A robust alternative to examine data dependency of fixed points of quasi-contractive operators: an efficient approach that relies on the collage theorem

Abstract

Usurelu et al. (Int J Comput Math 98:1049–1068, 2021) presented stability and data dependence results for a TTP (Thakur–Thakur–Postolache) iteration algorithm associated with quasi-strictly contractive mappings and contraction mappings, respectively, but these results were subject to strong conditions on the parametric control sequences used in the TTP iteration algorithm. This article aims to expand those results conducting a thorough analysis of the convergence of TTP and S iteration algorithms and improve those results by removing the restrictions on the parametric control sequences. Additionally, a data dependence result for the TTP iteration algorithm of quasi-strictly contractive mappings is established and several collage theorems are introduced to offer new insights into the data dependence of fixed points of quasi-strictly contractive mappings and to solve related inverse problems. In order to exhibit the dependability and effectiveness of all the results discussed in this work, a multitude of numerical examples are furnished, encompassing both linear and nonlinear differential equations (DEs) and partial differential equations (PDEs). This work can be viewed as an important refinement and complement to the study by Usurelu et al. (Int J Comput Math 98:1049–1068, 2021).