Properties of the solutions of those equations for which the Krasnoselskii iteration converges
Copyright policy )Properties of the solutions of those equations for which the Krasnoselskii iteration converges
Let (X, +, R, →) be a vectorial L-space, Y ⊂ X a nonempty convex subset of X and f : Y → Y be an operator with Ff := {x ∈ Y | f(x) = x} 6= ∅. Let 0 < λ < 1 and let fλ be the Krasnoselskii operator corresponding to f, i.e., fλ(x) := (1 − λ)x + λf(x), x ∈ Y. We suppose that fλ is a weakly Picard operator (see I. A. Rus, Picard operators and applications, Sc. Math. Japonicae, 58 (2003), No. 1, 191-219). The aim of this paper is to study some properties of the fixed points of the operator f: Gronwall lemmas and comparison lemmas (when (X, +, R, →, ≤) is an ordered L-space) and data dependence (when X is a Banach space). Some applications are also given.
- Babeș-Bolyai University Romania
Gronwall lemma, Numerical solutions to equations with nonlinear operators, Stability theory for difference equations, Picard operator, data dependence, iteration, Ulam-Hyers stability, comparison lemma, Krasnoselskij operator, Iterative procedures involving nonlinear operators, fixed point, \(\psi\)-Picard operator, \(\psi\)-weakly Picard operator, weakly Picard operator
Gronwall lemma, Numerical solutions to equations with nonlinear operators, Stability theory for difference equations, Picard operator, data dependence, iteration, Ulam-Hyers stability, comparison lemma, Krasnoselskij operator, Iterative procedures involving nonlinear operators, fixed point, \(\psi\)-Picard operator, \(\psi\)-weakly Picard operator, weakly Picard operator
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