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Convergence criterion of inexact methods for operators with Hölder continuous derivatives. (English) Zbl 1170.65042

Suppose f(x)=0 is a nonlinear equation where f is an operator between two Banach spaces X,Y with continuous Fréchet derivative f ' . The best known iterative method to solve approximatively the equation is the Newton method which has however two drawbacks: it requires the calculation of f ' and the solution of a linear equation. In order to avoid these drawbacks which make the method of Newton inefficient from the point of view of calculation, several authors have introduced the iterative inexact methods.

The purpose of the paper is to give a convergence criterion for inexact procedures. As example the authors apply their results to a Hammerstein integral equation of the second kind:

x(s)=l(s)+ a b G(s,t)[x(t) 1+p +μx(t)]dt

where l is continuous and positive on [a,b] and G is the Green’s function:

G(s,t)=(b-s)(t-a) b-ast,(s-a)(b-t) b-ats·

65J15Equations with nonlinear operators (numerical methods)
65R20Integral equations (numerical methods)
65Y20Complexity and performance of numerical algorithms
47H30Particular nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
45G10Nonsingular nonlinear integral equations