The Kac–Bernstein functional equation on Abelian groups

Aequationes mathematicae - We consider the Kac–Bernstein functional equation $$\begin{aligned} f(x+y)g(x-y)=f(x)f(y)g(x)g(-y), \quad x, y\in X, \end{aligned}$$ on an arbitrary Abelian group...     

A Levi–Civita functional equation on semigroups

Aequationes mathematicae - Let S be a semigroup. We describe the solutions $$f,g:S \rightarrow \mathbb {C}$$ of the functional equation $$\begin{aligned} f(xy) = f(x)g(y) + g(x)f(y) - g(x)g(y), \...     

On extensions of the generalized quadratic functions from “large” subsets of semigroups

Let $$(G,+)$$ be a commutative semigroup, $$\tau $$ be an endomorphism of G and involution, D be a nonempty subset of G, and $$(H,+)$$ be an abelian group, uniquely divisible by 2. Motivated by the extension problem of J. Aczél and the stability problem of S.M. Ulam, we show that if the set D is “sufficiently large”, then each function $$g{:} D\rightarrow H$$ such that $$g(x+y)+g(x+\tau (y))=2g(x)+2g(y)$$ for $$x,y\in D$$ with $$x+y,x+\tau (y)\in D$$ can be extended to a unique solution $$f{:} G\rightarrow H$$ of the functional equation $$f(x+y)+f(x+\tau (y))=2f(x)+2f(y)$$.     

OSCILLATION CRITERIA FOR NONLINEAR SECOND ORDER ELLIPTIC DIFFERENTIAL EQUATIONS

1.IntroductionOscillationtheoryforellipticdifferentialequationswithvariablecoefficielltshasbeenextensivelydevelopedinrecentyearsbymanyauthors(seee-g-,Allegretto[1,2l,Bugir['],Fiedler[5]'KitamuraandKusano['],Kural4]KusanoandNaitol1o]'NaitoandYoshidal12]'NoussairandSwanson[l3'14]'Swansonl15'l6]andthereferencescitedtherein).Inthispaper,weareconcernedwiththeoscillatorybehaviorofsolutionsofsecondorderellipticdifferelltialequatiollswithalternqtingcoefficients.Asusual,pointsinn-dimensionalEucli…     

On a quadratic difference assuming three values

Aequationes mathematicae - The aim of this work is to investigate the alternative quadratic functional equation $$\begin{aligned} f(x+y)+f(x-y)-2f(x)-2f(y)\in \{0,1,2\}, \end{aligned}$$ where $$f{:...     

ON THE EXISTENCE OF PERIODIC SOLUTIONS OF NONLINEAR OSCILLATION EQUATION

This paper deal with the existence of periodic solutions of the nonlinear cscillation equation $$\[\mathop x\limits^{ \cdot \cdot } + f(x)\varphi (x) + \psi (x)\eta (x) = 0\begin{array}{*{20}{c}} {}&{(3)} \end{array}\]$$ The author offers a method which can reduce (3) into system $$\[\mathop x\limits^ \cdot = h(y) - e(y)F(x),\mathop y\limits^ \cdot = - g(x)\begin{array}{*{20}{c}} {}&{(9)} \end{array}\]$$ Some sufficient condition for the existence of the limit cycles of (9) are obtained. These results generalize the results in [1,2,3,4,5,6].     

A note on certain integral equations associated with non-linear time series analysis

Probability Theory and Related Fields - We consider the integral equation inh $$h(y) = \int\limits_{\mathbb{R}^n } {f(y - \hat Tx)h(x)dx,} $$ wheref is a p.d.f. and $$\hat T:\mathbb{R}^n \to...     

The stability of additive $(\alpha,\beta)$-functional equations

In this paper, we investigate the following $(\alpha,\beta)$-functional equations $$ 2f(x)+2f(z)=f(x-y)+\alpha^{-1}f(\alpha (x+z))+\beta^{-1}f(\beta(y+z)),~~~~~~~~~(0.1) $$ $$ 2f(x)+2f(y)=f(x+y)+\alpha^{-1}f(\alpha(x+z)) +\beta^{-1}f(\beta(y-z)),~~~~~~~~~~~(0.2) $$ where $\alpha,\beta$ are fixed nonzero real numbers with $\alpha^{-1}+\beta^{-1}\neq 3$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the $(\alpha,\beta)$-functional equations $(0.1)$ and $(0.2)$ in non-Archimedean Banach spaces.     

