Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces

Let nonempty subsets E and F of a Banach space X be given, along with a mapping S:E∪F→E∪F defined as noncyclic when S(E)⊆E and S(F)⊆F. In this case, an optimal pair of fixed points is defined as a point (p,q)∈E×F where p and q are fixed points of S that estimate the distance between E and F. This article explores an extended version of Göhde’s fixed point problem to identify optimal fixed point pairs for noncyclic relatively nonexpansive maps in strictly convex Banach spaces, while introducing new classes of noncyclic Kannan contractions, noncyclic relatively Kannan nonexpansive contractions using the proximal projection mapping defined on union of proximal pairs, and proving additional existence results with supporting examples.