Semilocal convergence and \(R\)-order for modified Chebyshev-Halley methods
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.
nonlinear operator equation in Banach spaces
nonlinear integral equation of mixed Hammerstein type
scientific article; zbMATH DE number 6210996
scientific article