Abstract
In this paper we consider a fixed point problem where the mapping is supposed to satisfy a generalized contractive inequality involving rational terms. We first prove the existence of a fixed point of such mappings. Then we show that the fixed point is unique under some additional assumptions. We investigate four aspects of the problem, namely, error estimation and rate of convergence of the fixed point iteration, Ulam-Hyers stability, well-psoedness and limit shadowing property. In the existence theorem we use an admissibility condition. Two illustration are given. The research is in the line with developing fixed point approaches relevant to applied mathematics.
Highlights
For the metric space X and the mapping α as in Example 2.7, it can be verified that α has triangular property and X is regular with respect to α
Arguing as in proof of Theorem 3.1, we prove that {un} is a Cauchy sequence in X and there exists p ∈ X such that
Using the metric space X, mappings α and F as in Example 2.7, we see that α has triangular property and X is regular with respect to α and F is a α-dominated mapping
Summary
For the purpose of the following three definitions we formally state the following fixed point problem to which they are related. Problem (P): Let (X, d) be a metric space and F : X → X be a mapping. The problem (P) has the limit shadowing property in X if, for any sequence {xn} ∈ X for which d(xn, F xn) → 0 as n → ∞, it follows that there exists z ∈ X such that d(xn, F nz) → 0 as n → ∞. Let F : X → X and α : X × X → [0, ∞) be respectively defined as follows:. For the metric space X and the mapping α as in Example 2.7, it can be verified that α has triangular property and X is regular with respect to α. Let (X, d) be a metric space and α : X × X → [0, ∞) be a mapping.
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