• Journal of the Korean Mathematical Society
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• v.36 no.6
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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.

• Journal of applied mathematics & informatics
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• v.4 no.2
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• pp.573-581
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• 1997

In this paper we prove a weak common fixed point theo-rem in a normed almost linear space which is different from the result of S. P. Singh and B.A. Meade [9]. However for a Banach X our theorem is equal to the result of S. P. Singh and B. A. Meade.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.38 no.2
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• pp.293-302
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• 2001

We improve the greedy algorithm which is one of the general convergence criterion for certain iterative sequence in a given space by building a constructive greedy algorithm on a normed linear space using an arithmetic average of elements. We also show the degree of approximation order is still $Ο(1\sqrt{\n}$) by a bounded linear functional defined on a bounded subset of a normed linear space which offers a good approximation method for neural networks.

• Journal of the Korean Mathematical Society
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• v.21 no.2
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• pp.215-225
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• 1984

• Journal of the Chungcheong Mathematical Society
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• v.25 no.2
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• pp.201-215
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• 2012

In this paper, we prove the stability in random normed spaces via fixed point method for the functional equation $f(x+y+z+w)\;+\;2f(x)\;+\;2f(y)\;+\;2f(z)\;+\;2f(w)\;-\;f(x+y)\;-\;f(x+z)\;-\;f(x+w)\;-\;f(y+z)\;-\;f(y+w)\;-\;f(z+w)=0$.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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• pp.51-63
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• 2012

In this paper, we prove the stability in random normed spaces via fixed point method for the functional equation $$f(2x+y)+f(2x-y)+2f(x)-f(x+y)-f(x-y)-2f(2x)=0$$.

• Communications of the Korean Mathematical Society
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• v.25 no.1
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• pp.51-57
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• 2010

In this paper we introduce and study the lacunary strong Zweier sequence spaces $N_{\theta}^O[Z]$, $N_{\theta}[Z]$ consisting of all sequences x = $(x_k)$ such that (Zx) in the space $N_{\theta}$ and $N_{\theta}^O$ respectively, which is normed. Also, prove that $N_{\theta}^O[Z}$, $N_{\theta}[Z}$, are linearly isomorphic to the space $N_{\theta}^O$ and $N_{\theta}$, respectively. And we study some connections between lacunary strong Zweier sequence and lacunary statistical Zweier convergence sequence.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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• pp.97-103
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• 2012

In this paper, we introduce the notion of the statist-cally fuzzy convergent sequence and the statistical fuzzy ${\alpha}$-Cauchy sequence of fuzzy points in a fuzzy normed linear space. And investigate some related properties.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.3
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• pp.453-464
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• 2016

In this paper, we investigate the solution of the following functional inequality $$N(f(x)+f(y)+f(z),t){\geq}N(f(x+y+z),mt)$$ for some fixed real number m with $\frac{1}{3}$ < m ${\leq}$ 1 and using the fixed point method, we prove the generalized Hyers-Ulam stability for it in fuzzy Banach spaces.