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Infinitely Many Quasi-Coincidence Point Solutions of Multivariate Polynomial Problems

2013
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Abstract and Applied Analysis
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Let 𝐹 : R 𝑛 × R → R be a real-valued polynomial function of the form 𝐹(𝑥, 𝑦) = 𝑎 𝑠 (𝑥)𝑦 𝑠 + 𝑎 𝑠−1 (𝑥)𝑦 𝑠−1 + ⋅ ⋅ ⋅ + 𝑎 0 (𝑥) where the degree 𝑠 of 𝑦 in 𝐹(𝑥, 𝑦) is greater than 1. For arbitrary polynomial function 𝑓(𝑥) ∈ R[𝑥], 𝑥 ∈ R 𝑛 , 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 of

doi:10.1155/2013/307913
fatcat:lx3asxuv7zheblhytbey2pgwvu