The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $$f(t,vy,wy+uvz)=f(x,y,z)u^{2}v+g(t,x,u,v,w)vz+h(t,x,u,v,w)y+2uwz$$ is solved on $ℝ$ for $y\ne 0$, $v\ne 0.$

This paper deals with some characterizations of gradient-like continuous random dynamical systems (RDS). More precisely, we establish an equivalence with the existence of random continuous section or with the existence of continuous and strict Liapunov function. However and contrary to the deterministic case, parallelizable RDS appear as a particular case of gradient-like RDS.The obtained results are generalizations of well-known analogous theorems in the framework of deterministic dynamical systems....

A two-sided sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m}+c_{n-m}=2cₙcₘ$ for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence ${\left(cₙ\right)}_{n\in \mathbb{Z}}$ is bounded if $su{p}_{n\in \mathbb{Z}}\left|\right|cₙ\left|\right|<\infty$. A (bounded) group decomposition for a cosine sequence $c={\left(cₙ\right)}_{n\in \mathbb{Z}}$ is a representation of c as $cₙ=(bⁿ+b^{-n})/2$ for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying $su{p}_{n\in \mathbb{Z}}\left|\right|bⁿ\left|\right|<\infty$, respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, here referred...

Some $q$-analysis variants of Hardy type inequalities of the form $${\int }_0^b{\left(x^{\alpha -1}{\int }_0^x{t}^{-\alpha }f\left(t\right){\mathrm{d}}_{q}t\right)}^p{\mathrm{d}}_{q}x\le C{\int }_0^b{f}^p\left(t\right){\mathrm{d}}_{q}t$$ with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.

We consider the Abel equation φ[f(x)] = φ(x) + a on the plane ℝ², where f is a free mapping (i.e. f is an orientation preserving homeomorphism of the plane onto itself with no fixed points). We find all its homeomorphic and diffeomorphic solutions φ having positive Jacobian. Moreover, we give some conditions which are equivalent to f being conjugate to a translation.

It is shown that every almost linear Pexider mappings $f$, $g$, $h$ from a unital $C^{*}$-algebra $𝒜$ into a unital $C^{*}$-algebra $ℬ$ are homomorphisms when $f\left(2^{n}uy\right)=f\left(2^{n}u\right)f\left(y\right)$, $g\left(2^{n}uy\right)=g\left(2^{n}u\right)g\left(y\right)$ and $h\left(2^{n}uy\right)=h\left(2^{n}u\right)h\left(y\right)$ hold for all unitaries $u\in 𝒜$, all $y\in 𝒜$, and all $n\in \mathbb{Z}$, and that every almost linear continuous Pexider mappings $f$, $g$, $h$ from a unital $C^{*}$-algebra $𝒜$ of real rank zero into a unital $C^{*}$-algebra $ℬ$ are homomorphisms when $f\left(2^{n}uy\right)=f\left(2^{n}u\right)f\left(y\right)$, $g\left(2^{n}uy\right)=g\left(2^{n}u\right)g\left(y\right)$ and $h\left(2^{n}uy\right)=h\left(2^{n}u\right)h\left(y\right)$ hold for all $u\in \{v\in 𝒜\mid v=v^{*}\text{and}v\text{is}\text{invertible}\}$, all $y\in 𝒜$ and all $n\in \mathbb{Z}$. Furthermore, we prove the Cauchy-Rassias stability of $*$-homomorphisms between unital $C^{*}$-algebras, and $ℂ$-linear...