• Korean Journal of Mathematics
• /
• v.28 no.2
• /
• pp.205-221
• /
• 2020

Paranormed spaces are important as a generalization of the normed spaces in terms of having more general properties. The aim of this study is to introduce a new paranormed space |𝜙_z|(p) over the paranormed space ℓ(p) using Euler totient means, where p = (p_k) is a bounded sequence of positive real numbers. Besides this, we investigate topological properties and compute the α-, β-, and γ duals of this paranormed space. Finally, we characterize the classes of infinite matrices (|𝜙_z|(p), λ) and (λ, |𝜙_z|(p)), where λ is any given sequence space.

• Journal of the Chungcheong Mathematical Society
• /
• v.26 no.2
• /
• pp.287-296
• /
• 2013

In this paper, we investigate additive functional inequalities in paranormed spaces. Furthermore, we prove the Hyers-Ulam stability of additive functional inequalities in paranormed spaces.

• Korean Journal of Mathematics
• /
• v.29 no.1
• /
• pp.169-177
• /
• 2021

The aim of the present study is to introduce the concepts of deferred statistical convergence, deferred statistical Cauchy sequence and deferred Cesàro summability in paranormed spaces. We investigate some properties of these concepts and some inclusion relations with examples.

• Korean Journal of Mathematics
• /
• v.24 no.1
• /
• pp.41-49
• /
• 2016

In this paper, using xed point method, we prove the Hyers-Ulam stability of the following functional equation $$\hspace{15}f+(x+y+z)+f({\sigma}(x)+y+z)+f(x+{\sigma}(y)+z)+f(x+y+{\sigma}(z))\\=4f(x)+4f(y)+4f(z)$$ with involution in paranormed spaces.

• Kyungpook Mathematical Journal
• /
• v.63 no.2
• /
• pp.235-250
• /
• 2023

Let 1 ≤ p, q < ∞ be fixed, and let R = [r_jk] be an infinite scalar matrix such that 1 ≤ r_jk < ∞ and sup_j,k r_jk < ∞. Let 𝓑(𝑙_p, 𝑙_q) be the set of all bounded linear operator from 𝑙_p into 𝑙_q. For a fixed Banach algebra 𝐁 with identity, we define a new vector space S^R_p,q(𝐁) of infinite matrices over 𝐁 and a paranorm G on S^R_p,q(𝐁) as follows: let $$S^R_{p,q}({\mathbf{B}})=\{A:A^{[R]}{\in}{\mathcal{B}}(l_p,l_q)\}$$ and $G(A)={\parallel}A^{[R]}{\parallel}^{\frac{1}{M}}_{p,q}$, where $A^{[R]}=[{\parallel}a_{jk}{\parallel}^{r_{jk}}]$ and M = max{1, sup_j,k r_jk}. The existance of S^R_p,q(𝐁) equipped with the paranorm G(·) including its completeness are studied. We also provide characterizations of β -dual of the paranormed space.

• Kyungpook Mathematical Journal
• /
• v.55 no.4
• /
• pp.909-931
• /
• 2015

This paper presents some new paranormed sequence spaces $X(r,s,t,p;{\Delta})$ where $X{\in}\{l_{\infty}(p),c(p),c_0(p),l(p)\}$ defined by using generalized means and difference operator. It is shown that these are complete linear metric spaces under suitable paranorms. Furthermore, the ${\alpha}$-, ${\beta}$-, ${\gamma}$-duals of these sequence spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r,s,t,p;{\Delta})$ to X. Finally, it is proved that the sequence space $l(r,s,t,p;{\Delta})$ is rotund when $p_n$ > 1 for all n and has the Kadec-Klee property.

• Kyungpook Mathematical Journal
• /
• v.54 no.1
• /
• pp.73-86
• /
• 2014

In the present paper we introduce difference paranormed sequence spaces $c_0(\mathcal{M},{\Delta}^n_m,p,u,{\parallel}{\cdot},{\cdots},{\cdot}{\parallel})$, $c(\mathcal{M},{\Delta}^n_m,p,u,{\parallel}{\cdot},{\cdots},{\cdot}{\parallel})$ and $l_{\infty}(\mathcal{M},{\Delta}^n_m,p,u,{\parallel}{\cdot},{\cdots},{\cdot}{\parallel})$ defined by a Musielak-Orlicz function $\mathcal{M}$ = $(M_k)$ over n-normed spaces. We also study some topological properties and some inclusion relations between these spaces.