In this survey we present some recent applications of proof mining to the fixed point theory of (asymptotically) nonexpansive mappings and to the metastability (in the sense of Terence Tao) of ergodic averages in uniformly convex Banach spaces.

In this work, we first introduce almost contraction mappings for a pair of two in cone metric spaces over Banach algebras (CMSBA). Then, observe that the class such setting contains those many well known mappings. Finally, prove some fixed point theorems, and obtain $(S,T)$-stability results Jungck iterations CMSBA.

In this paper, a new viscosity iterative process, which converges strongly to a common element of the set of fixed points of a finite family of pseudo-contractive mappings more general than non-expansive mappings, is introduced in Banach spaces. Strong convergence theorems are obtained under milder conditions. The results presented in this paper extend and unify most of the results that have be...

The purpose of this paper is to introduce an implicit iteration process for approximating common fixed points of two asymptotically nonexpansive mappings and to prove strong convergence theorems in uniformly convex Banach spaces.

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly L-Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend

The purpose of this paper is to prove that the modified Mann iteration process can be applied to approximate the common fixed point of three strictly hemicontractive mappings in smooth Banach spaces.

We construct adaptive Mann iterations for finding fixed points of strongly pseudocontractive mappings and solving nonlinear strongly accretive (not necessary continuous) operator equation Sx = f in p-smooth Banach spaces.

We characterize the holomorphic mappings f between complex Banach spaces that may be written in the form f = T ◦ g, where g is another holomorphic mapping and T belongs to a closed surjective operator ideal.

We prove strong convergence theorems for three iterative algorithms which approximate solutions to systems of variational inequalities for mappings of monotone type. All the theorems are set in reflexive Banach spaces and take into account possible computational errors.