New Quasi-Coincidence Point Polynomial Problems

• Published in: Journal of Applied Mathematics Vol. 2013; no. 2013; pp. 525 - 532-854
• Main Authors: Chen, Yi-Chou, Lai, Hang-Chin
• Format: Journal Article
• Language: English
• Published: Cairo, Egypt Hindawi Limiteds 01.01.2013; Hindawi Puplishing Corporation; Hindawi Publishing Corporation; Hindawi Limited; Wiley
• Subjects: Fixed point theory; Mathematical research; Polynomials
• ISSN: 1110-757X; 1687-0042
• DOI: 10.1155/2013/959464

Let F:ℝ×ℝ→ℝ be a real-valued polynomial function of the form F(x,y)=as(x)ys+as-1(x)ys-1+⋯+a0(x), where the degree s of y in F(x,y) is greater than or equal to 1. For arbitrary polynomial function f(x)∈ℝ[x], x∈ℝ, we will find a polynomial solution y(x)∈ℝ[x] to satisfy the following equation: (*): F(x,y(x))=af(x), where a∈ℝ is a constant depending on the solution y(x), namely, a quasi-coincidence (point) solution of (*), and a is called a quasi-coincidence value. In this paper, we prove that (i) the leading coefficient as(x) must be a factor of f(x), and (ii) each solution of (*) is of the form y(x)=-as-1(x)/sas(x)+λp(x), where λ is arbitrary and p(x)=c(f(x)/as(x))1/s is also a factor of f(x), for some constant c∈ℝ, provided the equation (*) has infinitely many quasi-coincidence (point) solutions.