QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN NON-ARCHIMEDEAN NORMED SPACES
3
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
in this paper, we solve the following quadratic rho-functional inequalities \begin{eqnarray} \left\| f\left(\frac{x+y+z}{2}\right)+f\left(\frac{x-y-z}{2}\right)+f\left(\frac{y-x-z}{2}\right)+f\left(\frac{z-x-y}{2}\right) -f(x) -f(y) -f(z) \right\| \leq \| \rho (f(x+y+z) + f(x-y-z) +f(y-x-z)+f(z-x-y) -4f(x)-4f(y)-4f(z)\|, \end{eqnarray} and \begin{eqnarray} \| f(x+y+z) + f(x-y-z)+f(y-x-z)+f(z-x-y) -4f(x)-4f(y) -4f(z) \| \leq \left \| \rho \left( f\left(\frac{x+y+z}{2}\right)+f\left(\frac{x-y-z}{2}\right) +f\left(\frac{y-x-z}{2}\right)+f\left(\frac{z-x-y}{2}\right)-f(x)-f(y)-f(z) \right) \right\|,\end{eqnarray} where rho is a fixed non-Archimedean number with |rho| < |8|. Using the direct method, we prove the Hyers-Ulam stability of the quadratic rho-functional inequalities (0.1) and (0.2) in non-Archimedean Banach spaces and prove the Hyers-Ulam stability of quadratic ρ-functional equations associated with the quadratic rho-functional inequalities (0.1) and (0.2) in non-Archimedean Banach spaces.
Abstract
in this paper, we solve the following quadratic rho-functional inequalities \begin{eqnarray} \left\| f\left(\frac{x+y+z}{2}\right)+f\left(\frac{x-y-z}{2}\right)+f\left(\frac{y-x-z}{2}\right)+f\left(\frac{z-x-y}{2}\right) -f(x) -f(y) -f(z) \right\| \leq \| \rho (f(x+y+z) + f(x-y-z) +f(y-x-z)+f(z-x-y) -4f(x)-4f(y)-4f(z)\|, \end{eqnarray} and \begin{eqnarray} \| f(x+y+z) + f(x-y-z)+f(y-x-z)+f(z-x-y) -4f(x)-4f(y) -4f(z) \| \leq \left \| \rho \left( f\left(\frac{x+y+z}{2}\right)+f\left(\frac{x-y-z}{2}\right) +f\left(\frac{y-x-z}{2}\right)+f\left(\frac{z-x-y}{2}\right)-f(x)-f(y)-f(z) \right) \right\|,\end{eqnarray} where rho is a fixed non-Archimedean number with |rho| < |8|. Using the direct method, we prove the Hyers-Ulam stability of the quadratic rho-functional inequalities (0.1) and (0.2) in non-Archimedean Banach spaces and prove the Hyers-Ulam stability of quadratic ρ-functional equations associated with the quadratic rho-functional inequalities (0.1) and (0.2) in non-Archimedean Banach spaces.
참고문헌 (16)
- T. Aoki, "On the stability of the linear transformation in Banach spaces", J. Math. Soc. Japan, 2, pp.64~66 (1950)
- P.W. Cholewa, "Remarks on the stability of functional equations", Aequationes Math, 27, pp.76~86 (1984)
- G.Z. Eskandani, "Approximation of a general cubic functional equation in Felbin's type fuzzy normed linear spaces", Results Math, 66, pp.113~123 (2014)
- W. Fechner, "Stability of a functional inequalities associated with the Jordan-von Neu-mann functional equation", Aequationes Math, 71, pp.149~161 (2006)
- P. Gavruta, "A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings", J. Math. Anal. Appl, 184, pp.431~443 (1994)
- A. Gilanyi, "Eine zur Parallelogrammgleichung aquivalente Ungleichung", Aequationes Math, 62, pp.303~309 (2001)
- A. Gilanyi, "On a problem by K. Nikodem", Math. Inequal. Appl, 5, pp.707~710 (2002)
- D.H. Hyers, "On the stability of the linear functional equation", Proc. Natl. Acad. Sci. U.S.A, 27, pp.222~224 (1941)
- Pl. Kannappan, "Quadratic functional equation and inner product spaces", Results Math, 27, pp.368~372 (1995)
- J. Lee, "Stability of functional equations in matrix random normed spaces: a fixed point approach", Results Math, 66, pp.99~112 (2014)
- M.S. Moslehian, "A Mazur-Ulam theorem in non-Archimedean normed spaces", Nonlinear Anal.-TMA, 69, pp.3405~3408 (2008)
- C. Park, "Functional inequalities associated with Jordan-von Neumann-type additive functional equations", J. Inequal. Appl, 2007, pp.13~ (2007)
- Th.M. Rassias, "On the stability of the linear mapping in Banach spaces", Proc. Amer. Math. Soc, 72, pp.297~300 (1978)
- J. Ratz, "On inequalities associated with the Jordan-von Neumann functional equation", Aequationes Math, 66, pp.191~200 (2003)
- 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