Abstract
The aim of this paper is to introduce a viscosity iterative algorithm for the implicit midpoint rule of nonexpansive mappings in uniformly smooth spaces. Under some appropriate conditions on the parameters, we prove some strong convergence theorems. As applications, we apply our main results to solving fixed point problems of strict pseudocontractive mappings, variational inequality problems in Banach spaces and equilibrium problems in Hilbert spaces. Finally, we give some numerical examples for supporting our main results.
Highlights
1 Introduction Throughout this paper, we assume that E and E∗ is a real Banach space and the dual space of E, respectively
Can we extend and improve the main results of Xu et al [ ] from Hilbert space to general Banach space? For example we might consider a uniformly smooth Banach space
Under some suitable conditions on the parameters, we prove some strong convergence theorems
Summary
Throughout this paper, we assume that E and E∗ is a real Banach space and the dual space of E, respectively. It is well known that the duality mapping J : E → E∗ is defined by. When J is single-valued, we denote it by j. We notice that if E is a Hilbert space, J is the identity mapping and if E is smooth, J is single-valued. A mapping f : C → C is said to be a contraction, if there exists a constant α ∈ [ , ) satisfying f (x) – f (y) ≤ α x – y , ∀x, y ∈ C. We use C to denote the collection of all contractions from C into itself. We say that Banach space E is uniformly smooth if ρE(t) → as t →. It is well known that typical example of uniformly smooth
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