Some novel approaches in graphical modular 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 is an important concept in mathematics that studies situations where a function does not change the point to which it is applied. In other words, for a function 𝑆 a point ℑ that satisfies 𝑆(ℑ) = ℑ is called a fixed point. This theory is used in many areas such as analysis, topology, differential equations and numerical analysis . Finding a fixed point is sometimes equivalent to solving an equation, which makes the theory valuable in both theoretical and applied fields. 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, (𝔐, ℎ) is a complete metric space and 𝑆: 𝔐 → 𝔐 is a contraction mapping; that is, there exists a constant 0 ≤ 𝑘 < 1 and for each ℑ, ℌ ∈ 𝔐 ℎ(𝑆(ℑ), 𝑆( ℌ)) ≤ 𝑘ℎ(ℑ, ℌ) inequality is satisfied. Then, the mapping 𝑆 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 four chapters. The first and second chapters are in the form of a literature review, while the third and fourth chapter constitutes the original part of the thesis. The first chapter presents the fundamental definitions and theorems related to metric fixed point theory. Additionally, various spaces that generalize metric spaces are defined and their properties are examined. The aim is to contribute to the literature either by creating new spaces through modifications of the conditions on the metric or by altering contraction conditions on transformations, as generalizations of the Banach contraction principle also known as the Banach fixed point theorem put forth by many researchers. In the third chapter, the extended forms of classical fixed point theory under modular and graphical metric structures are discussed. The aim of the study is to establish fixed point results on graphical modular metric spaces, which are obtained by combining metric and modular approaches, and to demonstrate the analytical power of these structures, thereby extending the abstract theory to more general frameworks. Throughout the section, the definitions and fundamental topological properties of the new spaces arising from the combination of modular metric concepts and graphical structures will first be examined. Subsequently, fixed point theorems will be obtained for transformations satisfying the Banach, Kannan, and Reich contraction conditions within these spaces. The applicability of the fixed point results obtained in graphical modular metric spaces to various fields is also explored. Using transformations that satisfy the Reich-type contraction condition, the dynamic market equilibrium problem is addressed, and a fixed point formulation determining the stability of economic systems is developed. The same approach is then extended to a physical model — the heat conduction problem. Thus, the graphical modular–Reich contraction principle is evaluated within a unified framework that guarantees the existence of steady states in both economic and physical systems. First, the mathematical foundation of the heat conduction model will be presented, followed by a detailed examination of the physical interpretation of the fixed point of the operator defined in the graphical modular metric structure. This approach reveals the interdisciplinary strength of fixed point theory, demonstrating the applicability of abstract mathematical principles to the behavior of real-world systems. An example of a dynamic market equilibrium problem, which is an application of fixed point theory, will be presented. The proposed solutions to the dynamic market equilibrium problem are integrated into the dynamic market equilibrium framework commonly used in economic analyses, thereby forming a mathematical model through the solution of an initial value problem. Both consumer and producer markets are subject to significant fluctuations arising from daily price variations and pricing data. Despite the inherent volatility of prices, daily pricing trends have a substantial impact on the production market and the consumption market. Throughout this economic analysis, the economist aims to determine the prevailing price. In addition, the applicability of the fixed point results obtained in graphical modular metric spaces to a physical model is demonstrated. In particular, a one-dimensional nonlinear heat conduction problem is considered, and it will be shown that the solution of this problem is the fixed point of an operator satisfying the graphical modular–Reich contraction condition. Thus, the abstract theorems obtained in the previous sections become applicable to the analysis of real physical processes such as heat diffusion. In the concluding section, the findings derived from the study are presented, along with the recommedations concerning the pratical implementation of the theoretical frameworks discussed in the literatüre.