Stability of additive-quadratic $\rho$-functional equations in Banach spaces: a fixed point approach
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Authors
Choonkil Park
- Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea.
Sang Og Kim
- Department of Mathematics, Hallym University, Chuncheon 24252, Republic of Korea.
Cihangir Alaca
- Department of Mathematics, Celal Bayar University, Muradiye Campus, 45140 Manisa, Turkey.
Abstract
Let
\[M_1f(x,y):=\frac{3}{4}f(x+y)-\frac{1}{4}f(-x-y)+\frac{1}{4}f(x-y)+\frac{1}{4}f(y-x)-f(x)-f(y),\\
M_2f(x,y):=2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})-f(x)-f(y).\]
We solve the additive-quadratic \(\rho\)-functional equations
\begin{eqnarray}
M_1f(x,y):=\rho M_2f(x,y), \qquad\qquad (1)
\end{eqnarray}
and
\begin{eqnarray}
M_2f(x,y):=\rho M_1f(x,y), \qquad\qquad (2)
\end{eqnarray}
where \(\rho\) is a fixed nonzero number with \(\rho\neq 1\).
Using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic \(\rho\)-functional equations (1) and
(2) in Banach spaces.
Share and Cite
ISRP Style
Choonkil Park, Sang Og Kim, Cihangir Alaca, Stability of additive-quadratic $\rho$-functional equations in Banach spaces: a fixed point approach, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 3, 1252--1262
AMA Style
Park Choonkil, Kim Sang Og, Alaca Cihangir, Stability of additive-quadratic $\rho$-functional equations in Banach spaces: a fixed point approach. J. Nonlinear Sci. Appl. (2017); 10(3):1252--1262
Chicago/Turabian Style
Park, Choonkil, Kim, Sang Og, Alaca, Cihangir. "Stability of additive-quadratic $\rho$-functional equations in Banach spaces: a fixed point approach." Journal of Nonlinear Sciences and Applications, 10, no. 3 (2017): 1252--1262
Keywords
- Hyers-Ulam stability
- fixed point method
- additive-quadratic \(\rho\)-functional equation
- Banach space.
MSC
References
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