Convergence analysis of a new implicit iteration process for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces

We propose a new implicit iteration process for finding a common element of the sets of solutions of a finite family of non-Lipschitzian asymptotically nonexpansive mapping in the intermediate sense in Banach spaces. Furthermore, we modified Picard-Ishikawa hybrid scheme recently introduced by Okeke (Afr Math 30:817-835, 2019) to its new implicit iterative scheme. Then we used the new implicit iterative scheme to prove the strong convergence theorems, the data dependence and the stability results for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. Also, we apply the new implicit iterative process in finding the solution of delay differential equation. Moreover, we establish some numerical examples to illustrate our claims. The results presented in this paper extend and improve the result of Okeke (Afr Math 30:817-835, 2019), Saluja (Kragujevac J Math 2:237-249, 2012) and others in the literature.