Fixed point results of an implicit iterative scheme for fractal generations

<abstract><p>In this paper, we derive the escape criteria for general complex polynomial $ f(x) = \sum_{i = 0}^{p}a_{i}x^{i} $ with $ p\geq2 $, where $ a_{i} \in \mathbb{C} $ for $ i = 0, 1, 2, \dots, p $ to generate the fractals. Moreover, we study the orbit of an implicit iteration (i.e., Jungck-Ishikawa iteration with $ s $-convexity) and develop algorithms for Mandelbrot set and Multi-corn or Multi-edge set. Moreover, we draw some complex graphs and observe how the graph of Mandelbrot set and Multi-corn or Multi-edge set vary with the variation of $ a_{i} $'s.</p></abstract>

Implicit Iterative Method for Hierarchical Variational Inequalities

We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained.