Jaiok Roh

• Hallym University

In this paper, we present a uniqueness theorem obtained by using direct calculation. This theorem is applicable to stability problems of functional equations whose solutions are monomial or generalized polynomial mappings of degree n. The advantage of this uniqueness theorem is that it simplifies the proof by eliminating the need to repeatedly and...

In this paper, we introduce a way of representing a given mapping as the sum of odd and even mappings. Then, using this representation, we investigate the stability of various forms of the following general nonic functional equation: ∑i=01010Ci(−1)10−if(x+iy)=0.

In this paper, we introduce a way of representing a given mapping as the sum of odd and even mappings. Then, by using this representation, we investigate the stability of various forms for the general nonic functional equation.

In this article, we study the stability of various forms for the general octic functional equation ∑ i = 09 9 C i − 1 9 − i f x + i y = 0 . We first find a special way of representing a given mapping as the sum of eight mappings. And by using the above representation, we will investigate the hyperstability of the general octic functional equation....

In this paper, we investigate the transferred superstability

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If a mapping can be expressed by sum of a septic mapping, a sextic mapping, a quintic mapping, a quartic mapping, a cubic mapping, a quadratic mapping, an additive mapping, and a constant mapping, we say that it is a general septic mapping. A functional equation is said to be a general septic functional equation provided that each solution of that...

In this paper, we will consider the generalized sextic functional equation And by applying the fixed point theorem in the sense of Cdariu and Radu, we will discuss the stability of the solutions for this functional equation. 1. Introduction In 1940, Ulam [1] remarked the problem concerning the stability of group homomorphisms. In 1941, Hyers [2] g...

In this paper, we consider the generalized sextic functional equation \begin{align*} \sum_{i=0}^{7}{}_7 C_{i} (-1)^{7-i}f(x+iy) = 0. \end{align*} And by applying the fixed point theory in the sense of L. C\u adariu and V. Radu, we will discuss the stability of the solutions for this functional equation.

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u″(x)+αu′(x)+βu(x)=r(x), with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitab...

In this paper, we will apply the operator method to prove the generalized Hyers-Ulam stability of the wave equation, utt(x,t)−c2▵u(x,t)=f(x,t), for a class of real-valued functions with continuous second partial derivatives. Finally, we will discuss the stability more explicitly by giving examples.

In this paper, we will consider the stationary Stokes equations with the periodic boundary condition and we will study approximation property of the solutions by using the properties of the Fourier series. Finally, we will discuss that our estimation for approximate solutions is optimal .

In this paper, we want to see the properties of the smooth solutions u of the incompressible flows on an exterior domain Ω of R2. Specially, when the vorticity ω=∇×u has a bounded support, with suitable conditions we will show that there exists a constant C(p,q) such that ∥u∥Lp(Ω)≤C∥u∥Lq(Ω) for 1<p≤q≤∞.

In Jung and Roh (2017), we investigated some properties of approximate solutions of the second-order inhomogeneous linear differential equations. In this paper as an application of above paper we will study the stability of the time independent Schrödinger equation with a potential box of finite walls.

We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take ce...

We investigate some properties of approximate solutions for the second -order inhomogeneous linear differential equations, y"(x) + alpha y'(x) + beta y(x) = r(x), with complex constant coefficients. And, as an application of our results, we will see the time independent Schrodinger equations. This paper was motivated by the paper, Li and Shen (2010...

In this paper, we will obtain the optimal Hyers-Ulam’s constant for the first-order linear differential equations \(p(t)y'(t) - q(t)y(t) - r(t) = 0\).

We will consider a continuously differentiable function y : I → R satisfying the inequality p t y ′ t - q t y t - r t ≤ ε for all t ∈ I and y t 0 - α ≤ δ for some t 0 ∈ I and some α ∈ R . Then we will approximate y by a solution z of the linear equation p t z ′ t - q t z t - r ( t ) = 0 with z ( t 0 ) = α .

In this paper, we consider the smooth solutions , with suitable decay at infinity, of the Euler equations on . We assume that the initial vorticity has a compact support which leads by Marchioro (1994, 1996) to a consequence that the support of the vorticity for any finite time is a bounded set. Then, we will show that the decay(growth) rate, in sp...

We consider the derivations on noncommutative Banach algebras, and we will first study the conditions for a derivation on noncommutative Banach algebra. Then, we examine the stability of functional inequalities with a derivation. Finally, we take the derivations with the radical ranges on noncommutative Banach algebras.

We take into account the stability of ring homomorphism and ring derivation in intuitionistic fuzzy Banach algebra associated with the Jensen functional equation. In addition, we deal with the superstability of functional equation in an intuitionistic fuzzy normed algebra with unit.

