Stability theorems

1977 ◽  
Vol 9 (02) ◽  
pp. 336-361 ◽  
Author(s):  
Eugene Lukacs
Keyword(s):  
Stability Theorem ◽  
Stability Theorems ◽  
Primary Object ◽  
The Stability ◽  
Decomposition Theorems

A stability theorem determines the extent to which the conclusions of a given theorem are affected if the assumptions of the theorem are not exactly but only approximately satisfied. The meaning of the word ‘approximately’ has to be defined exactly. The stability of decomposition theorems, of characterizations by independence and by regression properties are the primary object of the paper.

Stability theorems

1977 ◽  
Vol 9 (2) ◽  
pp. 336-361 ◽  
Author(s):  
Eugene Lukacs
Keyword(s):  
Stability Theorem ◽  
Stability Theorems ◽  
Primary Object ◽  
The Stability ◽  
Decomposition Theorems

A stability theorem determines the extent to which the conclusions of a given theorem are affected if the assumptions of the theorem are not exactly but only approximately satisfied. The meaning of the word ‘approximately’ has to be defined exactly. The stability of decomposition theorems, of characterizations by independence and by regression properties are the primary object of the paper.

scholarly journals Generalized Hyers–Ulam Stability of the Additive Functional Equation

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 76 ◽  
Author(s):  
Yang-Hi Lee ◽  
Gwang Kim
Keyword(s):  
Functional Equation ◽  
Additive Function ◽  
Additive Functional ◽  
Stability Theorem ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
Stability Theorems ◽  
The Stability

We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x1 + y1, x2 + y2, …, xn + yn) = f(x1, x2, … xn) + f(y1, y2, …, yn). By restricting the domain of a mapping f that satisfies the inequality condition used in the assumption part of the stability theorem, we partially generalize the results of the stability theorems of the additive function equations.

scholarly journals Stability with Respect to a Part of Variables under Constant Perturbations of the Partial Equilibrium Position of Differential Equation Nonlinear Systems

2018 ◽  
Vol 28 (3) ◽  
pp. 344-351
Author(s):  
Pavel P. Lipasov ◽  
Vladimir N. Shchennikov
Keyword(s):  
Nonlinear Systems ◽  
Equilibrium Position ◽  
Lyapunov Method ◽  
Stability Theorem ◽  
Partial Equilibrium ◽  
Dynamic Processes ◽  
Stability Theorems ◽  
Research Objects ◽  
Controlled Motion ◽  
The Stability

Introduction. It is impossible to take into account all the forces acting in the process of mathematical modeling of dynamic processes. In order that mathematical models the most accurately describe the dynamic processes, they must include the terms that correspond the constant perturbations. These problems arise in applied tasks. In this paper we consider the case when the system allows for the partial equilibrium position. The aim of this work is to prove the stability theorem for the partial equilibrium position at constant perturbations, which are small at every instant. Materials and Methods. The research objects are nonlinear systems of differential equations that allow for a partial equilibrium position. Using the second Lyapunov method, there are proved the stability theorems for the constant perturbations of the partial equilibrium position, which are small at every instant. Results. Together with the introduction of stability for a part of the variables, it has become necessary to introduce stability for the part of phase variables under constant perturbations. The first stability theorem of the part of phase variables under constant perturbations was obtained by A. S. Oziraner. In this work, we prove a theorem of the stability of the constant perturbations of the partial equilibrium position, small at every instant. It should be noted that there is no stability theorems of constant perturbations for the partial equilibrium position. Thus, the theorem proved in this work is of a pioneer nature. Conclusions. The theorem 3 proved in the work is the development of the mathematical theory of stability. The results of this work are applicable in the mechanics of controlled motion, nonlinear system.

Pragmatical Asymptotical Stability Theorems on Partial Region and for Partial Variables with Applications to Gyroscopic Systems

2000 ◽  
Vol 16 (4) ◽  
pp. 179-187 ◽  
Author(s):  
Zheng-Ming Ge ◽  
Jung-Kui Yu
Keyword(s):  
Mathematical Model ◽  
Dynamical Systems ◽  
Zero Solution ◽  
Asymptotical Stability ◽  
Stability Theorem ◽  
Partial Region ◽  
Stability Theorems ◽  
Long Time ◽  
The Stability ◽  
Disturbed Motion

ABSTRACTFor a long time, all stability theorems are concerned with the stability of the zero solution of the differential equations of disturbed motion on the whole region of the neighborhood of the origin. But for various problems of dynamical systems, the stability is actually on partial region. In other words, the traditional mathematical model is unmatched with the dynamical reality and artificially sets too strict demand which is unnecessary. Besides, although the stability for many problems of dynamical systems may not be mathematical asymptotical stability, it is actual asymptotical stability — namely “pragmatical asymptotical stability” which can be introduced by the concept of probability. In order to fill the gap between the traditional mathematical model and dynamical reality of various systems, one pragmatical asymptotical stability theorem on partial region and one pragmatical asymptotical stability theorem on partial region for partial variables are given and applications for gyroscope systems are presented.

Stability of elliptical horizontally inhomogeneous rodons

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.

scholarly journals A General Theorem on the Stability of a Class of Functional Equations Including Quartic-Cubic-Quadratic-Additive Equations

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 282
Author(s):  
Yang-Hi Lee ◽  
Soon-Mo Jung
Keyword(s):  
Functional Equations ◽  
Direct Method ◽  
General Theorem ◽  
Stability Problems ◽  
General Stability ◽  
Stability Theorems ◽  
Additive Type ◽  
The Stability

We prove general stability theorems for n-dimensional quartic-cubic-quadratic-additive type functional equations of the form by applying the direct method. These stability theorems can save us the trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations.

scholarly journals HUR-approximation of an ELTA functional equation

Filomat ◽  
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.

The ?-stability theorem for flows

1970 ◽  
Vol 11 (2) ◽  
pp. 150-158 ◽  
Author(s):  
Charles Pugh ◽  
Michael Shub
Keyword(s):  
Stability Theorem ◽  
The Stability

The Stability Theorems for Subgroups of and

1994 ◽  
Vol 46 (5) ◽  
pp. 995-1006 ◽  
Author(s):  
Ali Lari-Lavassani ◽  
Yung-Chen Lu
Keyword(s):  
Large Class ◽  
Structural Stability ◽  
Singularity Theory ◽  
Stability Theorems ◽  
The Stability

AbstractIn singularity theory, J. Damon gave elegant versions of the unfolding and determinacy theorems for geometric subgroups of . and . In this work, we propose a unified treatment of the smooth stability of germs and the structural stability of versai unfoldings for a large class of such subgroups.

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