Fixed-Point and Common Fixed-Point Theorems for Rational and Almost Contractions in θ_R-Metric Space

In the present work, we establish fixed-point theorems for rational and almost contractions in the framework of a θ_R-metric space. The concept of θ_R-metric space is based on the incorporation of a control function θ:[0,∞)³→[0,∞) into the quadrilateral inequality of the generalized metric space introduced by Branciari. To prove our results, we impose certain natural assumptions on the control function, namely normalization, monotonicity, continuity, and iterative contractive control. These conditions ensure the well-posedness of the θ_R-metric structure and play a crucial role in establishing the convergence of the associated iterative sequences. Then, we ensure the existence (and uniqueness) of a common fixed point for a pair of self-mappings, satisfying rational and almost contractive conditions within this framework. Moreover, a detailed comparison is carried out between the results obtained in this study and several well-known fixed-point and common fixed-point theorems established in rectangular metric spaces and other related generalized metric spaces. This comparison clearly demonstrates that our results significantly extend, generalize, and refine many existing theorems available in the literature. To illustrate the applicability and wide scope of the established theorems, appropriate and carefully constructed examples are presented. These examples substantiate the theoretical results and illustrate that the established theorems constitute proper extensions and meaningful improvements over previously known findings.