In this paper, we take advantage of implicit relationships to come up with a new concept called " ${\mathcal{A}}_{\vartheta }$- $\alpha$-contraction mapping". We utilized our new notion to formulate and prove some common fixed point theorems for two and four self-mappings over complete extended quasi $b$-metric spaces under a set of conditions. Our main results widen and improve many existing results in the literature. To support our research, we present some examples as applications to our main findings.

In the present work, we will establish and prove some fixed point theorems for mappings that satisfy a set of conditions in controlled metric type spaces introduced by Mlaiki et al. [N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, Controlled metric type spaces and the related contraction principle. Mathematics 2018, 6,194]. Our technique in constructing our new contraction conditions is to insert the control function $\theta (u,l)$ that appears on the right hand side of the triangular inequality of the definition of the controlled metric spaces in the right hand side of our proposed contraction conditions. Our results enrich the field of fixed point theory with novel findings that generalize many findings found in the literature. We provide an example to show the usefulness of our results. Also, we present an application to our results to show their significance.

The construction of contraction conditions plays an important role in science for formulating new findings in fixed point theories of mappings under a set of specific conditions. The aim of this work is to take advantage of the idea of extended $b$-metric spaces in the sense introduced by Kamran et al. [A generalization of $b$-metric space and some fixed point theorems, Mathematics, 5 (2017), 1–7] to construct new contraction conditions to obtain new results related to fixed points. Our results enrich and extend some known results from $b$-metric spaces to extended b-metric spaces. We construct some examples to show the usefulness of our results. Also, we provide some applications to support our results.

In this study, we analyze a Casson micropolar hybrid nanofluid flow and heat transfer characteristics over a stretching sheet/cylinder. The analysis takes Joule heating and thermal radiation into account, as well as the variable thermal conductivity and the prescribed thermal conditions. The nanoparticles of $Ag$ and $CuO$ with base fluid $EG$ (Ethylene Glycol) are discussed. Additionally, the study explores the impact of an inclined magnetic field on the flow behavior. The governing partial differential equations are described, including the conservation of momentum, mass, and energy, which are transformed into a nonlinear ordinary differential equation using appropriate similarity transformations. Then, these equations are numerically cracked using a reliable computational technique. The study reveals significant influences of hybrid nanofluid properties on the velocity, temperature, and microrotation profiles. The inclined magnetic field significantly affects the fluid dynamics, leading to flow resistance and thermal performance variations. The results highlight the importance of these factors in enhancing the thermal efficiency of systems using hybrid nanofluids. The thermal thickness of the prescribed conditions (PHF and PST) for the temperature enhanced due to an increment in the factor of radiation. As more radiative heat is absorbed, the fluid internal energy increases, thus leading to a rise in the temperature because the absorbed radiation boosts the kinetic energy of the fluid molecules, thereby increasing the fluid temperature. The heat transfer of the sheet achieved more as compared to the stretching cylinder.