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In this paper, we construct a common iterative scheme that allows to unify two important results established recently by Ioffe and Ait Mansour, Bahraoui, El Bekkali, respectively. Our results rely on a weaker concept of metric regularity, called orbital regularity. Some applications are given to approximate and/or exact coincidence double fixed point problems as well as to the perturbation stability of approximate and/or exact Milyutin regularity of set-valued mappings in metric spaces. [ABSTRACT FROM AUTHOR]

Abstract We study an approximate coincidence point and a common fixed point problem for a hybrid pair of mappings with constraints in Menger PM-spaces, and obtain some new results. We derive interesting consequences of the main results by using the properties of a Menger–Hausdorff metric, and analogous results based on graphs instead of partial orders can be similarly formulated. Moreover, we construct two examples to reveal that the main results are valid, and show that the main results can be used to explore the existence of solutions to a system of nonlinear integral equations.

In the present paper, the notion of stability of fixed point iterative procedure of [A.M. Harder and T.L. Hicks: Stability results for fixed point iteration procedures, Math. Japonica, 33(1988), No. 5, 693-706] is extended from the fixed point setting to the approximate fixed point and approximate coincidence point considerations. These extensions are possible by introducing new notions of stability of iterative procedures; viz- (ε; T)-stability and (ε; S; T)-stability. The notion of approximate coincidence point was initiated in the article [M.O. Olatinwo: Some approximate fixed point and approximate coincidence point theorems in b-metric spaces, J. Adv. Math. Stud., 11(2018), No. 3, 576-586], and the notion in this article is employed in this present article. Some strong convergence results pertaining to some celebrated fixed point iterative procedures are also extended from the fixed point setting to the approximate fixed point (ε-fixed point) and approximate coincidence point (ε-coincidence point) situations. We obtain our results by employing two contractive inequality conditions. Our results generalize and extend a multitude of results in the literature. [ABSTRACT FROM AUTHOR]