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 aim of this research is to introduce two new notions, $\mathsf{\Theta }$-($\mathsf{\Xi }$,h)-contraction and rational ($\alpha$,$\eta )$-$\psi$-interpolative contraction, in the setting of $\mathfrak{F}$-metric space and to establish corresponding fixed point theorems. To reinforce understanding and highlight the novelty of our findings, we provide a non-trivial example that not only supports the obtained results but also illuminates the established theory. Finally, we apply our main result to discuss the existence and uniqueness of solutions for a fractional differential equation describing an economic growth model.

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.

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we study the existence of fixed points for contractive and nonexpansive Kannan maps in the setting of modular metric spaces.These are related to the successive approximations of fixed points (via orbits) which converge to the fixed points in the modular sense, which is weaker than the metric convergence.

This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal $\Theta$-contraction and orthogonal ($\alpha ,\Theta )$-contraction in the setting of complete orthogonal metric spaces and prove fixed point theorems for such contractions. In this way, we consolidate and amend innumerable celebrated results in fixed point theory. We provide a non-trivial example to show the legitimacy of the established results.

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.