New Quasi-Coincidence Point Polynomial Problems
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...
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Wiley
2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/959464 |
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| author | Yi-Chou Chen Hang-Chin Lai |
| author_facet | Yi-Chou Chen Hang-Chin Lai |
| author_sort | Yi-Chou Chen |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-ff21909f24204ea39bb7ef86917e91c9 |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-ff21909f24204ea39bb7ef86917e91c92025-02-03T01:03:46ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/959464959464New Quasi-Coincidence Point Polynomial ProblemsYi-Chou Chen0Hang-Chin Lai1Department of General Education, National Army Academy, Taoyuan 320, TaiwanDepartment of Mathematics, National Tsing Hua University, Hsinchu 300, TaiwanLet 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.http://dx.doi.org/10.1155/2013/959464 |
| spellingShingle | Yi-Chou Chen Hang-Chin Lai New Quasi-Coincidence Point Polynomial Problems Journal of Applied Mathematics |
| title | New Quasi-Coincidence Point Polynomial Problems |
| title_full | New Quasi-Coincidence Point Polynomial Problems |
| title_fullStr | New Quasi-Coincidence Point Polynomial Problems |
| title_full_unstemmed | New Quasi-Coincidence Point Polynomial Problems |
| title_short | New Quasi-Coincidence Point Polynomial Problems |
| title_sort | new quasi coincidence point polynomial problems |
| url | http://dx.doi.org/10.1155/2013/959464 |
| work_keys_str_mv | AT yichouchen newquasicoincidencepointpolynomialproblems AT hangchinlai newquasicoincidencepointpolynomialproblems |