Some fixed point and common fixed point theorems for single and multivalued mappings in generalized metric spaces

Fixed point theory represents a compelling and continually advancing domain within the mathematical sciences of the twenty-first century. It is characterized by a distinctive integration of nonlinear functional analysis, nonlinear operator theory, topology, and mathematical modeling, along with various applications. Recognized as a pivotal area of research in nonlinear analysis, fixed point theory serves as a foundational mathematical framework for establishing the existence of solutions to a diverse array of real-world problems. This underscores its significance as both a theoretical and applied discipline within mathematics, with its relevance as a burgeoning area of research. The breadth of fixed point theory extends beyond the geometric considerations of infinite-dimensional function spaces and operator-theoretic challenges; it encompasses a wide spectrum of interdisciplinary applications. These span various fields, including engineering, space science, hydromechanics, astrophysics, chemistry, biology, theoretical mechanics, biomechanics, economics, and stochastic game theory. The well-established concepts and methodologies intrinsic to fixed point theory furnish essential tools for the formulation of more realistic and precise models addressing phenomena encountered across a multitude of applied fields. Fixed point theory provides significant insights into methodologies for locating solutions to nonlinear equations of the form Tx=x, where T is a self-mapping defined on a subset of a metric space, a normed linear space, a topological vector space, or an appropriate mathematical structure. In this framework, fixed point theory encompasses a variety of applications within mathematical analysis and applied mathematics, thus rendering it an essential component of contemporary mathematical discourse. Metric fixed point theory constitutes a significant domain within mathematical analysis and topology, focusing on the exploration of fixed points of mappings defined on metric spaces. This theory originated with Banach's fixed point theorem, serving as a foundational result that has been progressively expanded to encompass a myriad of disciplines. Banach's fixed point theorem is one of the cornerstones of metric fixed point theory. This theorem guarantees the existence of fixed points of contraction maps defined on a metric space. As a formulation, (X,f) is a complete metric space and T:X→X is a contraction mapping; that is, there exists a constant 0≤k<1 for the mapping T and for each x,y∈ℶ f(T(x),T( y))≤kf(x,y) inequality is satisfied. Then, the mapping T has a unique fixed point and this fixed point can be found by iterative methods.Through the alteration of distance functions, the removal of certain properties from existing functions, or the introduction of novel properties, numerous topological structures have emerged in the literature. Initially rooted in Banach's fixed point theorem, metric fixed point theory has evolved into a critical component of contemporary mathematics, demonstrating its relevance across various mathematical disciplines. Beyond its theoretical implications in areas such as summability theory, sequence spaces, fuzzy set theory, and the geometry of Banach spaces, this theory exhibits extensive applicability in interdisciplinary realms. Fields such as engineering, space science, hydromechanics, astrophysics, chemistry, biology, theoretical mechanics, biomechanics, economics, and stochastic game theory have all benefited from the insights provided by metric fixed point theory, underscoring its versatility and importance in both pure and applied mathematics. This study consists of six chapters. While chapters one and two are literature reviews, chapters three, four and five constitute the original part of the thesis. The sixth chapter contains conclusions and recommendations. In the initial chapter of this study, we present fundamental definitions and theorems pertinent to metric fixed point theory. Furthermore, we delineate various spaces that extend the concept of metric spaces and analyze their properties comprehensively. As extensions of the Banach contraction principle commonly referred to as the Banach fixed point theorem proposed by numerous researchers, this work aims to either establish novel spaces through modifications of traditional metric conditions or to revise the contraction criteria applicable to transformations, thereby contributing significantly to the existing body of literature. The second chapter of the paper delineates several general metric structures that will be utilized throughout the subsequent discussions, along with a comprehensive analysis of their properties. In the third section, we introduce the concepts of asymmetric modular b-metric spaces and non-Archimedean modular b-metric space structures. A detailed examination of the topological characteristics inherent to these spaces is provided. Within the context of non-Archimedean asymmetric modular b-metric spaces, we establish fixed point theorems by formulating a novel contraction transformation, leveraging simulation and comparison functions for this purpose. Furthermore, we demonstrate the applicability of the derived fixed point theorems to the Ulam-Hyers stability problem. Building on these findings, we also derive a unique solution for a Caputo type nonlinear fractional differential equation, thereby illustrating the practical implications of our theoretical contributions. The equation under consideration is expressed as follows: D^δ ω(ϱ)+G(ϱ,ω(ϱ))=0, ϱ∈I=[0,1], 1<δ<2 ω(0)=0,ω(1)=0, where D^δ denotes the Caputo fractional derivative of order δ and ϱ∈(C[0,1],R), and also G:I×R→R is a continuous function. In the fourth section, the concept of a modular b-metric-like space is introduced, drawing upon the foundational concepts of modular metric-like space and modular b-metric space. This section encompasses an analysis of various topological properties associated with the newly proposed space. Furthermore, the framework of R-modular b-metric-like spaces is expanded to incorporate the binary relation R. By integrating the Z-simulation function and E-type contraction structures, we derive fixed point results that are pertinent to mappings of the Geraghty contraction type. Additionally, a novel graphical definition of modular b-metric-like space, informed by the binary relation R, is presented, thereby contributing to the existing literature. This framework facilitates the exploration of solutions to a specific class of functional equations that hold significant relevance in the contexts of dynamic programming and the resolution of initial value problems pertaining to electric current within an RLC parallel circuit. In the fifth section, the existence and uniqueness of the fixed point of the multivalued rational θ-type contraction mapping in the non-Archimedean modular metric space are proved. Additionally, it has been demonstrated that if a non-empty subset of non-Archimedean modular metric space is finitely ε-chainable, i.e.,if N,ε-chain can be formed between any two connected elements of the subset, the existence of the fixed point of the multivalued rational θ-type contraction mapping is guaranteed. These results are supported by generalizing them to the non-Archimedean modular metric space endowed with a graph, as one of the mathematical applications. In the concluding section, the findings derived from the study are presented, along with recommendations concerning the practical implementation of the theoretical frameworks discussed in the literature.