• 대한수학회보
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• 제50권6호
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• pp.2061-2070
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• 2013

In this paper, we investigate the generalized HyersUlam-Rassias stability of the following additive functional equation $$2\sum_{j=1}^{n}f(\frac{x_j}{2}+\sum_{i=1,i{\neq}j}^{n}\;x_i)+\sum_{j=1}^{n}f(x_j)=2nf(\sum_{j=1}^{n}x_j)$$, in quasi-${\beta}$-normed spaces.

• Korean Journal of Mathematics
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• 제27권1호
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• pp.81-92
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• 2019

We introduce a reciprocal-negative Fermat's equation generalized with constants coefficients and investigate its stability in a quasi-${\beta}$-normed space.

• 충청수학회지
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• 제32권1호
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• pp.1-14
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• 2019

We prove the generalized Hyers-Ulam-Rassias stability problem of the quadratic functional equation with involution in the fuzzy quasi ${\beta}$-normed space by using the fixed point method.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제26권2호
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• pp.85-97
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• 2019

In this paper we introduce the reciprocal-negative Fermat's equation induced by the famous equation in the Fermat's Last Theorem, establish the general solution in the simplest cases and the differential solution to the equation, and investigate, then, the generalized Hyers-Ulam stability in a $quasi-{\beta}-normed$ space with both the direct estimation method and the fixed point approach.

• 충청수학회지
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• 제35권2호
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• pp.149-159
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• 2022

In this note, we prove some theorems concerning the stability of functional inequality associated with additive mappings on quasi-𝛽-normed spaces.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1421-1433
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• 2011

In this paper, using a fixed point approach, the generalized Hyeres-Ulam stability of the following quadratic functional equation $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=3(f(x)+f(y)+f(z))$ will be studied, where f is a function from abelian group G into a quasi-${\beta}$-normed space and ${\sigma}$ is an involution on the group G. Next, we consider its pexiderized equation of the form $f(x+y+z)+f(x+{\sigma}(y))+f(y+{\sigma}(z))+f(x+{\sigma}(z))=g(x)+g(y)+g(z)$ and its generalized Hyeres-Ulam stability.

• 충청수학회지
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• 제32권1호
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• pp.29-46
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• 2019

We introduce a generalized cubic functional equation and investigate the Hyers-Ulam stability of the cubic functions as solutions to the generalized cubic functional equation on a quasi-fuzzy anti-${\beta}$-Banach space by both the direct method and the fixed point method.

• 대한수학회지
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• 제36권6호
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• pp.1061-1073
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• 1999

Let X be a real normed linear space. Let T : D(T) ⊂ X \longrightarrow X be a uniformly continuous and ∮-strongly quasi-accretive mapping. Let {${\alpha}$n}{{{{ { }`_{n=0 } ^{$\infty$ } }}}} , {${\beta}$n}{{{{ { }`_{n=0 } ^{$\infty$ } }}}} be two real sequences in [0, 1] satisfying the following conditions: (ⅰ) ${\alpha}$n \longrightarrow0, ${\beta}$n \longrightarrow0, as n \longrightarrow$\infty$ (ⅱ) {{{{ SUM from { { n}=0} to inf }}}} ${\alpha}$=$\infty$. Set Sx=x-Tx for all x $\in$D(T). Assume that {u}{{{{ { }`_{n=0 } ^{$\infty$ } }}}} and {v}{{{{ { }`_{n=0 } ^{$\infty$ } }}}} are two sequences in D(T) satisfying {{{{ SUM from { { n}=0} to inf }}}}∥un∥<$\infty$ and vn\longrightarrow0 as n\longrightarrow$\infty$. Suppose that, for any given x0$\in$X, the Ishikawa type iteration sequence {xn}{{{{ { }`_{n=0 } ^{$\infty$ } }}}} with errors defined by (IS)1 xn+1=(1-${\alpha}$n)xn+${\alpha}$nSyn+un, yn=(1-${\beta}$n)x+${\beta}$nSxn+vn for all n=0, 1, 2 … is well-defined. we prove that {xn}{{{{ { }`_{n=0 } ^{$\infty$ } }}}} converges strongly to the unique zero of T if and only if {Syn}{{{{ { }`_{n=0 } ^{$\infty$ } }}}} is bounded. Several related results deal with iterative approximations of fixed points of ∮-hemicontractions by the ishikawa iteration with errors in a normed linear space. Certain conditions on the iterative parameters {${\alpha}$n}{{{{ { }`_{n=0 } ^{$\infty$ } }}}} , {${\beta}$n}{{{{ { }`_{n=0 } ^{$\infty$ } }}}} and t are also given which guarantee the strong convergence of the iteration processes.

• 충청수학회지
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• 제33권3호
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• pp.319-326
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• 2020

In this paper, we establish a general solution of the following functional equation $$mf\({\sum\limits_{k=1}^{n}}x_k\)+{\sum\limits_{t=1}^{m}}f\({\sum\limits_{k=1}^{n-i_t}}x_k-{\sum\limits_{k=n-i_t+1}^{n}}x_k\)=2{\sum\limits_{t=1}^{m}}\(f\({\sum\limits_{k=1}^{n-i_t}}x_k\)+f\({\sum\limits_{k=n-i_t+1}^{n}}x_k\)\)$$ where m, n, t, i_t ∈ ℕ such that 1 ≤ t ≤ m < n. Also, we study Hyers-Ulam-Rassias stability for the generalized quadratic functional equation with n-variables and m-combinations form in quasi-𝛽-normed spaces and then we investigate its application.

• 충청수학회지
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• 제26권4호
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• pp.757-767
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• 2013

We investigate the general solution of the following functional equation and the generalized Hyers-Ulam-Rassias stability problem in quasi ${\beta}$-normed spaces and then the stability by using alternative fixed point method for the following quintic function $f:X{\rightarrow}Y$ such that f(3x+y)+f(3x-y)+5[f(x+y)+f(x-y)]=4[f(2x+y)+f(2x-y)]+2f(3x)-246f(x), for all $x,y{\in}X$.