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Semilocal convergence and \(R\)-order for modified Chebyshev-Halley methods. (English) Zbl 1279.65065

A nonlinear equation \(F(x)=0\) in Banach spaces is to solve on a nonempty open convex subset of space \(X\), where \(F\) has values in a Banach space \(Y\). Newton’s method converges quadratically. Third-order methods use the second Fréchet derivative of \(F\).
In this paper, a class of modified Chebyshev-Halley methods is proposed using recurrence relations. \(F'''\) is assumed to satisfy a certain weak continuity condition, but this operator has not to be Lipschitz-continuous or Hölder-continuous. Using the recurrence relations, a semilocal convergence theorem is proved. From this, the existence and uniqueness of the solution and a priori error bounds follow. The calculations in the proof are quite long, they are given in detail.
The \(R\)-order of the methods is analysed. If \(F'''\) is Lipschitz-continuous, the \(R\)-order becomes six, which is higher than the \(R\)-order of other known methods (Wang et al.). The methods are applied to a nonlinear integral equation of mixed Hammerstein type.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65R20 Numerical methods for integral equations
47J25 Iterative procedures involving nonlinear operators
47G10 Integral operators
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
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