Three problems of S. M. Ulam with solutions and generalizations
G. S. Stoller (1976)
Colloquium Mathematicae
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G. S. Stoller (1976)
Colloquium Mathematicae
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Yunbai Dong (2015)
Colloquium Mathematicae
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Let X,Y be real Banach spaces and ε > 0. Suppose that f:X → Y is a surjective map satisfying | ∥f(x)-f(y)∥ - ∥x-y∥ | ≤ ε for all x,y ∈ X. Hyers and Ulam asked whether there exists an isometry U and a constant K such that ∥f(x) - Ux∥ ≤ Kε for all x ∈ X. It is well-known that the answer to the Hyers-Ulam problem is positive and K = 2 is the best possible solution with assumption f(0) = U0 = 0. In this paper, using the idea of Figiel's theorem on nonsurjective isometries, we give a new...
Taneli Huuskonen, Jussi Väısälä (2002)
Studia Mathematica
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The best constant in the Hyers-Ulam theorem on isometric approximation in Hilbert spaces is equal to the Jung constant of the space.
Jin Rong Wang, Michal Fečkan (2017)
Mathematica Bohemica
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In this paper, we offer a new stability concept, practical Ulam-Hyers-Rassias stability, for nonlinear equations in Banach spaces, which consists in a restriction of Ulam-Hyers-Rassias stability to bounded subsets. We derive some interesting sufficient conditions on practical Ulam-Hyers-Rassias stability from a nonlinear functional analysis point of view. Our method is based on solving nonlinear equations via homotopy method together with Bihari inequality result. Then we consider nonlinear...
Li, Yongjin, Hua, Liubin (2009)
Banach Journal of Mathematical Analysis [electronic only]
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Jung, Soon-Mo (2007)
Abstract and Applied Analysis
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Nakmahachalasint, Paisan (2007)
International Journal of Mathematics and Mathematical Sciences
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Bidkham, M., Mezerji, H.A.Soleiman, Gordji, M.Eshaghi (2010)
Abstract and Applied Analysis
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Jung, Soon-Mo (2007)
Fixed Point Theory and Applications [electronic only]
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Kim, Byungbae, Jung, Soon-Mo (2007)
Journal of Inequalities and Applications [electronic only]
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Rafał Górak (2011)
Studia Mathematica
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We prove a very general theorem concerning the estimation of the expression ||T((a+b)/2) - (Ta+Tb)/2|| for different kinds of maps T satisfying some general perturbed isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization...
Jung, Soon-Mo, Brzdȩk, Janusz (2010)
Abstract and Applied Analysis
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Castro, L.P., Ramos, A. (2008)
Banach Journal of Mathematical Analysis [electronic only]
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Jung, Soon-Mo, Lee, Zoon-Hee (2008)
Fixed Point Theory and Applications [electronic only]
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Jung, Soon-Mo, Min, Seungwook (2009)
Fixed Point Theory and Applications [electronic only]
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Diana Otrocol, Veronica Ilea (2013)
Open Mathematics
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We study the Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability for a delay differential equation. Some examples are given.