In this paper, we study the stability of stationary solutions w for the Navier–Stokes flows in an exterior domain with zero velocity at infinity. With suitable assumptions of w, by the works of Chen (1993), Kozono–Ogawa (1994) and Borchers–Miyakawa (1995), if u0−w∈Lr(Ω)∩L3(Ω)u0−w∈Lr(Ω)∩L3(Ω) then one can obtain‖u(t)−w‖p=O(t−32(1r−1p))for 1<r<p<∞,‖∇...

Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) in L 2 space and by Shibata (1999) and Enomoto-Shibata (2005) in L p spaces for p ≥ 3 . However, their results did not include enough information to find the spatial decay. S...

We consider the stability of stationary solutions w for the exterior Navier-Stokes flows with a nonzero constant velocity u ∞ at infinity. For u ∞ = 0 with nonzero stationary solution w , Chen (1993), Kozono and Ogawa (1994), and Borchers and Miyakawa (1995) have studied the temporal stability in L p spaces for 1 < p and obtained good stability dec...

In this paper, we prove some decay properties of global solutions for the Navier–Stokes equations in an exterior domain Ω⊂Rn, n=2,3.When a domain has a boundary, the pressure term is troublesome since we do not have enough information on the pressure near the boundary. To overcome this difficulty, by multiplying a special form of test functions, we...

Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studie...

In the present paper, we establish the stability and the superstability of a functional inequality corresponding to the functional equation f n (xyx) = Σi+j+k=n f i (x)f j (y)f k(x). In addition, we take account of the problem of Jacobson radical ranges for such functional inequality.

For a strong solution u(x,t) of the Navier–Stokes equations in exterior domain Ω in Rn where n=2,3, we study the time decay of ‖|x|αu(t)‖Lp for αn. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in [H.-O. Bae...

The 2D g-Navier-Stokes equations have the following form, ∂u ∂tδ νδu + (u · ∇)u + ∇p = f, in Ω with the continuity equation ∇·(gu)=0, inΩ, where g is a smooth real valued function. We get the Navier-Stokes equations, for g = 1. In this paper, we investigate solutions {ug, pg} of the g-NavierStokes equations, as g → 1 in some suitable spaces.

The functional inequality ‖f(x)+2f(y)+2f(z)‖≤‖2f(x/2+y+z)‖+ϕ  (x,y,z) (x,y,z∈G) is investigated, where G is a group divisible by 2,f:G→X and ϕ:G3→[0,∞) are mappings, and X is a Banach space. The main result of the paper states that the assumptions above together with (1) ϕ(2x,âˆ...

It is well-known that the concept of Hyers-Ulam-Rassias stability originated by Th. M. Rassias (Proc. Amer. Math. Soc. 72(1978), 297-300) and the concept of Ulam-Gavruta-Rassias stability by J. M. Rassias (J. Funct. Anal. U.S.A. 46(1982), 126-130; Bull. Sc. Math. 108 (1984), 445-446; J. Approx. Th. 57 (1989), 268-273) and P. Gavruta (``An answer t...

In this paper, we prove that a function satisfying the following inequality for all x, y, z X and for , is Cauchy additive. Moreover, we will investigate for the stability in Banach spaces.

The g-Navier–Stokes equations in spatial dimension 2 were introduced by Roh as∂u∂t−νΔu+(u⋅∇)u+∇p=f, with the continuity equation∇⋅(gu)=0, where g is a suitable smooth real valued function. Roh proved the existence of global solutions and the global attractor, for the spatial periodic and Dirichlet boundary conditions. Roh also proved that the globa...

The 2D g-Navier-Stokes equations are a certain modi-fied Navier-Stokes equations and have the following form, ∂u ∂t − ν∆u + (u · + = f , in Ω with the continuity equation · (gu) = 0, in Ω, where g is a suitable smooth real valued function. In this paper, we will derive 2D g-Navier-Stokes equations from 3D Navier-Stokes equations. In addition, we wi...

In this paper, we study the two dimensional g-Navier?Stokes equations on some unbounded domain {\Omega}\;{\subset}\;R^2. We prove the existence of the global attractor for the two dimensional g-Navier?Stokes equations under suitable conditions. Also, we estimate the dimension of the global attractor. For this purpose, we exploit the concept of asym...

The 2D g-Navier–Stokes equations has the following form:∂u∂t-νΔu+(u·∇)u+∇p=fwith the continuity equation∇·(gu)=0,where g is a suitable smooth real-valued function. For the restricted function g, Roh showed the existence of the global attractors for the periodic boundary conditions. One note that we get the 2D Navier–Stokes equations for g=1.Therefo...

The g-Navier-Stokes equations in spatial dimension 2 are the following equations introuduced in [3] ∂u/∂t - vΔu + (u · ∇)u + ∇p = f, with the continuity equation 1/g∇·(gu) = 0. Here, we show the existence and uniqueness of solutions of g-Navier-Stokes equations on Rn for n = 2, 3.