scholarly journals Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Set ◽  
Variational Problems ◽  
Generalized Projection ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Nonconvex Sets ◽  
Generalized Projections ◽  
Uniformly Smooth Banach Spaces

The present paper is devoted to the study of the generalized projectionπK:X∗→K, whereXis a uniformly convex and uniformly smooth Banach space andKis a nonempty closed (not necessarily convex) set inX. Our main result is the density of the pointsx∗∈X∗having unique generalized projection over nonempty close sets inX. Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.

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 Inertial Krasnoselskii–Mann Method in Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 638
Author(s):  
Yekini Shehu ◽  
Aviv Gibali
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Splitting Method ◽  
Uniformly Convex ◽  
Inclusion Problems ◽  
Uniformly Smooth ◽  
Mann Algorithm ◽  
Uniformly Smooth Banach Spaces

In this paper, we give a general inertial Krasnoselskii–Mann algorithm for solving inclusion problems in Banach Spaces. First, we establish a weak convergence in real uniformly convex and q-uniformly smooth Banach spaces for finding fixed points of nonexpansive mappings. Then, a strong convergence is obtained for the inertial generalized forward-backward splitting method for the inclusion. Our results extend many recent and related results obtained in real Hilbert spaces.

scholarly journals Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

2016 ◽  
Vol 25 (1) ◽  
pp. 107-120
Author(s):  
T. M. M. SOW ◽  
◽  
C. DIOP ◽  
N. DJITTE ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Process ◽  
Real Banach Space ◽  
Monotone Mappings ◽  
Uniformly Convex ◽  
Hammerstein Equation ◽  
Uniformly Smooth ◽  
Monotone Maps

For q > 1 and p > 1, let E be a 2-uniformly convex and q-uniformly smooth or p- uniformly convex and 2-uniformly smooth real Banach space and F : E → E∗, K : E∗ → E be bounded and strongly monotone maps with D(K) = R(F) = E∗. We construct a coupled iterative process and prove its strong convergence to a solution of the Hammerstein equation u + KF u = 0. Futhermore, our technique of proof is of independent of interest.

scholarly journals A New Hybrid Algorithm forλ-Strict Asymptotically Pseudocontractions in 2-Uniformly Smooth Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Xin-dong Liu ◽  
Shih-sen Chang
Keyword(s):  
Banach Spaces ◽  
Hybrid Algorithm ◽  
Metric Projection ◽  
Projection Algorithm ◽  
Strong Convergence Theorem ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Hybrid Projection Algorithm ◽  
Uniformly Smooth Banach Spaces ◽  
Asymptotically Pseudocontractive Mapping

A new hybrid projection algorithm is considered for aλ-strict asymptotically pseudocontractive mapping. Using the metric projection, a strong convergence theorem is obtained in a uniformly convex and 2-uniformly smooth Banach spaces. The result presented in this paper mainly improves and extends the corresponding results of Matsushita and Takahashi (2008), Dehghan (2011) Kang and Wang (2011), and many others.

scholarly journals Iterative solutions ofK-positive definite operator equations in real uniformly smooth Banach spaces

2001 ◽  
Vol 27 (3) ◽  
pp. 155-160
Author(s):  
Zeqing Liu ◽  
Shin Min Kang ◽  
Jeong Sheok Ume
Keyword(s):  
Banach Space ◽  
Iterative Method ◽  
Banach Spaces ◽  
Positive Definite ◽  
Operator Equations ◽  
Uniformly Smooth ◽  
Definite Operator ◽  
Iterative Solutions ◽  
Uniformly Smooth Banach Spaces ◽  
Positive Definite Operator

LetXbe a real uniformly smooth Banach space and letT:D(T)⫅X→Xbe aK-positive definite operator. Under suitable conditions we establish that the iterative method by Bai (1999) converges strongly to the unique solution of the equationTx=f,f∈X. The results presented in this paper generalize the corresponding results of Bai (1999), Chidume and Aneke (1993), and Chidume and Osilike (1997).

Algorithm for Hammerstein equations with monotone mappings in certain Banach spaces

2016 ◽  
Vol 25 (1) ◽  
pp. 107-120
Author(s):  
T. M. M. SOW ◽  
◽  
C. DIOP ◽  
N. DJITTE ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Process ◽  
Real Banach Space ◽  
Monotone Mappings ◽  
Uniformly Convex ◽  
Hammerstein Equation ◽  
Uniformly Smooth ◽  
Monotone Maps

For q > 1 and p > 1, let E be a 2-uniformly convex and q-uniformly smooth or p- uniformly convex and 2-uniformly smooth real Banach space and F : E → E∗, K : E∗ → E be bounded and strongly monotone maps with D(K) = R(F) = E∗. We construct a coupled iterative process and prove its strong convergence to a solution of the Hammerstein equation u + KF u = 0. Futhermore, our technique of proof is of independent of interest.

A CONVERGENCE THEOREM FOR COMMON ELEMENTS OF EQUILIBRIUM PROBLEMS AND MAPPINGS SATISFYING CONDITION (Φ-Eµ) IN UNIFORMLY CONVEX AND UNIFORMLY SMOOTH BANACH SPACES

2018 ◽  
Vol 1 (30) ◽  
pp. 67-77
Author(s):  
Hieu Trung Nguyen ◽  
Tien Cam Truong
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Satisfying Condition ◽  
Common Element ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Uniformly Smooth ◽  
Point Set ◽  
Uniformly Smooth Banach Spaces

In this paper, we propose a new hybrid iteration for finding a common element of solution set of equilibrium problems and the fixed point set of mappings  satisfying condi- tion (Φ-Eµ), and establish the convergence of this iteration in uniformly convex and uniformly smooth Banach spaces. From this theorem, we geta corollary for the convergence for equilibrium problems and mappings satisfying condition (Eµ) in real Hilbert spaces. In addition, an example is provided to illustrate for the convergence of equilibrium problems and mappings satisfying condition (Φ-Eµ). These results are the generations and improvements of some  existing results in the literature

The Equivalence of Mann and Implicit Mann Iterations for Uniformly Pseudocontractive Mappings in Uniformly Smooth Banach Spaces

2011 ◽  
Vol 50-51 ◽  
pp. 718-722
Author(s):  
Cheng Wang ◽  
Zhi Ming Wang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Convex Subset ◽  
Real Banach Space ◽  
Nonempty Closed Convex Subset ◽  
Mann Iteration ◽  
Pseudocontractive Mappings ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

In this paper, suppose is an arbitrary uniformly smooth real Banach space, and is a nonempty closed convex subset of . Let be a generalized Lipschitzian and uniformly pseudocontractive self-map with . Suppose that , are defined by Mann iteration and implicit Mann iteration respectively, with the iterative parameter satisfying certain conditions. Then the above two iterations that converge strongly to fixed point of are equivalent.

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