General theory of iterative projection methods for operator equations in abstract metric and normed spaces

2005 ◽  
pp. 13-86
Keyword(s):  
General Theory ◽  
Projection Methods ◽  
Normed Spaces ◽  
Operator Equations

scholarly journals The generalized projection methods in countably normed spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sarah Tawfeek ◽  
Nashat Faried ◽  
H. A. El-Sharkawy
Keyword(s):  
Banach Spaces ◽  
Projection Operator ◽  
Metric Projection ◽  
Projection Methods ◽  
Generalized Projection ◽  
Uniformly Convex ◽  
Normed Spaces ◽  
Generalized Projection Operator ◽  
Set Valued Mapping ◽  
Uniformly Smooth

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

scholarly journals PROJECTION METHODS FOR INTEGRAL EQUATIONS IN EPIDEMIC

2002 ◽  
Vol 7 (2) ◽  
pp. 229-240 ◽  
Author(s):  
L. Hacia
Keyword(s):  
Mathematical Modeling ◽  
Numerical Methods ◽  
General Theory ◽  
Integral Equations ◽  
Projection Methods ◽  
Volterra Integral Equations ◽  
Temporal Development ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Spatio Temporal

In this paper numerical methods for mixed integral equations are presented. Studied equations arise in the mathematical modeling of the spatio‐temporal development of an epidemic. The general theory of these equations is given and used in the projection methods. Projection methods lead to a system of algebraic equations or to a system of Volterra integral equations. The considered theory is illustrated by numerical examples.

scholarly journals The projection methods in countably normed spaces

2015 ◽  
Vol 2015 (1) ◽  
Author(s):  
Nashat Faried ◽  
Hany A El-Sharkawy
Keyword(s):  
Projection Methods ◽  
Normed Spaces
Sign in / Sign up

Export Citation Format

Share Document