Strong Convergence Theorem for Finite Family of Asymptotically Nonexpansive in the Intermediate Sense Nonself Maps

: Let K be a nonexpansive retract of a uniformly convex Banach space X with retraction P. Let Ti: K → X (i= 1,...,m) be a finite family of uniformly continuous asymptotically nonexpansive in the intermediate sense maps with a nonempty common fixed points set F. Sufficient conditions for the strong convergence of a sequence of successive approximations generated by an m-step algorithm to a point of F are proved. MSC(2010): 47H10, 47J25


Introduction
Let K be a nonempty subset of a normed linear space E. K is said to be (sequentially) compact if every bounded sequence in K has a subsequence that converges in K. K is said to be boundedly compact if every bounded subset of K is compact. In finite dimensional spaces, closed subsets are boundedly compact.
Given a subset S of K, we shall denote by Co(S) and ccl(S) the convex hull and the closed convex hull of S respectively. If K is boundedly compact convex and S is bounded, then co(S) and hence ccl(S) are compact convex subsets of K.
The class of total asymptotically nonexpansive type mappings includes the class of mappings which are asymptotically nonexpansive in the intermediate sense and Generalised asymptotically nonexpansive maps. These classes of mappings have been studied by several authors (see e.g. Goebel and Kirk (1972), Lim and Xu (1994), Alber et al (2008) ,Zegeye and Shahzad (2013), Nnubia and Bishop (2020)).

Preliminaries
We shall make use of the following in the sequel.
A Banach space E is said to satisfy Opial's condition if for each sequence {x _n} subset E which converges weakly to a point z in E, we have that liminf_{n → ∞} ||x_n -z|| < liminf_{n → ∞} ||x_n -y||, forall y in E; y not equal to z.
It is well known that every Hilbert space satisfies

Proposition 2.1
Suppose that there exist c > 0, k > 0 constants such that Q(t) ≤ c t forall t ≥ k, then T is total asymptotically nonexpansive if and only if T is asymptotically nonexpansive in the itermediate sense.

Theorem 3.3
Let K, X, P,T_i's, F, {x_n} be as in Theorem 3.1 Then, {x_n} converges strongly to a common fixed point of T_i 's if one of the T_i 's satisfies any of the following condition: (a) condition B (b ). semi-compact (c). demi compact at 0 in K, (d). completely continuous.

Theorem 3.4
Let K, X, P,T_i's, F, {x_n} be as in Theorem 3.1, then, {x_n} converges strongly to a point of F if (a) K is compact. (b). K is boundedly compact.

Conclusion
Our iterative process generalizes some of the existing ones, our theorems improve, unify and extend several known results and our method of proof is of independent interest.