• The Pure and Applied Mathematics
• /
• v.29 no.2
• /
• pp.189-199
• /
• 2022

In this paper, we introduce a new generalized (a, b)-cubic Euler-Lagrange-Jensen functional equation and obtain its general solution. Furthermore, we prove the Hyers-Ulam stability of the new generalized (a, b)-cubic Euler-Lagrange-Jensen functional equation in orthogonality normed spaces.

• The Pure and Applied Mathematics
• /
• v.8 no.2
• /
• pp.153-162
• /
• 2001

In this Paper, the notion of $\varepsilon$-Birkhoff orthogonality introduced by Dragomir [An. Univ. Timisoara Ser. Stiint. Mat. 29(1991), no. 1, 51-58] in normed linear spaces has been extended to metric linear spaces and a decomposition theorem has been proved. Some results of Kainen, Kurkova and Vogt [J. Approx. Theory 105 (2000), no. 2, 252-262] proved on e-near best approximation in normed linear spaces have also been extended to metric linear spaces. It is shown that if (X, d) is a convex metric linear space which is pseudo strictly convex and M a boundedly compact closed subset of X such that for each $\varepsilon$>0 there exists a continuous $\varepsilon$-near best approximation $\phi$ : X → M of X by M then M is a chebyshev set .

• Bulletin of the Korean Mathematical Society
• /
• v.45 no.3
• /
• pp.601-606
• /
• 2008

The purpose of this paper is to introduce and discuss the concept of ${\omega}$-Chebyshev subspaces in Banach spaces. The concept of quasi Chebyshev in Banach space is defined. We show that ${\omega}$-Chebyshevity of subspaces are a new class in approximation theory. In this paper, also we consider orthogonality in normed spaces.

• Korean Journal of Mathematics
• /
• v.20 no.1
• /
• pp.33-46
• /
• 2012

Using the direct method, we prove the Hyers-Ulam stability of the orthogonally additive-quartic functional equation (0.1) $f(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+10f(x)+14f(-x)-3f(y)-3f(-y)$ for all $x$, $y$ with $x{\perp}y$, in non-Archimedean Banach spaces. Here ${\perp}$ is the orthogonality in the sense of R$\ddot{a}$tz.

• Korean Journal of Mathematics
• /
• v.21 no.1
• /
• pp.1-21
• /
• 2013

Using the fixed point method, we prove the Ulam-Hyers stability of the orthogonally additive and orthogonally quadratic functional equation $$f(\frac{x}{2}+y)+f(\frac{x}{2}-y)+f(\frac{x}{2}+z)+f(\frac{x}{2}-z)=\frac{3}{2}f(x)-\frac{1}{2}f(-x)+f(y)+f(-y)+f(z)+f(-z)$$ (0.1) for all $x$, $y$, $z$ with $x{\bot}y$, in orthogonality Banach spaces and in non-Archimedean orthogonality Banach spaces.

• Bulletin of the Korean Mathematical Society
• /
• v.43 no.2
• /
• pp.377-387
• /
• 2006

In this paper, some characterizations of representation for continuous linear functionals on $2{\kappa}$-inner product spaces in terms of best approximations and orthogonalities are given.