On approximating the solutions of equations in metric spaces

Let \(\left( X,\rho \right)\) be a complete metric space, \(f:X\rightarrow X\) a nonlinear mapping. In order to solve the equation \(x=f\left( x\right) \) we consider a multistep method \[x_{n+k+1}=G(x_{n},x_{n+1},…,x_{n+k}), \quad n=1,2,… \] generated by a mapping \(G:X^{k+1}\rightarrow X\), whose diagonal restriction coincides with \(f\): \(G(x,…,x)=f(x)\). Under Lipschitz assumption on \(G\) we determine the algebraic equation whose unique positive solution leads to the convergence order of the iterations. We also study the case when the operator \(G\) replaced by an approximation of it.

Ion Păvăloiu

Original title (in French)

Sur l’approximation des racines des equations dans une espace métrique

English translation of the title

On approximating the solutions of equations in metric spaces

multistep iterative methods; convergence; successive approximations

[1] I. Pavaloiu, I., Serb, Sur des methodes iteratives optimales, Research Seminars, Seminar on Functional Analysis and Numerical Methods, Preprint Nr.1 (1983), 175–182.

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[3] Weinischke, J.H., Uber eine klasse von Iterationverfahren, Numerische Mathematik 6 (1964), 395–404.

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I. Păvăloiu, Sur l’approximation des racines des equations dans une espace métrique, Seminar on functional analysis and numerical methods, Preprint no. 1 (1989), pp. 95-104 (in French).

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