Infinitely many quasi-coincidence point solutions of multivariate polynomial problems. (English) Zbl 1381.39020

Summary: Let \(F\:\mathbb R^n\times\mathbb R\rightarrow\mathbb R\) be a real-valued polynomial function of the form \(F(\overline x,y)=a_s(\overline x)y^s+a_{s-1}(\overline x)y^{s-1}+\dots+a_0(\overline x)\) where the degree \(s\) of \(y\) in \(F(\overline x,y)\) is greater than 1. For arbitrary polynomial function \(f(\overline x)\in\mathbb R[\overline x]\), \(\overline x\in\mathbb R^n\), we will find a polynomial solution \(y(\overline x)\in\mathbb R[\overline x]\) to satisfy the following equation \((*)\): \(F(\overline x,y(\overline x))=af(\overline x)\) where \(a\in\mathbb R\) is a constant depending on the solution \(y(\overline x)\), namely a quasi-coincidence (point) solution of \((*)\), and \(a\) is called a quasi-coincidence value of \((*)\). In this paper, we prove that (i) the number of all solutions in \((*)\) does not exceed \(\deg_yF(\overline x,y)((2^{\deg f(\overline x)} +s+3)\cdot2^{\deg f(\overline x)}+1)\) provided those solutions are of finitely many exist, (ii) if all solutions are of infinitely many exist, then any solution is represented as the form \(y(\overline x)=-a_{s-1}(\overline x)/sa_s(\overline x)+ \lambda p(\overline x)\) where \(\lambda\) is arbitrary and \(p(\overline x)= (f(\overline x)/a_s(\overline x))^{1/s}\) is also a factor of \(f(\overline x)\), provided the equation \((*)\) has infinitely many quasi-coincidence (point) solutions.


39B22 Functional equations for real functions
47H10 Fixed-point theorems
Full Text: DOI


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