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The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates. (English) Zbl 0627.65069

Let X, Y be Banach spaces, B(x 0 ,r) the closed ball with the centre x 0 and radius r in X. Suppose that F:B(x 0 ,r)Y, G:B(x 0 ,r)Y are operators such that F is Fréchet differentiable on Int B(x 0 ,r), F ' (x 0 ) is continuously invertible and F ' (x 1 )-F ' (x 2 )k(r)x 1 -x 2 ,G(x 1 )-G(x 2 )ϵ(r)x 1 -x 2 , x 1 ,x 2 B(x 0 ,r), where k(r) and ϵ (r) are nondecreasing functions on [0,R].

Put a=[F ' (x 0 )] -1 (F(x 0 )+G(x 0 )), b=[F ' (x 0 )] -1 , ω(r)= 0 r k(t)dt,ϕ(r)=a+b 0 r ω(t)dt-r,ψ(r)=a 0 r ϵ(t)dt·

Theorem. Suppose that the function α(r)=ϕ(r)+ψ(r) has a unique zero ρ in [0,R] and α (R)0. Then the equation F(x)+G(x)=0 has a solution x * in B(x 0 ,ρ) and this solution is unique in the ball B(x 0 ,R). Moreover, the approximations x n+1 =x n -[F ' (x n )] -1 (F(x n )+G(x n )) are defined for all n, belong to B(x 0 ,ρ) and satisfy the estimates x n+1 -x n ρ n+1 -ρ n ,x * -x n ρ-ρ n , where (ρ n ) is monotonically increasing sequence convergent to ρ and defined by the formula ρ n+1 =ρ n -α(ρ n )[ϕ ' (ρ n )] -1 · These error estimates generalize the estimates of V. Pták [Comment. Math. Univ. Carol. 16, 699-705 (1975; Zbl 0314.65023), and Numer. Math. 25, 279-285 (1976; Zbl 0304.65037)] for the usual Newton-Kantorovich method.

Reviewer: J.Kolomý

65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)