In this paper, we discuss some Iterations and apply it to $d/BCK$-algebras and semigroups. Given a $BCK$-algebra $(X,*, 0)$, we construct an edge $d$-algebra by using the notion of Iterations.

This paper extends the notion of weak convergence in metric spaces to the case of S-metric spaces. Moreover, some results on the weak convergence of fixed point Iterations of Banach's, Kannan's, Chatterjee's, Reich's, Hardy and Roger's types of contractions on S-metric spaces are obtained. In addition, an example is presented to demonstrate our primary result.

We introduce a modified Noor iteration scheme generated by an infinite family of strict pseudo-contractive mappings and prove the strong convergence theorems of the scheme in the framework of q-uniformly smooth and strictly convex Banach space. Results shown here are extensions and refinements of previously known results.

In this paper, we propose a novel Mann iterative algorithm for discovering a shared fixed point of two nonexpansive mappings in real Hilbert spaces. We establish the weak convergence of this fixed-point approach under new conditions, and additionally demonstrate its strong convergence with the inclusion of an additional requirement. Our findings expand upon previous results presented by Kanzow and Shehu, as well as by Cho et al. Furthermore, we showcase the versatility of our main results through the presentation of various applications in the last section, accompanied by illustrative examples.

In this paper, we study the Iterations of quasi $\phi$-nonexpansive mappings and its applications in Banach spaces. At the first, we prove strong convergence of the sequence generated by the hybrid proximal point method to a common fixed point of a family of quasi $\phi$-nonexpansive mappings.  Then, we give  applications of our main results in equilibrium problems.