2013 New Quasi-Coincidence Point Polynomial Problems
Yi-Chou Chen, Hang-Chin Lai
J. Appl. Math. 2013(SI21): 1-8 (2013). DOI: 10.1155/2013/959464

## Abstract

Let $F:\Bbb R×\Bbb R\to \Bbb R$ be a real-valued polynomial function of the form $F(x,y)={a}_{s}(x){y}^{s}+{a}_{s-\mathrm{1}}(x){y}^{s-\mathrm{1}}+\cdots +{a}_{\mathrm{0}}(x)$, where the degree $s$ of $y$ in $F(x,y)$ is greater than or equal to $\mathrm{1}$. For arbitrary polynomial function $f(x)\in \Bbb R[x]$, $x\in \Bbb R$, we will find a polynomial solution $y(x)\in \Bbb R[x]$ to satisfy the following equation: ($\mathrm{\ast}$): $F(x,y(x))=af(x)$, where $a\in \Bbb R$ is a constant depending on the solution $y(x)$, namely, a quasi-coincidence (point) solution of ($\mathrm{\ast}$), and $a$ is called a quasi-coincidence value. In this paper, we prove that (i) the leading coefficient ${a}_{s}(x)$ must be a factor of $f(x)$, and (ii) each solution of ($\mathrm{\ast}$) is of the form $y(x)=-{a}_{s-\mathrm{1}}(x)/s{a}_{s}(x)+\lambda p(x)$, where $\lambda$ is arbitrary and $p(x)=c(f(x)/{a}_{s}(x){)}^{\mathrm{1}/s}$ is also a factor of $f(x)$, for some constant $c\in \Bbb R$, provided the equation $(\mathrm{\ast})$ has infinitely many quasi-coincidence (point) solutions.

## Citation

Yi-Chou Chen. Hang-Chin Lai. "New Quasi-Coincidence Point Polynomial Problems." J. Appl. Math. 2013 (SI21) 1 - 8, 2013. https://doi.org/10.1155/2013/959464

## Information

Published: 2013
First available in Project Euclid: 14 March 2014

zbMATH: 06950959
MathSciNet: MR3100826
Digital Object Identifier: 10.1155/2013/959464