arXiv subjects
Publications and source records attributed to Choonkil Park.
This paper explores the Hyers-Ulam stability of generalized Jensen additive and quadratic functional equations in \(β\)-homogeneous \(F\)-space, showing that approximately satisfying mappings have a unique exact approximating counterpart within a specific bound.
In this paper, we introduce the idea of $\ast$-homomorphism on a Hilbert $C^{*}$-module. Furthermore, we prove the Hyers-Ulam stability of homomorphisms and $\ast$-homomorphisms on Hilbert $C^{*}$-modules using the fixed point method.
In this paper, we present a new concept of interpolative contraction mappings in $C^{\ast}$-algebra valued complete metric space and we prove the existence of fixed points and common fixed points for Kannan-Riech type contractions.
Using the direct method, we prove the generalised Hyers-Ulam stability of the following functional equation \begin{equation} ϕ(x+y, z+w)+ϕ(x-y, z-w)-2 ϕ(x, z)-2 ϕ(x, w)=0 \end{equation} in modular space satisfying the Fatou property or $Δ_2$-condition.
Recently, Bemrose et al. \cite{BE} developed a theory of weaving frames, which was motivated by a problem regarding distributed signal processing. In this present article, we introduce the atomic $g$-system and we generalize some of the known results in continuous $L$-frames, weaving continuous and weaving continuous $ g$-frames, also we study weaving continuous $ L$-$g$-frames in Hilbert spaces. Moreover, we study the behaviour continuous $ L$-$g$-frames under some perturbations, and we show that approximate $L$-duals are stable under small perturbation and that it is possible to remove some elements of a woven continuous $ L$-$g$-frame and still have a woven continuous $ L$-$g$-frame.
In this paper, we present controlled finite continuous frames in a finite dimensional Hilbert space and we study some properties of them. Parseval controlled integral frames are presented and we characterize operators that construct controlled integral finite frames.
Let $\mathcal{A}$ and $\mathcal{B}$ be two algebras and let $n$ be a positive integer. A linear mapping $D:\mathcal{A} \rightarrow \mathcal{B}$ is called a \emph{strongly generalized derivation of order $n$} if there exist families of linear mappings $\{E_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$, $\{F_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$, $\{G_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$ and $\{H_k:\mathcal{A} \rightarrow \mathcal{B}\}_{k = 1}^{n}$ which satisfy $D(ab) = \sum_{k = 1}^{n}\left[E_k(a) F_k(b) + G_k(a)H_k(b)\right]$ for all $a, b \in \mathcal{A}$. The purpose of this article is to study the automatic continuity of such derivations on Banach algebras and $C^{\ast}$-algebras.
The goal of our work is to investigate the oscillation and asymptotic properties of a class of difference equations with a condition. In contrast to most previous studies, the oscillation of the investigated equation is obtained with only one condition.
The concept of operator frame can be considered as a generalization of frame. Firstly, we introduce the notion of operator frame for the set of all adjointable operators $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$ on a Hilbert pro-$C^{\ast}$-module $\mathcal{X}.$ The analysis operator, the synthesis operator and the frame operator are presented. Secondly, we study the stability of operator frame under small perturbations. We also study the tensor product of operator frame for Hilbert pro-$C^{\ast}$-modules. Finally, we establish its dual and some properties.
In this paper, we introduce and investigate general 3-variable Jensen $ρ$-functional equation, and prove the Hyers-Ulam stability of the Jensen functional equations associated with the general 3-variable Jensen $ρ$-functional inequalities in complex Banach spaces.
The main purpose of this paper is to determine the solution of generalized convex set-valued mappings satisfying certain functional equation. Some conclusions of stability of set-valued functional equations are obtained.
In this paper we establish the Ulam stability of Jensen functional inequality in some classes of groups.
In this paper, we solve the following tri-additive $s$-functional inequalities \begin{eqnarray}\label{0.1} && \nonumber \| f(x+y, z-w, a+b) + f(x-y, z+w, a-b) \\ && \nonumber\qquad -2 f(x, z, a) + 2 f(x, w, b) -2f(y, z, b) +2 f(y, w, a)\| \\ && \quad \le \left \|s \left(2f\left(\frac{x+y}{2}, z-w, a+b \right) + 2f\left(\frac{x-y}{2}, z+w, a-b\right) \right. \right. \\ && \qquad \left. \left. -2 f(x, z, a) + 2 f(x, w, b) -2f(y, z, b) +2 f(y, w, a)\right)\right\| , \nonumber \end{eqnarray} \begin{eqnarray}\label{0.2} && \nonumber \left\|2f\left(\frac{x+y}{2}, z-w, a+b \right) + 2f\left(\frac{x-y}{2}, z+w, a-b\right) \right. \\ && \nonumber \qquad \left. -2 f(x, z, a) + 2 f(x, w, b) -2f(y, z, b) +2 f(y, w, a)\right\| \\ && \quad \le \|s ( f(x+y, z-w, a+b) + f(x-y, z+w, a-b) \\ && \nonumber\qquad -2 f(x, z, a) + 2 f(x, w, b) -2f(y, z, b) +2 f(y, w, a) )\| , \end{eqnarray} where $s$ is a fixed nonzero complex number with $|s |< 1$. Moreover, we prove the Hyers-Ulam stability and hyperstability of permuting triderivations and permuting trihomomorphisms in Banach algebras and unital $C^*$-algebras, associated with the tri-additive $s$-functional inequalities {\rm (\ref{0.1})} and {\rm (\ref{0.2})}.
By using the Ishikawa iterative algorithm, we approximate the fixed points and the best proximity points of a relatively non expansive mapping. Also, we use the von Neumann sequence to prove the convergence result in a Hilbert space setting. A comparison table is prepared using a numerical example which shows that the Ishikawa iterative algorithm is faster than some known iterative algorithms such as Picard and Mann iteration.
In this paper, we introduce and solve the following additive-additive $(s,t)$-functional inequality \begin{eqnarray}\label{0.1} && \|g\left(x+y\right) -g(x) -g(y)\| +\| h(x+y) + h(x-y) -2 h(x) \| && \le \left\|s\left( 2 g\left(\frac{x+y}{2}\right)-g(x)-g(y)\right)\right\|+ \left\|t \left( 2h\left(\frac{x+y}{2}\right)+ 2h \left(\frac{x-y}{2}\right)- 2h (x)\right) \right\| , \nonumber \end{eqnarray} where $s$ and $t$ are fixed nonzero complex numbers with $|s| <1$ and $ |t| <1$. Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of Lie bracket derivation-derivations in complex Banach algebras, associated to the additive-additive $(s,t)$-functional inequality (\ref{0.1}) and the following functional inequality \begin{eqnarray} \label{0.2}\| [g, h](xy)-[g,h](x) y- x [g,h](y) \| +\| h(xy) - h(x) y -x h(y) \| \le φ(x,y). \end{eqnarray}
In this paper, we proved the generalized Hyers-Ulam stability of homomorphisms in $C^*$- ternary algebras and of derivations on $C^*$-ternary algebras for the following Cauchy- Jensen functional equation $$3f\bigg(\frac{x+y+z}{3}\bigg)=2f\bigg(\frac{x+y}{2}\bigg)+f(z).$$ These were applied to investigate isomorphisms between $C^*$-ternary algebras.