Related papers: A Malmquist-Yosida type theorem for Schwarzian dif…

A novel method for finding the eigenvalues of a Sturm-Liouville problem is developed. Following the minimalist approach the problem is transformed to a single first-order differential equation with appropriate boundary conditions. Although…

In this paper we have studied the growth of meromorphic solutions of higher order homogeneous and non-homogeneous linear difference equations with entire and meromorphic coefficients. We have extended and improved some results of Zhou and…

Let $f=\sum_{n=0}^\infty f_n x^n \in \overline{\mathbb Q}[[x]$ be a solution of an algebraic differential equation $Q(x,y(x), \ldots, y^{(k)}(x))=0$, where $Q$ is a multivariate polynomial with coefficients in $\overline{\mathbb Q}$. The…

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \begin{eqnarray*} \Delta^{2}u-M(\|\nabla u\|_{2}^{2})\Delta u+V(x)u=f(x,u),\ \ \ \ \ x\in \mathbb{R}^{N},…

Let $$ S_{m,n}(q):=\sum_{k=1}^{n}\frac{1-q^{2k}}{1-q^2} (\frac{1-q^k}{1-q})^{m-1}q^{\frac{m+1}{2}(n-k)}. $$ Generalizing the formulas of Warnaar and Schlosser, we prove that there exist polynomials $P_{m,k}(q)\in\mathbb{Z}[q]$ such that $$…

In this paper we give a conditional improvement to the Elekes-Szab\'{o} problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family of…

In this paper we investigate the growth of meromorphic solutions of homogeneous and non-homogeneous linear difference equations with entire or meromorphic coefficients. We further extend and improve few results on the order of meromorphic…

This paper is concerned with two properties of positive weak solutions of quasilinear elliptic equations with nonlinear gradient terms. First, we show a Liouville-type theorem for positive weak solutions of the equation involving the…

This paper is devoted to the study of meromorphic solutions of nonlinear differential equations, specifically the equation \[ (f^n)^{(k)}(g^n)^{(k)} = \alpha^2, \] where $k$ and $n$ are positive integers with $n>2k$, and $\alpha$ is a…

We present a new construction related to systems of polynomials which are consistent on a cube. The consistent polynomials underlie the integrability of discrete counterparts of integrable partial differential equations of Korteweg- de…

Some of important univalence criteria for a non-constant meromorphic function $f(z)$ on the unit disk $\ID$ involve its pre-Schwarzian or Schwarzian derivative. We consider an appropriate norm for the pre-Schwarzian derivative, and discuss…

Let $s_0,s_1,\dots,s_{m-1}$ be complex numbers and $r_0,\dots,r_{m-1}$ rational integers in the range $0\le r_j\le m-1$. Our first goal is to prove that if an entire function $f$ of sufficiently small exponential type satisfies…

In this paper, we combine Bochner formula, Saloff-Coste's Sobolev inequality and the Nash-Moser iteration method to study the local and global behaviors of solutions to the nonlinear elliptic equation $\Delta_pu+\Delta_qu+h(u,|\nabla…

We study the existence and multiplicity of solutions for a class of fractional Schr\"{o}dinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \begin{gather*} \begin{cases}…

We consider the problem of the reconstruction of a Schwarz matrix from exactly one given eigenvalue. This inverse eigenvalue problem leads to the Jacobi orthogonal polynomials~$\{P_k^{(-n,n)}\}_{k=0}^{n-1}$ that can be treated as a discrete…

We prove a new Elekes-Szab\'o type estimate on the size of the intersection of a Cartesian product $A\times B\times C$ with an algebraic surface $\{f=0\}$ over the reals. In particular, if $A,B,C$ are sets of $N$ real numbers and $f$ is a…

We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,...,fm belonging to k[x1,...,xn], where k is a field of characteristic zero and m=1,...,n. We express the generalized Jacobian condition in terms of…

This paper presents two new classes of M\"untz functions which are called Jacobi-M\"untz functions of the first and second types. These newly generated functions satisfy in two self-adjoint fractional Sturm-Liouville problems and thus they…

We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence…

For differential equations $P(y^{(k)},y)=0,$ where $P$ is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate.