In mathematical analysis, the Hausdorff derivatives or the fractal derivatives play an important role. Fixed-point theorems and metric fixed-point theory have varied applications in establishing a unique common solution to differential equations and integral equations. In the present work, some fixed-point theorems using the extension of Meir–Keeler contraction in the setting of bipolar metric spaces have been proved. The derived results have been supplemented with non-trivial examples. Our results extend and generalise the results established in the past. We have provided an application to find an analytical solution to an Integral Equation to supplement the derived result.

The purpose of this paper is to establish some common fixed point theorems of w-compatible maps in bipolar metric spaces by employing a comparison function ϕ instead of some altering distance functions. We employ generalized type contraction conditions involving the comparison function ϕ to enunciate common fixed point theorems. Further, we provide illustrative examples to uphold our results.

In this article, we introduce the concept of compatible mappings of type $(A)$ and weaken the same in the setting of Bipolar metric spaces and established fixed point results in the setting of Bipolar metric spaces, using implicit relation function. The derived results have been supplemented with suitable non-trivial examples. Our results have extended and generalized some results proven in the past and some open problems for future research has been given.

Introduction/purpose: A new three-step iteration process, which converges faster than the Mann iteration and the S-iteration, is introduced, as well as some convergence results for approximation of fixed points of the Suzuki generalized nonexpansive mappings and nearly asymptotically nonexpansive mappings have been established. Methods: The authors provide a specific three-step iterative method {xn} in a Banach space, defined as a sequence of convex combinations of the current iterate and its images under the mapping T, with control sequences {αn}, {βn}, {γn} ⊆ (0, 1). The results are proved in the setting of uniformly convex Banach spaces, where T is assumed to be either a Suzuki generalized nonexpansive mapping or a nearly asymptotically nonexpansive mapping. The authors obtain both weak and strong convergence theorems by using demiclosedness principles, Suzuki generalized nonexpansive mapping properties, and suitable lemmas on the behavior of the iterates. To compare the rates of convergence, they perform numerical experiments (usually implemented in MATLAB) where the proposed three-step iteration is run in parallel with the Thakur and S-iterations. The iterates are graphed to display the error convergence per iteration. Results: The new iteration scheme converges faster than the S-iteration scheme if the mapping is a contraction. The new iteration scheme converges to a fixed point of a Suzuki generalized nonexpansive mapping under suitable conditions. The new iteration scheme converges to a fixed point of a nearly asymptotically nonexpansive mapping under suitable conditions. Conclusions: The three-step iteration algorithm is proved, both theoretically and numerically, to converge faster than the Mann iteration and the S iteration (and sometimes faster than several other existing methods) for the considered types of mappings. The authors prove weak and strong convergence theorems for fixed points of the Suzuki generalized nonexpansive mappings and nearly asymptotically nonexpansive mappings in uniformly convex Banach spaces, thus generalizing, extending, and unifying several existing fixed-point approximation results in the literature.