Afrah Ahmad Noman Abdou

In this research article, we introduce and develop the notion of complex-valued b-suprametric spaces as a natural generalization of existing metric-type structures. Fundamental concepts, including convergence, Cauchy sequences, and completeness, are examined in this new setting. We establish new common fixed point theorems for generalized and cyclic rational contractive mappings. The obtained results extend and unify various known fixed point theorems available in the current literature. To demonstrate the applicability and effectiveness of our theoretical findings, illustrative nontrivial examples are provided. As an application, we investigate the existence and uniqueness of solutions for Caputo fractional differential equations, which naturally arise in systems with hereditary and memory effects, particularly in biomedical modeling of viscoelastic biological tissues such as arteries, cartilage, and brain tissue. This demonstrates both the mathematical strength and the practical relevance of the proposed framework.

The primary objective of this study is to establish common fixed-point theorems for single-valued and multivalued mappings in the setting of complex-valued suprametric spaces. A new contraction is introduced for single-valued mappings, while a generalized Hausdorff distance aids in proving results for multivalued mappings. Additionally, fixed-point theorems are extended to complex-valued metric spaces. An example illustrates the findings, and an application to Fredholm integral equations demonstrates their role in image processing.

This study explores the concept of complex-valued suprametric spaces, a recent and generalized framework in fixed point theory. This concept extends both complex-valued spaces and classical metric spaces. The article aims to establish common fixed point results for rational contractions governed by control functions dependent on two variables within the setting of complex-valued suprametric spaces. These findings generalize several well-known results in the field. An illustrative example is provided to validate the novelty of the key result. Furthermore, the study derives common fixed point theorems for rational contractions with control functions of a single variable under the same framework. As a practical application, the findings are applied to study the solution of a nonlinear mixed Volterra–Fredholm integral equation within the context of predator-prey dynamics.

The article outlines a creative approach by fusing ideas from fuzzy contractions, graph mappings, and Kannan mappings to create a brand-new notion known as “Kannan-graph-fuzzy contraction”. This combination of concepts makes it possible to create a unique framework for resolving real-world issues controlled by nonlinear models. It is anticipated that using Kannan-graph-fuzzy contractions will offer a fresh approach to discussing solution existence in situations where these ideas overlap. This strategy might have a significant impact on a lot of real-world applications, proving how practical multidisciplinary techniques are for tackling issues in the real world originating from technology and engineering. The article includes several illustrated examples based on computer simulations, making the results more explicit.

The article outlines a creative approach by fusing ideas from fuzzy contractions, graph mappings, and Kannan mappings to create a brand-new notion known as "Kannan-graph-fuzzy contraction." This combination of concepts makes it possible to create a unique framework for resolving real-world issues controlled by nonlinear models. It is anticipated that using Kannan-graph-fuzzy contractions will offer a fresh approach to discussing solution existence in situations where these ideas overlap. This strategy might have a significant impact on a lot of real-world applications, proving how practical multidisciplinary techniques are for tackling issues in the real world originating from technology and engineering. The article includes several illustrated examples based on computer simulations, making the results more explicit.

The purpose of this research work is to explore the solution of the Fredholm integral equation by common fixed point results in bicomplex valued metric spaces. In this way, we develop some common fixed point theorems for generalized contractions containing point-dependent control functions in the context of bicomplex valued metric spaces. An illustrative and practical example is also given to show the novelty of the most important result.

In this article, we develop a new notion that combines fixed-point theory and graph theory: graphical bipolar b-metric spaces. We demonstrate fixed-point solutions in the framework of graphical bipolar b-metric spaces, employing covariant and contravariant mapping contractions, which is a new addition to this end. This article also features illustrative examples drawn from various contexts to further demonstrate our findings. This is a significant study since it melds ideas from graph theory with those from generalized bipolar metric spaces, and considers that the symmetry of the edges of the underlying graphs connected with the enunciated metric spaces is essential in the graphical metric spaces.