Additive ρ-functional inequalities in β-homogeneous F-spaces
4
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In this paper, we solve the additive $\rho$-functional inequalities \begin{eqnarray}\label{0.1} \|f(2x-y)+ f(y-x)-f(x)\| \le \left\|\rho\left( f(x+y) - f(x)- f(y) \right)\right\|, \end{eqnarray} where $\rho$ is a fixed complex number with $|\rho|<1$, and \begin{eqnarray}\label{0.2} \left\| f(x+y)-f(x)-f(y)\right\| \le \|\rho(f(2x-y)+f(y-x)-f(x))\| , \end{eqnarray} where $\rho$ is a fixed complex number with $|\rho|<\frac{1}{2}$. Using the direct method, we prove the Hyers-Ulam stability of the additive $\rho$-functional inequalities (\ref{0.1}) and (\ref{0.2}) in $\beta$-homogeneous $F$-spaces.
Abstract
In this paper, we solve the additive $\rho$-functional inequalities \begin{eqnarray}\label{0.1} \|f(2x-y)+ f(y-x)-f(x)\| \le \left\|\rho\left( f(x+y) - f(x)- f(y) \right)\right\|, \end{eqnarray} where $\rho$ is a fixed complex number with $|\rho|<1$, and \begin{eqnarray}\label{0.2} \left\| f(x+y)-f(x)-f(y)\right\| \le \|\rho(f(2x-y)+f(y-x)-f(x))\| , \end{eqnarray} where $\rho$ is a fixed complex number with $|\rho|<\frac{1}{2}$. Using the direct method, we prove the Hyers-Ulam stability of the additive $\rho$-functional inequalities (\ref{0.1}) and (\ref{0.2}) in $\beta$-homogeneous $F$-spaces.
참고문헌 (25)
- M. Adam, "On the stability of some quadratic functional equation", J. Nonlinear Sci. Appl, 4, pp.50~59 (2011)
- T. Aoki, "On the stability of the linear transformation in Banach spaces", J. Math. Soc. Japan, 2, pp.64~66 (1950)
- L. Cadariu, "On the stability of an a±ne functional equation", J. Nonlinear Sci. Appl, 6, pp.60~67 (2013)
- A. Chahbi, "On the generalized stability of d'Alembert functional equa-tion", J. Nonlinear Sci. Appl, 6, pp.198~204 (2013)
- P.W. Cholewa, "Remarks on the stability of functional equations", Aequationes Math, 27, pp.76~86 (1984)
- G. Z. Eskandani, "Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces", J. Nonlinear Sci. Appl, 5, pp.459~465 (2012)
- P. G·avruta, "A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings", J. Math. Anal. Appl, 184, pp.431~436 (1994)
- D.H. Hyers, "On the stability of the linear functional equation", Proc. Natl. Acad. Sci. U.S.A, 27, pp.222~224 (1941)
- C. Park, "Orthogonal stability of a cubic-quartic functional equation", J. Nonlinear Sci. Appl, 5, pp.28~36 (2012)
- C. Park, "Additive ½-functional inequalities and equations", J. Math. Inequal, 9, pp.17~26 (2015)
- C. Park, "Additive ½-functional inequalities in non-Archimedean normed spaces", J. Math. Inequal, 9, pp.397~407 (2015)
- C. Park, "Approximate n-Jordan ¤-homomorphisms in C¤-algebras", J. Comput. Anal. Appl, 15, pp.365~368 (2013)
- C. Park, "Fixed points and fuzzy stability of an additive-quadratic functional equation", J. Comput. Anal. Appl, 15, pp.452~462 (2013)
- M. Rassias, "On the stability of the linear mapping in Banach spaces", Proc. Amer. Math. Soc, 72, pp.297~300 (1978)
- K. Ravi, "Solution and stability of a reciprocal type functional equation in several variables", J. Nonlinear Sci. Appl, 7, pp.18~27 (2014)
- S. Rolewicz, "Metric Linear Spaces" (1972), PWN-Polish Scientic Publishers
- S. Schin, "Random stability of quadratic functional equa-tions: a ¯xed point approach", J. Nonlinear Sci. Appl, 4, pp.37~49 (2011)
- S. Shagholi, "Nearly ternary cubic homo-morphism in ternary Fr¶echet algebras", J. Comput. Anal. Appl, 13, pp.1106~1114 (2011)
- S. Shagholi, "Stability of ternary quadratic derivation on ternary Banach algebras", J. Comput. Anal. Appl, 13, pp.1097~1105 (2011)
- D. Shin, "On the superstability of ternary Jordan C¤-homomorphisms", J. Comput. Anal. Appl, 16, pp.964~973 (2014)
- D. Shin, "Stability and superstability of J¤-homomorphisms and J¤-derivations for a generalized Cauchy-Jensen equation", J. Comput. Anal. Appl, 17, pp.125~134 (2014)
- F. Skof, "Propriet locali e approssimazione di operatori", Rend. Sem. Mat. Fis. Milano, 53, pp.113~129 (1983)
- S.M. Ulam, "A Collection of the Mathematical Problems" (1960), Interscience Publ
- C. Zaharia, "On the probabilistic stability of the monomial functional equation", J. Non-linear Sci. Appl, 6, pp.51~59 (2013)
- S. Zolfaghari, "Approximation of mixed type functional equations in p-Banach spaces", J. Nonlinear Sci. Appl, 3, pp.110~122 (2010)