In this paper, we combine the subgradient extragradient method with the Halpern method for finding a solution of a variational inequality involving a monotone Lipschitz mapping in Banach spaces. By using the generalized projection operator and the Lyapunov functional introduced by Alber, we prove a strong convergence theorem. We also consider the problem of finding a common element of the set o...

Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators. For examp...

In this paper, we introduce a new iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inversestrongly-monotone mapping in a Banach space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common el...

We study convergences of Mann and Ishikawa iteration processes for mappings of asymptotically quasi-nonexpansive type in Banach spaces. 1. Introduction and preliminaries. Let D be a nonempty subset of a real Banach space X and T : D → D a nonlinear mapping. The mapping T is said to be asymptotically quasi-nonexpansive (see [5]) if F(T) = ∅ and there exists a sequence {k n } in [0, ∞) with lim n...

In this article, a modified gradient-projection algorithm (GPA) is introduced, which combines Xu’s idea of an alternative averaged mapping approach to the GPA and the general iterative method for nonexpansive mappings in Hilbert space introduced by Marino and Xu. Under suitable conditions, it is proved that the strong convergence of the sequences generated by implicit and explicit schemes to a ...

In this work we use the Noor iteration process for total asymptotically nonexpansive mapping to establish the strong and $Delta$-convergence theorems in the framework of CAT(0) spaces. By doing this, some of the results existing in the current literature  generalize, unify and extend.

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