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

The “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.

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

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.

The Mixed Cubic-Quartic Functional Equation

In 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.

Approximation on the Quadratic Reciprocal Functional Equation

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.

Simple Harmonic Oscillator Equation and Its Hyers-Ulam Stability

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.