scholarly journals Approximation on the Quadratic Reciprocal Functional Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Abasalt Bodaghi ◽  
Sang Og Kim
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Functional Equations ◽  
Ulam Stability ◽  
Real Numbers ◽  
Ulam Stability Problem

The quadratic reciprocal functional equation is introduced. The Ulam stability problem for anϵ-quadratic reciprocal mappingf:X→Ybetween nonzero real numbers is solved. The Găvruţa stability for the quadratic reciprocal functional equations is established as well.

HEAT KERNEL METHOD FOR THE LEVI-CIVITÁ EQUATION IN DISTRIBUTIONS AND HYPERFUNCTIONS

2015 ◽  
Vol 92 (1) ◽  
pp. 77-93
Author(s):  
JAEYOUNG CHUNG ◽  
PRASANNA K. SAHOO
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Kernel Method ◽  
Functional Inequality ◽  
Complex Numbers ◽  
Ulam Stability ◽  
Real Numbers ◽  
Positive Real ◽  
Image Position ◽  
Ulam Stability Problem

Let$G$be a commutative group and$\mathbb{C}$the field of complex numbers,$\mathbb{R}^{+}$the set of positive real numbers and$f,g,h,k:G\times \mathbb{R}^{+}\rightarrow \mathbb{C}$. In this paper, we first consider the Levi-Civitá functional inequality$$\begin{eqnarray}\displaystyle |f(x+y,t+s)-g(x,t)h(y,s)-k(y,s)|\leq {\rm\Phi}(t,s),\quad x,y\in G,t,s>0, & & \displaystyle \nonumber\end{eqnarray}$$where${\rm\Phi}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$is a symmetric decreasing function in the sense that${\rm\Phi}(t_{2},s_{2})\leq {\rm\Phi}(t_{1},s_{1})$for all$0<t_{1}\leq t_{2}$and$0<s_{1}\leq s_{2}$. As an application, we solve the Hyers–Ulam stability problem of the Levi-Civitá functional equation$$\begin{eqnarray}\displaystyle u\circ S-v\otimes w-k\circ {\rm\Pi}\in {\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})\quad [\text{respectively}\;{\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})] & & \displaystyle \nonumber\end{eqnarray}$$in the space of Gelfand hyperfunctions, where$u,v,w,k$are Gelfand hyperfunctions,$S(x,y)=x+y,{\rm\Pi}(x,y)=y,x,y\in \mathbb{R}^{n}$, and$\circ$,$\otimes$,${\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$and${\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$denote pullback, tensor product and the spaces of bounded distributions and bounded hyperfunctions, respectively.

Solution of the Ulam stability problem for Euler–Lagrange k-quintic mappings

2020 ◽  
Vol 27 (4) ◽  
pp. 585-592
Author(s):  
Syed Abdul Mohiuddine ◽  
John Michael Rassias ◽  
Abdullah Alotaibi
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Functional Equations ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Stability Problems ◽  
Quartic Functional Equation ◽  
Ulam Stability Problem

AbstractThe “oldest quartic” functional equationf(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y)was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly, an interesting “quintic” functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form:2f(2x+y)+2f(2x-y)+f(x+2y)+f(x-2y)=20[f(x+y)+f(x-y)]+90f(x).In this paper, we generalize this “Cho–Kang–Koh equation” by introducing pertinent Euler–Lagrange k-quintic functional equations, and investigate the “Ulam stability” of these new k-quintic functional mappings.

scholarly journals On the stability problem of quadratic functional equations in 2-Banach spaces

2017 ◽  
pp. 5054-5061
Author(s):  
Seong Sik Kim ◽  
Ga Ya Kim ◽  
Soo Hwan Kim
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Stability Problem ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Normed Spaces ◽  
The Stability ◽  
Stability Results

In this paper, we investigate the stability problem in the spirit of Hyers-Ulam, Rassias and G·avruta for the quadratic functional equation:f(2x + y) + f(2x ¡ y) = 2f(x + y) + 2f(x ¡ y) + 4f(x) ¡ 2f(y) in 2-Banach spaces. These results extend the generalized Hyers-Ulam stability results by thequadratic functional equation in normed spaces to 2-Banach spaces.

On Hyers–Ulam Stability of Cauchy and Wilson Equations

2004 ◽  
Vol 11 (1) ◽  
pp. 69-82
Author(s):  
Elhoucien Elqorachi ◽  
Mohamed Akkouchi
Keyword(s):  
Topological Group ◽  
Compact Support ◽  
Stability Problem ◽  
Functional Equations ◽  
Identity Element ◽  
Continuous Functions ◽  
Dirac Measure ◽  
Ulam Stability ◽  
Natural Extensions ◽  
Ulam Stability Problem

Abstract We study the Hyers–Ulam stability problem for the Cauchy and Wilson integral equations where 𝐺 is a topological group, 𝑓, 𝑔 : 𝐺 → ℂ are continuous functions, μ is a complex measure with compact support and σ is a continuous involution of 𝐺. The result obtained in this paper are natural extensions of the previous works concerning the Hyers–Ulam stability of the Cauchy and Wilson functional equations done in the particular case of μ=δe : The Dirac measure concentrated at the identity element of 𝐺.

scholarly journals The Mixed Cubic-Quartic Functional Equation

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bodaghi ◽  
D. Kang ◽  
J.M. Rassias
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Stability Problem ◽  
Ulam Stability ◽  
Quartic Functional Equation ◽  
Ulam Stability Problem

AbstractIn this paper, we obtain the general solution of the following generalized mixed cubic and quartic functional equation f(x + kx) + f(x − ky) = k2{f(x + y) + f(x−y)}−2(k2−1)f(x)−2k2(k2−1)f(y)+ 1/4 k2(k2−1)f(2y), for fixed integers k with k ≠ 0,±1. The Hyers-Ulam stability problem for the mentioned functional equation is also proved.

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