Positive periodic solutions for a nonlinear differential system with two parameters

In this article, we investigate a nonlinear system of differential equations with two parameters $$\left\{ \begin{array}{l} x"(t)=a(t)x(t)-\lambda f(t, x(t), y(t)),\y"(t)=-b(t)y(t)+\mu g(t, x(t), y(t)),\end{array}\right.$$ where $a,b \in C(\textbf{R},\textbf{R}_+)$ are $\omega-$periodic for some period $\omega > 0$, $a,b \not\equiv 0$, $f,g \in C(\textbf{R} \times \textbf{R}_+ \times \textbf{R}_+ ,\textbf{R}_+)$ are $\omega-$periodic functions in $t$, $\lambda$ and $\mu$ are positive parameters. Based upon a new fixed point theorem, we establish sufficient conditions for the existence and uniqueness of positive periodic solutions to this system for any fixed $\lambda,\mu>0$. Finally, we give a simple example to illustrate our main result.     

Generalized sine equations,II

In this paper we solve the equations

ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR （I）

ＯＮＡＭＵＬＴＩＬＩＮＥＡＲＯＳＣＩＬＬＡＴＯＲＹＳＩＮＧＵＬＡＲＩＮＴＥＧＲＡＬＯＰＥＲＡＴＯＲ（Ｉ）ＣＨＥＮＷＥＮＧＵＨＵＧＵＯＥＮＬＵＳＨＡＮＺＨＥＮＭａｎｕｓｃｒｉｐｔｒｅｃｅｉｖｅｄＯｃｔｏｂｅｒ１８，１９９４．ＲｅｖｉｓｅｄＤｅｃｅ...     

A functional equation and its application to the characterization of gamma distributions

The functional equation $$ f\left(x\right)g\left(y\right)=p\left(x+y\right)q\left(\frac{x}{y} \right) $$ is investigated for almost all ${\left(x,\,y\right)\in\mathbb{R}^{2}_{+}}$ and for the measurable functions ${f,\,g,\,p,\,q:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}}$ . This equation is related to the Lukács characterization of gamma distribution.     

一类振荡积分算子在Wiener 共合空间上的有界性

假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~p,L~q)上的映射性质.本文的结论足以表明,Wiener共合空间是Lebesgue空间的一个很好的替代.     

On the stability of the generalized sine functional equations

The aim of this paper is to study the stability problem of the generalized sine functional equations as follows：
g（x）f（y）=f（x＋y/2）^2-f（x-y/2）^2 f（x）g（y）=f（x＋y/2）^2-f（x-y/2）^2,g（x）g（y）=f（x＋y/2）^-f（x-y/2）^2
Namely, we have generalized the Hyers Ulam stability of the （pexiderized） sine functional equation.     

On a class of Hammerstein integral equations

By a monotone representation of the nonlinearity we derive sufficient (and partly necessary) conditions for the unique existence of positive solutions of the Hammerstein integral equation $$\begin{array}{*{20}c} {u(x) = \int\limits_D {f(y,u(y)) k(x,y) dy ,} } & {x \in D,} \\ \end{array} $$ and for the convergence of successive approximations towards the solution. Further we study the corresponding nonlinear eigenvalue problem. Essentially we assume that the integral kernel k satisfies appropriate positivity conditions and that, for the nonlinearity f and any y ∈ D, rf(y,r) strictly monotone increases and f(y,r)/r strictly monotone decreases as r>0 increases.     

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problem {p′(s)=f′(s)-λp^β(s)for s∈(0,1);β∈(0,1),p(0)=p(1)=0,p(s)>0 if s∈(0,1),has a solution(λ^-，p^-（s）),where |λ^-| is the smallest parameter,under the minimal stringent restrictions on f(s), by applying the shooting and regularization methods. In a classic paper, Kohmogorov et.al.studied in 1937 a problem which can be converted into a special case of the above problem. The author also use the solution(λ^-,p^-(s)) to construct a weak travelling wave front solution u(x,t)=y(ξ),ξ=x-Ct,C=λ^-N/(N+1),of the generalized diffusion equation with reaction δ/δx（k（u）|δu/δx|^n-1 δu/δx)-δu/δt=g（u），where N>0,k(s)>0 a.e.on(0,1),and f(a):=n+1/N∫0ag(t)k^1/N(t)dt is absolutely continuous ou[0,1],while y(ξ) is increasing and absolutely continuous on (-∞,+∞) and (k(y(ξ))|y′(ξ)|^N)′=g(y(ξ))-Cy′(ξ)a.e.on(-∞，+∞）,y(-∞)=0,y(+∞)=1.     

On a multilinear singular integral operator

In this paper, we prove that the maximal operatorsatisfiesis homogeneous of degree 0, has vanishing moment up to order M and satisfies Lq-Dini condition for some     

Arithmetic properties of polynomials

Periodica Mathematica Hungarica - First, we prove that the Diophantine system $$\begin{aligned} f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q) \end{aligned}$$ has infinitely many integer solutions for...     

Rough bilinear fractional integrals with variable kernels

We study the rough bilinear fractional integral