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E-grāmata: Stability of Functional Equations in Random Normed Spaces

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Stability of Functional Equations in Random Normed SpacesPalielināt
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This book discusses the rapidly developing subject of mathematical analysis that deals primarily with stability of functional equations in generalized spaces. The fundamental problem in this subject was proposed by Stan M. Ulam in 1940 for approximate homomorphisms. The seminal work of Donald H. Hyers in 1941 and that of Themistocles M. Rassias in 1978 have provided a great deal of inspiration and guidance for mathematicians worldwide to investigate this extensive domain of research.

The book presents a self-contained survey of recent and new results on topics including basic theory of random normed spaces and related spaces; stability theory for new function equations in random normed spaces via fixed point method, under both special and arbitrary t-norms; stability theory of well-known new functional equations in non-Archimedean random normed spaces; and applications in the class of fuzzy normed spaces. It contains valuable results on stability in random normed spaces, and is geared toward both graduate students and research mathematicians and engineers in a broad area of interdisciplinary research.

Recenzijas

The book should interest any professional mathematician whose research is connected with functional equations, especially their stability in random spaces; I also can recommend it for graduate students interested in the subject. It could serve as a complete and independent introduction to the field of stability of functional equations in random spaces and as an excellent source of references for further study. (Janusz Brzdk, SIAM Review, Vol. 57 (1), March, 2015)

The book under review is essentially a collection of several recent papers related to the stability of functional equations in the framework of fuzzy and random normed spaces. useful for graduate students who are interested in the Hyers-Ulam-Rassias stability of functional equations. (Mohammad Sal Moslehian, zbMATH, Vol. 1281, 2014)

1 Preliminaries
1(10)
1.1 Triangular Norms
1(4)
1.2 Triangular Norms on Lattices
5(1)
1.3 Distribution Functions
6(1)
1.4 Fuzzy Sets
7(4)
2 Generalized Spaces
11(36)
2.1 Random Normed Spaces
11(6)
2.2 Random Topological Structures
17(14)
2.3 Random Functional Analysis
31(10)
2.4 Non-Archimedean Random Normed Spaces
41(2)
2.5 Fuzzy Normed Spaces
43(4)
3 Stability of Functional Equations in RN-Spaces Under Spacial t- Norm
47(16)
3.1 Cauchy Additive Equations
47(4)
3.2 Quadratic Functional Equations
51(5)
3.3 Cubic Functional Equations
56(4)
3.4 Quartic Functional Equations
60(3)
4 Stability of Functional Equations in RN-Spaces Under Arbitrary t-Norms
63(18)
4.1 Cauchy Additive Equations
63(2)
4.2 Cubic Functional Equations
65(6)
4.3 Quartic Functional Equations
71(10)
5 Stability of Functional Equations in RN-Spaces via Fixed Point Methods
81(44)
5.1 m-Variable Additive Functional Equations
82(5)
5.2 Quartic Functional Equations
87(2)
5.3 ACQ Functional Equations
89(15)
5.3.1 The Generalized Hyers-Ulam Stability of the Functional Equation (5.3.1): An Odd Case
89(11)
5.3.2 The Generalized Hyers-Ulam Stability of the Functional Equation (5.3.1): An Even Case
100(4)
5.4 AQCQ Functional Equations
104(21)
5.4.1 The Generalized Hyers-Ulam Stability of the Functional Equation (5.4.1): An Odd Case via Fixed Point Method
104(12)
5.4.2 The Generalized Hyers-Ulam Stability of the Functional Equation (5.4.1): An Even Case via Fixed Point Method
116(9)
6 Stability of Function Equations in Non-Archimedean Random Spaces
125(28)
6.1 Cubic Functional Equations
125(4)
6.2 Quartic Functional Equations
129(5)
6.3 Another Quartic Functional Equations
134(5)
6.4 Mixed AQCQ Functional Equations
139(14)
6.4.1 The Generalized Hyers-Ulam Stability of the Quartic Functional Equations (6.4.1) in Non-Archimedean RN-Spaces: An Odd Case
139(8)
6.4.2 The Generalized Hyers-Ulam Stability of the Functional Equation (6.4.1) in Non-Archimedean RN-Spaces: An Even Case
147(6)
7 Stability of Functional Equations Related to Inner Product Spaces
153(22)
7.1 AQ Functional Equations
153(10)
7.1.1 The Generalized Hyers-Ulam Stability of the Functional Equation (7.1.1): An Odd Case
154(5)
7.1.2 The Generalized Hyers-Ulam Stability of the Functional Equation (7.1.1): An Even Case
159(4)
7.2 Non-Archimedean Lattice RN-Spaces
163(12)
7.2.1 The Hyers-Ulam Stability of Functional Equations in Non-Archimedean Lattice Random Spaces
163(12)
8 Random Banach Algebras and Stability Results
175(32)
8.1 Random Homomorphisms
175(9)
8.1.1 The Stability of Random Homomorphisms in Random Normed Algebras
176(5)
8.1.2 The Stability of Random Derivations on Random Normed Algebras
181(3)
8.2 Cauchy-Jensen Functional Equations in Banach *-Algebras
184(14)
8.2.1 The Hyers-Ulam Stability of Cauchy-Jensen Functional Equations in Induced Random C*-algebras
191(3)
8.2.2 The Hyers-Ulam Stability of Cauchy-Jensen Functional Inequalities in Random Banach *-Algebras and Induced Random C*-Algebras
194(4)
8.3 Random *-Derivations in Banach *-Algebras
198(9)
8.3.1 The Stability of Homomorphisms and Derivations in Non-Archimedean Random C*-Algebras
199(4)
8.3.2 The Stability of Homomorphisms and Derivations in Non-Archimedean Lie C*-Algebras
203(4)
9 Related Results on Stability of Functional Inequalities and Equations
207(28)
9.1 Latticetic Stability of the Functional Inequalities
207(8)
9.2 Systems of QC and AQC Functional Equations
215(12)
9.3 Couple Functional Equations
227(8)
References 235(10)
Index 245
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