The implicit midpoint rule (IMR) for nonexpansive mappings is established in Banach spaces. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a uniformly convex Banach space which either satisfies Opial’s property or has a Fréchet differentiable norm. Consequently, this algorithm applies in both `p and Lp for 1 < p < ∞.

We prove path convergence theorems and introduce a new iterative sequence for a countably infinite family of m-accretive mappings and prove strong convergence of the sequence to a common zero of these operators in uniformly convex real Banach space. Consequently, we obtain strong convergence theorems for a countably infinite family of pseudocontractive mappings. Our theorems extend and improve ...

*Correspondence: [email protected] 3Department of Mathematics, GDCW, Bosan Road, Multan, Pakistan Full list of author information is available at the end of the article Abstract In this paper, we first introduce a cyclic generalized contraction map in metric spaces and give an existence result for a best proximity point of such mappings in the setting of a uniformly convex Banach space. Then we...

vol. 162, No. 2, 1991 k-β and k-Nearly Uniformly Convex Banach Spaces Denka Kutzarova Different uniform geometrical properties have been defined between the uniform convexity and the reflexivity of Banach spaces. In the present paper we introduce other properties of this type, namely k-β and k-nearly uniform convexity. The definitions, as well as some of the results presented here, are announce...

In this paper, we introduce a one-step iterative scheme for finding a common fixed point of a finite family of multivalued quasi-nonexpansive mappings in a real uniformly convex Banach space. We establish weak and strong convergence theorems of the proposed iterative scheme under some appropriate conditions.

An analytic function f(z)= z+an+1zn+1+··· , defined on the unit disk = {z : |z|< 1}, is in the class Sp if zf ′(z)/f(z) is in the parabolic region Rew > |w −1|. This class is closely related to the class of uniformly convex functions. Sufficient conditions for function to be in Sp are obtained. In particular, we find condition on λ such that the function f(z), satisfying (1−α)(f(z)/z)μ+αf ′(z)(...

In this paper, we propose a viscosity iteration process for semigroup of asymptotically pseudocontractive mappings, and prove a strong convergence theorem in uniformly convex Banach space for the proposed iteration process.