Constructing Solutions to $\int_0^1 {f(j)} {{A(j,x)} / {f(x)}}dj = \int_0^1 {f(x){{A(x,j)} / {f(j)}}dj} $
| Autor: | G. Edgar Parker, Terry J. Walters |
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| Rok vydání: | 1982 |
| Předmět: | |
| Zdroj: | SIAM Journal on Mathematical Analysis. 13:856-865 |
| ISSN: | 1095-7154 0036-1410 |
| DOI: | 10.1137/0513059 |
| Popis: | Given a continuous function A from $[0,1] \times [0,1]$ into $(0,\infty )$, a function f so that if $x \in [0,1]$, \[ \int_0^1 {f(j) {{A(j,x)} / {f(j)dj}}} = \int {f(x){{A(x,j)} / {f(j)dj}}} \] is constructed. Approximations to f are made by forming the $n \times n$ matrix $a_{ik} = A({(i - 1) / n},{(k - 1) / n})$ and constructing the n-vector d, a solution to the nonlinear system $\sum_{k = 1}^n {d_i } a_{ik} d_k^{ - 1} = \sum_{k = 1}^n {d_k } a_{ki} d_i^{ - 1} $. For $x \in [0,1]$, n large enough and ${{(k - 1)} / n}$ close to x, $d_k $ approximates $f(x)$. The construction is adaptable for use on the computer and provides solutions in a situation where previously only a fixed-point proof was known. |
| Databáze: | OpenAIRE |
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