Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results
Let Fx, y=asxys+as-1xys-1+⋯+a0x be a real-valued polynomial function in which the degree s of y in Fx, y is greater than or equal to 1. For any polynomial yx, we assume that T:Rx→Rx is a nonlinear operator with Tyx=Fx, yx. In this paper, we will find an eigenfunction yx∈Rx to satisfy the following e...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2015-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2015/516159 |
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| Summary: | Let Fx, y=asxys+as-1xys-1+⋯+a0x be a real-valued polynomial function in which the degree s of y in Fx, y is greater than or equal to 1. For any polynomial yx, we assume that T:Rx→Rx is a nonlinear operator with Tyx=Fx, yx. In this paper, we will find an eigenfunction yx∈Rx to satisfy the following equation: Fx, yx=ayx for some eigenvalue a∈R and we call the problem Fx, yx=ayx a fixed point like problem. If the number of all eigenfunctions in Fx, yx=ayx is infinitely many, we prove that (i) any coefficients of Fx, y, asx, as-1x,…, a0x, are all constants in R and (ii) yx is an eigenfunction in Fx, yx=ayx if and only if yx∈R. |
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| ISSN: | 1110-757X 1687-0042 |