Semilocal convergence of Stirling’s method under Hölder continuous first derivative in Banach spaces. (English) Zbl 1208.65074
The article deals with the following iterative method
\[ x_{n+1} = x_n - (I - F'(F(x_n)))^{-1}(x_n - F(x_n)), \quad n = 0,1,2,\dots\tag{1} \]
of approximative solving of nonlinear operator equation
\[ x - F(x) = 0,\tag{2} \]
where \(F\) is an operator defined at the ball \(B(x_0,r)\) of a Banach space \(X\) with values in \(X\) (Stirling’s method). The authors consider the case when \(F'\) is Hölder continuous (with the exponent \(p\), \(0 < p \leq 1\)) and prove (under some natural additional assumptions) the convergence of Stirling iteration (1) to the unique solution of the equation (2); moreover, they obtain natural error estimates of type \(\|x_n - x^*\| = O(S^{((p+1)^n - 1)/p})\). In the end of the article a numerical example is presented (Hammerstein integral equation in the space of continuous functions).
\[ x_{n+1} = x_n - (I - F'(F(x_n)))^{-1}(x_n - F(x_n)), \quad n = 0,1,2,\dots\tag{1} \]
of approximative solving of nonlinear operator equation
\[ x - F(x) = 0,\tag{2} \]
where \(F\) is an operator defined at the ball \(B(x_0,r)\) of a Banach space \(X\) with values in \(X\) (Stirling’s method). The authors consider the case when \(F'\) is Hölder continuous (with the exponent \(p\), \(0 < p \leq 1\)) and prove (under some natural additional assumptions) the convergence of Stirling iteration (1) to the unique solution of the equation (2); moreover, they obtain natural error estimates of type \(\|x_n - x^*\| = O(S^{((p+1)^n - 1)/p})\). In the end of the article a numerical example is presented (Hammerstein integral equation in the space of continuous functions).
Reviewer: Peter Zabreiko (Minsk)
MSC:
| 65J15 | Numerical solutions to equations with nonlinear operators |
| 47J25 | Iterative procedures involving nonlinear operators |
| 45G10 | Other nonlinear integral equations |
| 65R20 | Numerical methods for integral equations |
Keywords:
Stirling method; Hölder continuity conditions; Fréchet derivative; nonlinear operator equations; iterative method; Banach space; convergence; numerical exampleReferences:
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