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An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations. (English) Zbl 1327.47066

Summary: Let \((X,\to)\) be an \(L\)-space, \(G:X\times X\to X\) and \(f:X\to X\) be two operators. Let \(f_G:X\to X\) be defined by \(f_G(x):=G(x,f(x))\). If the operator \(G\) satisfies the following conditions:
\((A_1)\)
\(G(x,x)=x \text{ for all }x\in X\);
\((A_2)\)
\(G(x,y)=x\Rightarrow y=x\),
then we call \(f_G\) admissible perturbation of \(f\).
We introduce some iterative algorithms in terms of admissible perturbations. We suppose that these algorithms are convergent.
In this paper, we study the impact of this hypothesis on the theory of fixed point equations: Gronwall lemmas (when \((X,\to,\leq)\) is an ordered \(L\)-space), data dependence, stability and shadowing property (when \((X,d)\) is a metric space). Some open problems are presented.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
65J15 Numerical solutions to equations with nonlinear operators
54H25 Fixed-point and coincidence theorems (topological aspects)
37N30 Dynamical systems in numerical analysis
39A30 Stability theory for difference equations
39B12 Iteration theory, iterative and composite equations
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