In this paper, we introduce the concept of $\left(\alpha ,\text{BK}\right)$ -contractions and prove the existence of fixed points for contravariant mappings on bipolar metric spaces.

In this article, we introduce the neighborhood versions of two classical topological indices, namely neighborhood geometric–arithmetic and neighborhood atom bond connectivity indices. We study the graph-theoretic properties of these new topological indices for some known graphs, e.g., complete graph $K_n$, regular graph $R_n$, cycle graph $C_n$, star graph $S_n$, pendant graph, and irregular graph and further establish their respective bounds. We note that the neighbourhood geometric–arithmetic index of $K_n$, $R_n$, $C_n$, and $S_n$ is equal to the number of edges. The neighborhood atom bond connectivity index of an arbitrary simple graph $\mathcal{G}$ is strictly less than the number of edges. Our results contribute to the literature in this direction.

In this article, we propose two Banach-type fixed point theorems on bipolar metric spaces. More specifically, we look at covariant maps between bipolar metric spaces and consider iterates of the map involved. We also propose a generalization of the Banach fixed point result via Caristi-type arguments.

The main purpose of this paper is to propose and study a two-step inertial anchored version of the forward–reflected–backward splitting algorithm of Malitsky and Tam in a real Hilbert space. Our proposed algorithm converges strongly to a zero of the sum of a set-valued maximal monotone operator and a single-valued monotone Lipschitz continuous operator. It involves only one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration; a feature that is absent in many other available strongly convergent splitting methods in the literature. Finally, we perform numerical experiments involving image restoration problem and compare our algorithm with known related strongly convergent splitting algorithms in the literature.

This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply s-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem that determines the condition when a sequence diverges to infinity, known as the escape criterion, for the transcendental sine function $sin\left(u^m\right)-\alpha u+\beta$, where $u,\alpha ,$$\beta \in \mathbb{C}$ and $m\ge 2$. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Visual representations are included to demonstrate how various parameters influence the coloration and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set’s structure.