On the other hand, notice that the formalism of contractive cyclic self-maps in uniformly convex Banach spaces requires for the distances in-between any pairs of non-adjacent subsets to be identical.
Kirk and Xu [2] studied the asymptotic nonexpansive mapping in uniformly convex Banach spaces.
Now we show that the firmly nonexpansive mapping is a subclass of ψ-firmly nonexpansive mapping in uniformly convex Banach space.
Motivated by the works mentioned above, in this paper, we study the convergence of implicit viscosity iteration process (1.8) constructed from the pseudocontractive semigroup and -strongly pseudocontractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differential norms.
We also study the convergence of the explicit viscosity iteration process (1.12) constructed from the pseudocontractive semigroup and -strongly pseudocontractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differential norms.
Recently, Homaeipour and Razani [33] studied the following iterative scheme for a fixed point of relatively nonexpansive multi-valued mapping in uniformly convex and uniformly smooth Banach space E: { x 0 ∈ C, chosen arbitrary, x n + 1 = Π C J − 1 ( α n J x n + ( 1 − α n ) J z n ), z n ∈ T x n, n ≥ 0, (1.4).
In the following, we shall prove that a continuous mapping of asymptotically nonexpansive type in UCW-hyperbolic space with a monotone modulus of uniform convexity is demiclosed as it was noticed by Cöhde [25] for non-expansive mapping in uniformly convex Banach spaces.
Recently, Kangtunyakarn [11] introduced a new mapping in uniformly convex and 2-smooth Banach spaces to prove a strong convergence theorem for finding a common element of the set of fixed points of finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of solutions of variational inequality problems as follows.
In this paper, we study an inertial algorithm for approximating a common fixed point for a countable family of relatively nonexpansive maps in a uniformly convex and uniformly smooth real Banach space.
Later on, Kirk and Xu [23] introduced the concept of asymptotic pointwise nonexpansive mappings which generalizes the concept of asymptotically nonexpansive mappings and proved the existence of fixed points for such maps in a uniformly convex Banach space.
