Related papers: On Popoviciu-Ionescu functional equation in one an…
In 1922 Ritt described polynomial solutions of the functional equation P(f)=Q(g). In this paper we describe solutions of the equation above in the case when P,Q are polynomials while f,g are allowed to be arbitrary entire functions. In…
Statistical systems with time-periodic spatially non-uniform forces are of immense importance in several areas of physics. In this paper, we provide an analytical expression of the time-periodic probability distribution function of…
We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a…
The theory of first-order differential subordination developed by Miller and Mocanu was recently extended to functions with fixed initial coefficient by R. M. Ali, S. Nagpal and V. Ravichandran [Second-order differential subordination for…
Properties of the mappings \begin{align*} C&\mapsto\frac1{(2\pi i)^2}\int_{\Gamma_1}\int_{\Gamma_2}f(\lambda,\mu)\,R_{1,\,\lambda}\,C\, R_{2,\,\mu}\,d\mu\,d\lambda, C&\mapsto\frac1{2\pi i}\int_{\Gamma}g(\lambda)R_{1,\,\lambda}\,C\,…
We consider a nonlocal functional equation that is a generalization of the mathematical model used in behavioral sciences. The equation is built upon an operator that introduces a convex combination and a nonlinear mixing of the function…
It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a…
In this paper we present a two-component generalization of the C-integrable Calogero equation (see [1]). This system is C-integrable as well, and moreover we show that the Calogero equation and its two-component generalization are solvable…
This monograph is devoted to the theory of entire functions of several variables. A definition of bounded index was supposed by B. Lepson. We generalised his definition for several variables and obtained criteria of L-index boundedness in…
By applying Ricceri's variational principle, we demonstrate the existence of solutions for the following Robin problem \begin{equation*}\left\{ \begin{array}{cc}-\func{div}\left( \omega _{1}(x)\left\vert \nabla u\right\vert^{p(x)-2}\nabla…
The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy-Riemann…
Necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials on lattices are stated. Moreover, the functional Rodrigues formula and a closed…
In this paper, as an application of Zalcman's lemma in $\mathbb{C}^n$, we give a sufficient condition for normality of holomorphic functions of several complex variables, which generalizes previous known one-dimensional criterion of A.J.…
Cain, Clark and Rose defined a function $f\colon \RR^n \to \RR$ to be \emph{vertically rigid} if $\graph(cf)$ is isometric to $\graph (f)$ for every $c \neq 0$. It is \emph{horizontally rigid} if $\graph(f(c \vec{x}))$ is isometric to…
We verify the critical case $p=p_0(n)$ of Strauss' conjecture (1981) concerning the blow-up of solutions to semilinear wave equations with variable coefficients in $\mathbf{R}^n$, where $n\geq 2$. The perturbations of Laplace operator are…
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general…
We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x)=F(G(x),...,G(x)), where F is a given function of n real variables. We provide necessary and sufficient conditions on F for the…
The method of separation of variables is significant, it has been applied to physics, engineering , chemistry and other fields. It allows to reduce the diffculity of problems by separating the variables from partial differential equation…
This paper is dedicated to studying matrix solutions of the cubic Szeg\H{o} equation on the line in Pocovnicu [arXiv:1001.4037, arXiv:1012.2943] and G\'erard--Pushnitski [arXiv:2307.06734], leading to the following matrix Szeg\H{o} equation…
For $\Omega\subseteq\mathbb{R}^{n}$ an open and bounded region we consider solutions $u\in W_{\text{loc}}^{1,p(x)}\big(\Omega;\mathbb{R}^{N}\big)$, with $N>1$, of the $p(x)$-Laplacian system \begin{equation}…