scholarly journals Legendre's Differential Equation and Its Hyers-Ulam Stability

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Soon-Mo Jung
Keyword(s):  
Differential Equation ◽  
Stability Problem ◽  
Partial Solution ◽  
Ulam Stability ◽  
Ulam Stability Problem

We solve the nonhomogeneous Legendre's differential equation and apply this result to obtaining a partial solution to the Hyers-Ulam stability problem for the Legendre's equation.

scholarly journals Simple Harmonic Oscillator Equation and Its Hyers-Ulam Stability

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Soon-Mo Jung ◽  
Byungbae Kim
Keyword(s):  
Harmonic Oscillator ◽  
Stability Problem ◽  
Partial Solution ◽  
Ulam Stability ◽  
Simple Harmonic Oscillator ◽  
Ulam Stability Problem

We solve the inhomogeneous simple harmonic oscillator equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the simple harmonic oscillator equation.

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.

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.

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