Our aim of the research article was to obtain fixed point results for generalized contractions equipped with precisely defined control functions in the framework of $\mathfrak{F}$-bipolar metric spaces. As a consequence, some well-known results from the literature were derived as special cases of our leading theorems. To validate the theoretical findings and demonstrate their practical relevance, a non-trivial illustrative example is included. Furthermore, we explored the applicability of our major results by addressing the existence and uniqueness of solutions to Volterra integral equations, a class of problems frequently arising in mathematical modeling of real-world systems. In addition, we extended the utility of our fixed point theorems to homotopy theory, where they served as a foundational tool in analyzing the existence of continuous deformations between mappings, thereby contributing to the study of topological invariants and the structural properties of function spaces.

In this paper, we introduce the notion of a generalized ( $\alpha$, ${\mathrm{\Theta }}_{\mathcal{F}})$-contraction in the context of an orthogonal $\mathcal{F}$-complete metric space and obtain some new fixed point results for this newly introduced contraction. A nontrivial example is also provided to satisfy the validity of the established results. As consequences of our obtained results, we derive the leading results in [Fixed Point Theory Appl., 2015,185, 2015] and [Symmetry, 2020, 12,832]. As an application, we investigate the existence and uniqueness of the solution for a nonlinear fractional differential equation.

The aim of this paper is to define a Berinde type ( $\rho$, $\mu$)- $\vartheta$ contraction and establish some fixed point results for self mappings in the setting of complete metric spaces. We derive new fixed point results, which extend and improve some results in the literature. We also supply a non trivial example to support the obtained results. Finally, we investigate the existence of solutions for the nonlinear fractional differential equation.

In this paper, we solve the existence and uniqueness of a solution for a fractional differential equation by introducing some new fixed point results for rational ( $\alpha$, $\beta$, $\psi$)-contractions in the framework of orthogonal $\mathcal{F}$-metric spaces. We derive some well-known results in literature as consequences of our leading result.

The primary objective of this study is to establish novel fixed point theorems for single-valued mappings satisfying generalized contraction conditions in the framework of extended $b$ -suprametric spaces. These results not only generalize and unify existing fixed point results in the literature but also provide a broader context for analysis. Building upon this foundation, we further derive corresponding fixed point theorems in suprametric and $b$-suprametric spaces as special cases of our main results. To highlight the relevance and originality of the proposed theorems, we present a concrete example that validates the applicability and sharpness of the key findings. Finally, we demonstrate the practical significance of our theoretical contributions by applying the main result to establish the existence of solutions for a nonlinear Volterra-Fredholm integral equation.

The aim of this research article is to explore the notion of extended cone $b$-metric spaces over Banach algebras and to establish new fixed point theorems for both single-valued mappings and multi-valued mappings. These results extend and generalize several well-known findings in metric fixed point theory. To demonstrate the applicability of our theoretical results, we provide illustrative examples that highlight the distinct features of our approach. Additionally, we showcase the practical relevance of our primary theorem by solving a Fredholm integral equation of the second kind, which arises in the context of radar and sonar signal processing.