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Stability of a quadratic functional equation in quasi--Banach spaces
Abbas Najati,Fridoun Moradlou 대한수학회 2008 대한수학회보 Vol.45 No.3
In this paper we establish the general solution and investigate the Hyers–Ulam–Rassias stability of the following functional equation in quasi-Banach spaces. [수식] where Iij = {1, 2, 3, 4}\{i, j} for all 1 ≤ i < j ≤ 4. The concept of Hyers-Ulam-Rassias stability 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. In this paper we establish the general solution and investigate the Hyers–Ulam–Rassias stability of the following functional equation in quasi-Banach spaces. [수식] where Iij = {1, 2, 3, 4}\{i, j} for all 1 ≤ i < j ≤ 4. The concept of Hyers-Ulam-Rassias stability 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.
Stability of a mixed quadratic and additive functional equation in quasi-Banach spaces
Abbas Najati,Fridoun Moradlou 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5
In this paper we establish the general solution of the functional equation f(2x + y) + f(x − 2y) = 2f(x + y) + 2f(x− y) + f(−x) + f(−y) and investigate the Hyers–Ulam–Rassias stability of this equation in quasi- Banach spaces. The concept of Hyers-Ulam-Rassias stability 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. In this paper we establish the general solution of the functional equation f(2x + y) + f(x − 2y) = 2f(x + y) + 2f(x− y) + f(−x) + f(−y) and investigate the Hyers–Ulam–Rassias stability of this equation in quasi- Banach spaces. The concept of Hyers-Ulam-Rassias stability 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.
STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN QUASI-BANACH SPACES
Najati, Abbas,Moradlou, Fridoun Korean Mathematical Society 2008 대한수학회보 Vol.45 No.3
In this paper we establish the general solution and investigate the Hyers-Ulam-Rassias stability of the following functional equation in quasi-Banach spaces. $${\sum\limits_{{{1{\leq}i<j{\leq}4}\limits_{1{\leq}k<l{\leq}4}}\limits_{k,l{\in}I_{ij}}}\;f(x_i+x_j-x_k-x_l)=2\;\sum\limits_{1{\leq}i<j{\leq}4}}\;f(x_i-x_j)$$ where $I_{ij}$={1, 2, 3, 4}\backslash${i, j} for all $1{\leq}i<j{\leq}4$. The concept of Hyers-Ulam-Rassias stability 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.
HYERS-ULAM-RASSIAS STABILITY OF A SYSTEM OF FIRST ORDER LINEAR RECURRENCES
Xu, Mingyong Korean Mathematical Society 2007 대한수학회보 Vol.44 No.4
In this paper we discuss the Hyers-Ulam-Rassias stability of a system of first order linear recurrences with variable coefficients in Banach spaces. The concept of the Hyers-Ulam-Rassias stability 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. As an application, the Hyers-Ulam-Rassias stability of a p-order linear recurrence with variable coefficients is proved.
STABILITY OF A MIXED QUADRATIC AND ADDITIVE FUNCTIONAL EQUATION IN QUASI-BANACH SPACES
Najati, Abbas,Moradlou, Fridoun The Korean Society for Computational and Applied M 2009 Journal of applied mathematics & informatics Vol.27 No.5
In this paper we establish the general solution of the functional equation f(2x+y)+f(x-2y)=2f(x+y)+2f(x-y)+f(-x)+f(-y) and investigate the Hyers-Ulam-Rassias stability of this equation in quasi-Banach spaces. The concept of Hyers-Ulam-Rassias stability 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.
On the Hyers-Ulam-Rassias stability of Jensen's equation
Dongyan Zhang,Jian Wang 대한수학회 2009 대한수학회보 Vol.46 No.4
J. Wang [21] proposed a problem: whether the Hyers-Ulam- Rassias stability of Jensen’'s equation for the case p, q, r, s ∈(β,1/β)\{1} holds or not under the assumption that G and E are β-homogeneous F-space (0<β<1). The main purpose of this paper is to give an answer to Wang’'s problem. Furthermore, we proved that the stability property of Jensen’'s equation is not true as long as p or q is equal to β,1/β, or ¯β_(2)/β_(1) (0<β_(1),β_(2)≤1). J. Wang [21] proposed a problem: whether the Hyers-Ulam- Rassias stability of Jensen’'s equation for the case p, q, r, s ∈(β,1/β)\{1} holds or not under the assumption that G and E are β-homogeneous F-space (0<β<1). The main purpose of this paper is to give an answer to Wang’'s problem. Furthermore, we proved that the stability property of Jensen’'s equation is not true as long as p or q is equal to β,1/β, or ¯β_(2)/β_(1) (0<β_(1),β_(2)≤1).
