Superstability of adjointable mappings on Hilbert c∗-modules

We define the notion of ?-perturbation of a densely defined adjointable mapping and prove that any such mapping f between Hilbert A-modules over a fixed C*-algebra A with densely defined corresponding mapping g is A-linear and adjointable in the classical sense with adjoint g. If both f and g are every- where defined then they are bounded. Our work concerns with the concept of Hyers-Ulam-Rassias stability originated from the Th. M. Rassias' stability theorem that appeared in his paper [On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300]. We also indicate complementary results in the case where the Hilbert C?-modules admit non-adjointable C*-linear mappings.

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HUR-approximation of an ELTA functional equation

Filomat ◽

10.2298/fil2013311s ◽

2020 ◽

Vol 34 (13) ◽

pp. 4311-4328

Author(s):

A.R. Sharifi ◽

Azadi Kenary ◽

B. Yousefi ◽

R. Soltani

Keyword(s):

Functional Equation ◽

Banach Spaces ◽

Linear Mapping ◽

Stability Theorem ◽

Normed Spaces ◽

The Stability ◽

Rassias Stability

The main goal of this paper is study of the Hyers-Ulam-Rassias stability (briefly HUR-approximation) of the following Euler-Lagrange type additive(briefly ELTA) functional equation ?nj=1f (1/2 ?1?i?n,i?j rixi- 1/2 rjxj) + ?ni=1 rif(xi)=nf (1/2 ?ni=1 rixi) where r1,..., rn ? R, ?ni=k rk?0, and ri,rj?0 for some 1? i < j ? n, in fuzzy normed spaces. The concept of HUR-approximation originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

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On the Stability of Quadratic Functional Equations

Abstract and Applied Analysis ◽

10.1155/2008/628178 ◽

2008 ◽

Vol 2008 ◽

pp. 1-8 ◽

Cited By ~ 9

Author(s):

Jung Rye Lee ◽

Jong Su An ◽

Choonkil Park

Keyword(s):

Functional Equation ◽

Positive Integer ◽

Banach Spaces ◽

Functional Equations ◽

Vector Spaces ◽

Quadratic Functional ◽

The Stability ◽

Fixed Positive Integer ◽

Rassias Stability

LetX,Ybe vector spaces andka fixed positive integer. It is shown that a mappingf(kx+y)+f(kx-y)=2k2f(x)+2f(y)for allx,y∈Xif and only if the mappingf:X→Ysatisfiesf(x+y)+f(x-y)=2f(x)+2f(y)for allx,y∈X. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

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On the stability of the linear mapping in Banach spaces

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1978-0507327-1 ◽

1978 ◽

Vol 72 (2) ◽

pp. 297-297 ◽

Cited By ~ 1146

Author(s):

Themistocles M. Rassias

Keyword(s):

Banach Spaces ◽

Linear Mapping ◽

The Stability

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Fuzzy approximation of an additive functional equation

Journal of Function Spaces and Applications ◽

10.1155/2011/941731 ◽

2011 ◽

Vol 9 (2) ◽

pp. 205-215 ◽

Cited By ~ 1

Author(s):

G. Zamani Eskandani ◽

Ali Reza Zamani ◽

H. Vaezi

Keyword(s):

Banach Space ◽

Functional Equation ◽

Banach Spaces ◽

Normed Space ◽

Additive Functional ◽

Additive Functional Equation ◽

Fuzzy Approximation ◽

The Stability ◽

Rassias Stability

In this paper, we investigate the generalized Hyers– Ulam– Rassias stability of the functional equation∑i=1mf(mxi+∑j=1, j≠imxj)+f(∑i=1mxi)=2f(∑i=1mmxi)in fuzzy Banach spaces and some applications of our results in the stability of above mapping from a normed space to a Banach space will be exhibited.

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Stability of universal unfoldings: a correction

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410005862x ◽

1981 ◽

Vol 90 (1) ◽

pp. 195-196 ◽

Cited By ~ 1

Author(s):

R. J. Magnus

Keyword(s):

Banach Spaces ◽

Bifurcation Theory ◽

Singularity Theory ◽

Stability Theorem ◽

Dimensional Theory ◽

Infinite Dimensional ◽

Finite Dimensional ◽

Potential Case ◽

The Stability ◽

Imperfect Bifurcation

The author is grateful to Les Lander for pointing out an error in the stability section of (1). In fact Theorems 5 and 7 are incorrect. Recently Arkeryd proved a stability theorem for the infinite-dimensional case in the context of the imperfect bifurcation theory of Golubitsky and Schaeffer(3). In his result finitely many derivatives are controlled, the number depending on the codimension of the singularity unfolded. In this note we shall present a stability theorem involving the determinacy of the singularity. The context is the parameter-free potential case, that is, catastrophe theory. The proof is without recourse to the finite-dimensional results, and the theorem concludes an account of a part of singularity theory in Banach spaces, in which the author has tried to use as little as possible of the finite-dimensional theory (1, 2).

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Existence and Ulam stability for implicit fractional q-difference equations

Advances in Difference Equations ◽

10.1186/s13662-019-2411-y ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

Saïd Abbas ◽

Mouffak Benchohra ◽

Nadjet Laledj ◽

Yong Zhou

Keyword(s):

Fixed Point ◽

Banach Spaces ◽

Difference Equations ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Ulam Stability ◽

Uniqueness Of Solutions ◽

Stability Results ◽

Rassias Stability

AbstractThis paper deals with some existence, uniqueness and Ulam–Hyers–Rassias stability results for a class of implicit fractional q-difference equations. Some applications are made of some fixed point theorems in Banach spaces for the existence and uniqueness of solutions, next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Two illustrative examples are given in the last section.

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Stability of elliptical horizontally inhomogeneous rodons

Journal of Fluid Mechanics ◽

10.1017/s0022112000008569 ◽

2000 ◽

Vol 416 ◽

pp. 29-43

Author(s):

RENÉ PINET ◽

E. G. PAVÍA

Keyword(s):

Formal Analysis ◽

Detailed Examination ◽

Stability Theorem ◽

Homogeneous Case ◽

Density Distributions ◽

Formal Stability ◽

Inhomogeneous Density ◽

Loss Of Mass ◽

The Stability ◽

Main Effects

The stability of one-layer vortices with inhomogeneous horizontal density distributions is examined both analytically and numerically. Attention is focused on elliptical vortices for which the formal stability theorem proved by Ochoa, Sheinbaum & Pavía (1988) does not apply. Our method closely follows that of Ripa (1987) developed for the homogeneous case; and indeed they yield the same results when inhomogenities vanish. It is shown that a criterion from the formal analysis – the necessity of a radial increase in density for instability – does not extend to elliptical vortices. In addition, a detailed examination of the evolution of the inhomogeneous density fields, provided by numerical simulations, shows that homogenization, axisymmetrization and loss of mass to the surroundings are the main effects of instability.

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