The generalized projection methods in countably normed spaces

Sarah Tawfeek,H A El-Sharkawy,Nashat Faried

doi:10.1186/s13660-021-02701-z

Sarah Tawfeek, H A El-Sharkawy + Show 1 more

Open Access

https://doi.org/10.1186/s13660-021-02701-z

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Abstract

Let E be a Banach space with dual space E^{*}, and let K be a nonempty, closed, and convex subset of E. We generalize the concept of generalized projection operator “Pi _{K}: E rightarrow K” from uniformly convex uniformly smooth Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties. We show the relation between J-orthogonality and generalized projection operator Pi _{K} and give examples to clarify this relation. We introduce a comparison between the metric projection operator P_{K} and the generalized projection operator Pi _{K} in uniformly convex uniformly smooth complete countably normed spaces, and we give an example explaining how to evaluate the metric projection P_{K} and the generalized projection Pi _{K} in some cases of countably normed spaces, and this example illustrates that the generalized projection operator Pi _{K} in general is a set-valued mapping. Also we generalize the generalized projection operator “pi _{K}: E^{*} rightarrow K” from reflexive Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties in these spaces.

Highlights

Let E be a Banach space with dual space E∗, and let K be a nonempty, closed, and convex subset of E
We extend the concept of generalized projection operators “ K : E → K ” from uniformly convex uniformly smooth Banach spaces to uniformly convex uniformly smooth countably normed spaces and “πK : E∗ → K ” from reflexive Banach spaces to uniformly convex uniformly smooth countably normed spaces
4 Conclusion In this paper we extend the concept of the generalized projection operator “ K : E → K ” from uniformly convex uniformly smooth Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties

Summary

Introduction

Let E be a Banach space with dual space E∗, and let K be a nonempty, closed, and convex subset of E. In 1996, Alber [1] introduced the generalized projection operators “ K : E → K ” and “πK : E∗ → K ” in uniformly convex and uniformly smooth Banach spaces, which are a natural extension of the classical metric projection operators of Hilbert spaces, and studied their properties in detail. Alber [1] presented two of the most important applications of the generalized projection operators: solving variational inequalities by iterative projection methods and finding a common point of closed convex sets by iterative projection methods in Banach spaces. In 2005, Li [3] extended the generalized projection operator πK : E∗ → K from uniformly convex uniformly smooth Banach spaces to reflexive Banach spaces and studied the properties and applications of the generalized projection operator. We present a comparison between metric projection and generalized projection in uniformly convex uniformly smooth complete countably normed spaces

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Conclusion