Theoretical fixed point results involving double averaged mappings with new enriching techniques and applications

This dissertation delves into the intricate development of the classical fixed point theorem by delving deep into the fixed point theory in metric spaces. The main focus of this research is on the enriching technique, which was first introduced by Berinde and P\u{a}curar [Approximating fixed points of enriched contraction mappings in Banach spaces. J. Fixed Point Theory Appl. 22, 38 (2020)] published in 2020. In this study, we aim to explore the limitations of classical fixed point theorems and propose new ways to enhance its efficacy. To this end, we are the first to introduce a novel class of mappings called double averaged mappings and conduct an in-depth investigation of their properties. Our findings on double averaged mappings provide a fresh perspective on the development of classical fixed point theory. By utilizing a variety of the enriching technique, which we refer to as the new enriching technique, we are able to demonstrate how our results on double averaged mappings can be utilized to refine classical fixed point theorems with fewer strictly necessary conditions. Lastly, we apply our newly-developed fixed point results to the existence theory of nonlinear fractional integral equations, demonstrating the practical relevance of our research. Overall, this dissertation represents a significant contribution to the field of fixed point theory and lays the foundation for future research in this area.

本论文通过深入研究度量空间（metric spaces）中的不动点理论（fixed point theory），探讨经典不动点定理（fixed point theorem）的复杂发展历程。本研究的核心聚焦于由Berinde与Păcurar于2020年首次提出的富集技巧（enriching technique），其相关成果发表于论文《Approximating fixed points of enriched contraction mappings in Banach spaces》，见*J. Fixed Point Theory Appl.* 22, 38 (2020)。本研究旨在探究经典不动点定理的局限性，并提出提升其应用有效性的新路径。为此，我们首次引入一类全新的映射，命名为双平均映射（double averaged mappings），并对其性质展开深入系统的研究。我们针对双平均映射的研究结论，为经典不动点理论的发展提供了全新视角。通过运用我们所界定的新型富集技巧相关手段，我们得以阐明如何借助双平均映射的研究成果，以更少的严格必要条件优化经典不动点定理。最后，我们将新获证的不动点研究结论应用于非线性分数阶积分方程（nonlinear fractional integral equations）的存在性理论，以此彰显本研究的实际应用价值。总体而言，本论文对不动点理论领域做出了重要贡献，并为该领域的后续研究奠定了坚实基础。