299
Hyers-Ulam Stability of Some Differential Equations and Differential Operators....Pages 301-322
Results and Problems in Ulam Stability of Operatorial Equations and Inclusions....Pages 323-352
Superstability of Generalized Module Left Higher Derivations on a Multi-Banach Module....Pages 353-365
D’Alembert’s Functional Equation and Superstability Problem in Hypergroups....Pages 367-396

2015-02-17