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

2018 ◽  
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 \begin{eqnarray*} \sum_{i=1}^\ell c_i f \big( a_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n \big) = 0 \end{eqnarray*} by applying the direct method. These stability theorems can save us much trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations.

Robust Stability of Nonlinear Feedback Systems With Parameter Deviations and Perturbed Nonlinearities

1997 ◽  
Vol 119 (1) ◽  
pp. 133-135
Author(s):  
Hayao Miyagi ◽  
Kimiko Kawahira ◽  
Norio Miyagi
Keyword(s):  
Robust Stability ◽  
Direct Method ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Individual Parameter ◽  
Nominal System ◽  
Additive Type ◽  
The Stability ◽  
Parameter Deviation ◽  
Nonlinear Feedback Systems

Robust stability of perturbed nonlinear feedback systems subjected to plant variations is investigated by using the direct method of Lyapunov. To establish the stability of the nominal system, the multivariable Popov criterion is utilized first. Then the stability of the system with parameter deviations and perturbed nonlinearities is studied. In this paper, an additive-type of parameter deviations are considered. The feature of the proposed method is that the tolerable range of individual parameter deviation and the conditions for the perturbed nonlinearities are simultaneously obtainable.

scholarly journals On Approximate Solutions of Functional Equations in Vector Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Bogdan Batko
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Spectral Representation ◽  
General Theorem ◽  
Approximate Solutions ◽  
Quadratic Functional ◽  
Cauchy Equation ◽  
Riesz Spaces ◽  
The Stability ◽  
Class Of Functions

We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra). The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equationF(x+y)+F(x)+F(y)≠0⇒F(x+y)=F(x)+F(y)in Riesz spaces, the Cauchy equation with squaresF(x+y)2=(F(x)+F(y))2inf-algebras, and the quadratic functional equationF(x+y)+F(x-y)=2F(x)+2F(y)in Riesz spaces.

A Study of the Influence of Initial Shape Imperfections on the Stability of Lattice Shells by Direct and Shell Analogy Methods

1987 ◽  
Vol 2 (4) ◽  
pp. 223-230 ◽  
Author(s):  
Jozef Sumec ◽  
Antonino E. Zingali
Keyword(s):  
Mathematical Model ◽  
Direct Method ◽  
Initial Shape ◽  
Analogy Method ◽  
General Stability ◽  
Reduction Coefficient ◽  
The Stability ◽  
Comparison Of The Results ◽  
A Direct Method ◽  
Lattice Shells

This paper describes the influence of initial shape imperfections of lattice shells on their general stability. An adequate mathematical model for the analysis of the shell instability by using a direct method (FEM) and a shell analogy method is presented. The reduction coefficient k for calculation of the critical load is derived. An illustrative example is used for comparison of the results calculated by the two numerical methods.

scholarly journals A General Uniqueness Theorem concerning the Stability of Additive and Quadratic Functional Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Yang-Hi Lee ◽  
Soon-Mo Jung
Keyword(s):  
Functional Equation ◽  
Uniqueness Theorem ◽  
Functional Equations ◽  
Additive Functional ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Additive Functional Equation ◽  
Additive Type ◽  
Stability Of Functional Equations ◽  
The Stability

We prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. This uniqueness theorem can replace the repeated proofs for uniqueness of the relevant solutions of given equations while we investigate the stability of functional equations.

scholarly journals Ulam Stability of a Functional Equation in Various Normed Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1119
Author(s):  
Krzysztof Ciepliński
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Functional Equations ◽  
Direct Method ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Quadratic Mappings ◽  
The Stability

In this note, we study the Ulam stability of a general functional equation in four variables. Since its particular case is a known equation characterizing the so-called bi-quadratic mappings (i.e., mappings which are quadratic in each of their both arguments), we get in consequence its stability, too. We deal with the stability of the considered functional equations not only in classical Banach spaces, but also in 2-Banach and complete non-Archimedean normed spaces. To obtain our outcomes, the direct method is applied.

scholarly journals Fuzzy Stability Results of Generalized Quartic Functional Equations

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 120
Author(s):  
Sang Og Kim ◽  
Kandhasamy Tamilvanan
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Control Function ◽  
Direct Method ◽  
Normed Spaces ◽  
Quartic Functional Equation ◽  
New Type ◽  
The Stability ◽  
Fixed Point Techniques ◽  
Stability Results

In the present paper, we introduce a new type of quartic functional equation and examine the Hyers–Ulam stability in fuzzy normed spaces by employing the direct method and fixed point techniques. We provide some applications in which the stability of this quartic functional equation can be controlled by sums and products of powers of norms. In particular, we show that if the control function is the fuzzy norm of the product of powers of norms, the quartic functional equation is hyperstable.

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