• Journal of the Chungcheong Mathematical Society
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• v.29 no.2
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• pp.287-328
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• 2016

In the current work, we define and find the general solution of the decic functional equation g(x + 5y) - 10g(x + 4y) + 45g(x + 3y) - 120g(x + 2y) + 210g(x + y) - 252g(x) + 210g(x - y) - 120g(x - 2y) + 45g(x - 3y) - 10g(x - 4y) + g(x - 5y) = 10!g(y) where 10! = 3628800. We also investigate and establish the generalized Ulam-Hyers stability of this functional equation in Banach spaces, generalized 2-normed spaces and random normed spaces by using direct and fixed point methods.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.2
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• pp.207-211
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• 2006

We prove the Riesz theorem in linear n-normed spaces.

• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.339-352
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• 2012

We prove the stability for the systems of quadratic-cubic and additive-quadratic-cubic functional equations with constant coefficients on the probabilistic normed spaces (briefly PN spaces).

• Journal of the Chungcheong Mathematical Society
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• v.22 no.1
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• pp.65-71
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• 2009

Let f be an 2-isometry on a non-Archimedean 2-normed space. In this paper, we prove that the barycenter of triangle is invariant for f up to the translation by f(0), in this case, needless to say, we can imply naturally the Mazur-Ulam theorem in non-Archimedean 2-normed spaces.

• Honam Mathematical Journal
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• v.46 no.2
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• pp.291-306
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• 2024

The concept of lacunary statistical convergence of triple sequences with respect to gradual 2-normed linear spaces is introduced in this research. We learn about its link to some inclusion and fundamental properties. The notion of lacunary statistical Cauchy triple sequences is introduced in the conclusion, and it is demonstrated that it is equivalent to the idea of lacunary statistical convergence.

• Kyungpook Mathematical Journal
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• v.49 no.2
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• pp.289-297
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• 2009

We establish a new method to prove Hyers-Ulam-Rassias stability of the quartic functional equation f(2x + y) + f(2x - y) + 6f(y) = 4[f(x + y) + f(x - y) + 6f(x)] in non-Archimedean normed linear spaces.

• Bulletin of the Korean Mathematical Society
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• v.21 no.1
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• pp.1-5
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• 1984

The notion of linear 2-normed spaces was introduced by S. Gahler ([8,9,10,11]), and these space have been extensively studied by C. Diminnie, R. Ehret, S. Gahler, K. Iseki, A. White, Jr, and others. For nonzero vectors x,y in X, let V(x,y) denote the subspace of X generated by x and y. A linear 2-normed space (X,v) is said to be strictly convex ([3]) if v(x+y,z)=v(x,z)+v(y+z) and z not.mem.V(x,y) imply that y=ax for some a>0. Some characterizations of strict convexity for linear 2-normed spaces are given in [1,3,4,5,12]. Also, a linear 2-normed space (X,v) is said to be strictly 2-convex ([6]) if v(x,y)=v(x,z)=v(y,z)=1/3v(x+z, y+z)=1 implies that z=x+y. These space have been studied in [2,4,6,13]. It is easy to see that every strictly convex linear 2-normed space is always strictly 2-convex but the converse is not necessarily true. Throughout this paper, let (X,v) denote a linear 2-normed space.

• Honam Mathematical Journal
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• v.35 no.1
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• pp.93-100
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• 2013

In this paper, we investigate the stability for the functional equation $$2f(x+y)+f(x-y)+f(y-x)-f(2x)-f(2y)=0$$ in non-Archimedean normed spaces.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.367-375
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• 2013

In this paper, we investigate the stability for the functional equation $$f(x+y+z)+f(x-y)+f(x-z)-f(x-y-z)-f(x+y)-f(x+z)=0$$ in non-Archimedean normed spaces.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1631-1637
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• 2013

This paper generalizes the Aleksandrov problem and Mazur Ulam theorem to the case of $n$-normed spaces. For real $n$-normed spaces X and Y, we will prove that $f$ is an affine isometry when the mapping satisfies the weaker assumptions that preserves unit distance, $n$-colinear and 2-colinear on same-order.