STABILITY OF DERIVATIONS ON PROPER LIE CQ<sup>*</sup>-ALGEBRAS
Najati, Abbas,Eskandani, G. Zamani Korean Mathematical Society 2009 대한수학회논문집 Vol.24 No.1
In this paper, we obtain the general solution and the generalized Hyers-Ulam-Rassias stability for a following functional equation $$\sum\limits_{i=1}^mf(x_i+\frac{1}{m}\sum\limits_{{i=1\atop j{\neq}i}\.}^mx_j)+f(\frac{1}{m}\sum\limits_{i=1}^mx_i)=2f(\sum\limits_{i=1}^mx_i)$$ for a fixed positive integer m with $m\;{\geq}\;2$. This is applied to investigate derivations and their stability on proper Lie $CQ^*$-algebras. 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.
ON THE STABILITY OF EULER-LAGRANGE TYPE FUNCTIONAL EQUATION
SHIN, KEE YOUNG,CHOI, BUUNG MUN,LEE, YOUNG WHAN,LEE, JI YEON 대전대학교 기초과학연구소 2003 自然科學 Vol.14 No.1
1장에서는 이미 알려진 함수 방정식의 안전성에 대하여 소개하였고, 2장에서는 오일러-라그랑주 타입 함수 방정식의 해는 이차 함수 형태로서 안정성을 갖는다는 성질을 밝혔다. 특히 일반화된 안정성 성질을 알기 위해서 Gavruta의 관점에서 안정성을 밝혔으며, 이것으로부터 Hyers-Ulam-Rassias 안전성 정리를 얻었다. TH. M. Rassias obtained the Hyers-Ulam stability of the general Euler-Lagrange functional equation. In this paper we prove the stability of an another Euler-Lagrange type functional equation in the spirit of Hyers, Ulam, Rassias and Gavruta.
ON THE RASSIAS-TABOR PROBLEM FOR A GENERALIZED CAUCHY FUNCTIONAL EQUATION
KIM,GWANG HUI,LEE,YOUNG WHAN,CHO,DAE YEON 대전대학교 기초과학연구소 1999 自然科學 Vol.10 No.2
Rassias-Tabor가 제시한 일반화된 함수방정식의 안정성에 대한 미해결문제를 연구하여 부분적인 해결을 얻었다. 우리는 Borelli의 정리로부터 ∥f(ax+by+v)-af(x)-bf(x)-w∥≤△(x, y) 을 만족하는 다양한 a, b가 주어질 때, g(axby+v)=ag(x)+bg(y)+w 을 만족하는 g가 f 가까이에 존재함을 보였다. In this paper, we investigatethe stability problems (especially, the modified Hyers-Ulam-Rassias stability) of a generalized Cauchy functional equation by Hyers-Ulam stability for a class of functional equations. Further we obtain a partial answer for the open problem, which was posed by Th. M. Rassias and J. Tabor in their paper[5], on the stability for a special type of generalized Cauchy function equations.
On a general hyers-ulam stability of gamma functional equation
Jung, Soon-Mo Korean Mathematical Society 1997 대한수학회보 Vol.34 No.3
In this paper, the Hyers-Ulam stability and the general Hyers-Ulam stability (more precisely, modified Hyers-Ulam-Rassias stability) of the gamma functional equation (3) in the following setings $$ \left$\mid$ f(x + 1) - xf(x) \right$\mid$ \leq \delta and \left$\mid$ \frac{xf(x)}{f(x + 1)} - 1 \right$\mid$ \leq \frac{x^{1+\varepsilon}{\delta} $$ shall be proved.