Rus, Ioan A. An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations. (English) Zbl 1327.47066 Fixed Point Theory 13, No. 1, 179-192 (2012). 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. Cited in 4 ReviewsCited in 11 Documents 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 Keywords:fixed point; admissible perturbation; iterative method; Gronwall lemma; comparison lemma; data dependence; stability; Ulam-Hyers stability; limit shadowing property; open problem PDFBibTeX XMLCite \textit{I. A. Rus}, Fixed Point Theory 13, No. 1, 179--192 (2012; Zbl 1327.47066) Full Text: Link