Mathematics

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Browsing Mathematics by Author "Dr. Abdallah A. Hakawati"

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    Best Approximation in General Normed Spaces
    (2001) Mu'tas Hasan Mahmoud Al-Sayed; Dr. Abdallah A. Hakawati
    Let X=(X,||.||) be a normed space and suppose that any given x in X is to be approximated by an element Y in Y, where Y is a fixed subspace of X. We let d denoted the distance from x to Y. By definition, d = d(x, Y) = infyєY ||x – y|| .Clearly, d depends on both x and Y, which we keep, fixed, so that the simple notation d is in order. If there exists a YoєY such that ||x - Yo|| =d. then Yo is called a best approximation of Y to x or a best approximant of x in Y. We see that a best approximation Yo is an element of minimum distance from the given x. Such a YoєY mayor may not exist; this raises the problem of existence. The problem of uniqueness is of practical interest, too, since for a given x and Y there may be more than one best approximation. My thesis consists of three chapters. In chapter one we summarize some of the essential and basic concepts which shall be needed in the following chapters, this chapter consists of two sections; in the first one we present metric, normed, Banach spaces, and the last one we present inner product, and Hilbert spaces. This chapter is absolutely fundamental. In chapter two, we define best approximations in section one. In section two we study some properties of the set of all best approximations P(x,Y). In section three we study some properties of the proximinaI set and show that compact subspace and finite-dimensional subspace are proximal. In section four we consider the problem of uniqueness of best approximation. In section five we review the properties of OrIicz spaces in which we introduce some of the basic theory of proximinality In chapter three, which is the main body of our thesis, we, in section one, study the main characterizations and properties of best approximations and some consequences of the characterization in arbitrary normed linear spaces. In sections two and three we gives some application in several spaces like L ¹(T,v), C(K) and CR(K).
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    Cone Metric Spaces
    (2016) Haitham Darweesh Abu Sarries; Dr. Abdallah A. Hakawati
    Cone metric spaces were introduced in [1] by means of partially ordering real Banach spaces by specified cones. In [4] and [8] , the nation of cone – normed spaces was introduced. cone- metric spaces, and hence, cone- normed spaces were shown to be first countable topological spaces. The reader may consult [5] for this development. In [6], it was shown that, in a sense, cone- metric spaces are not, really, generalizations of metric spaces. This was the motive to do further investigations. Now, we put things in order. 1. Definition:[1] Let (E ,‖∙‖) be a real Banach space and P a subset of E then P is called a cone if : (a) P is closed, convex, nonempty, and P ≠ {0}. (b) a,b ∈ ℝ ; a,b ≥ 0 ; x, y ∈ P ⇒ ax+by ∈ P. (c) x ∈ P and –x ∈ P ⇒ x = 0. 2. Example: [13] Let E= ℓ¹, the absolutely summable real sequences. Then the set P = {x ∈ E : xn ≥ 0 , n} is a cone in E. In our project, we will attempt to enforce the feeling that cone metric spaces are not real generalization of metric spaces by the necessary theory and examples. In the meantime, we will keep it conceivable to arrive at generalization aspects.
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    Convexity, Fixed Point Theorems and Walrasian Equilibrium
    (2002) Abdul-Rahim Omar Amin Nur; Dr. Abdallah A. Hakawati; Dr. Rimon Abduh' Y. Jadoun
    In this thesis, I will deal with an application of fixed-point theorem of set valued map [Let X and Y be two subsets of Rⁿ: A set-valued map F from X to Y, is a map that associates with any x є X a subset F(x) of Y, A fixed point x for F exists if x є F(x) ], and convexity to prove the existence of Walrasian Equilibrium under sufficient conditions for both pure exchange economy and private ownership economy. Then I will show how to modify these theorems in more general cases under uncertainty and externalities.
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    The S-Property and Best Approximation
    (2000) Sawsan Azmi Sabri Al-Dwaik; Dr. Abdallah A. Hakawati; Dr. Waleed Deeb
    The problem of best approximation is the problem of finding , for a given point x and a given set G in a normed space (X,||.||) , a point go in G which should be nearest to x among all points of the set G . However , in our study , we shall mainly take as X not an arbitrary normed space but Orlicz space , we shall denote by P(x,G) , the set of all elements of best approximants of x in G. i.e P(x,G)= { gₒ є G llx- g= inf{||x-g||: g є G } The problem of best approximation began , in 1853 , with P. L. Chebyshev who considered the problem in the space of all real valued continuous function defined on [a,b] , a closed real interval in R . My theses consist of four chapters. Each chapter is divided into sections. A number like 2.1.3 indicates item (definition, theorem, corollary or lemma) number 3 in section 1 of chapter 2. Each chapter begins with a clear statement of the pertinent definitions and theorems together with illustrative and descriptive material. At the end of this thesis we present a collection of references. In chapter (1) we introduce the basic results and definitions which shall be needed in the following chapters. The topics include projection, normed space, compactness , Hilbert space and measure theory . This chapter is absolutely fundamental. The results have been stated without proofs, for theory may be looked up in any standard text book in Functional Analysis. A reader who is familiar with these topics may skip this chapter and refer to it only when necessary. Chapter (2) will be devoted to give an introduction to fundamental ideas of Best Approximation in Normed Space. We will start by introducing the definition of best approximants of x є X in a closed subspace G of X . We denote the set of all best approximation of x in G by P(x,G) . In section (2) we study the properties of P(x,G) . In section (3) we define proximinal set and Chebyshev subspace , and we mention some conditions that can assure that G is proximinal in X . Finally , we define Lᵖ- summand and give a simpler proof for the fact that “every a closed subspace of a Hilbert space is proximinal ". Chapter (3) has two purposes .First, we review the properties of Orlicz spaces. Second, we introduce some ofthe basic theory ofproximinality in Orlicz space . This material was designed to meet the needs of chapter (4). W. Deeb and R. Khalil proved the following results. (1) If G is 1-complemented in X, then G is proximinal in X. [1, p.529] . (2) If Lᶲ(μ,G) is proximinal in Lᶲ(μ,X) ,then G is proximinal in X. [3 , p.8] , [2 ,p.297] , [4 , p.37] (3) If L¹(μ,G) is proximinal in L¹(μ,X), then L∞(μ,G) is proximinal in L∞(μ,X) . [1 , p.528] Some questions about proximinality in Lᶲ (μ,G now suggest themselves. (1) Let X be a Banach space and let G be proximinal in X. Under what conditions can it be asserted that G is l-complemented in X? (2)If G is proximinal in X, Under what conditions can it be concluded that Lᶲ (μ,G) is proximinal in Lᶲ(μ,X)? In particular, is the proximinatily of G in X a sufficient condition? (3)If L∞(μ,G) is proximinal in L∞(μ,X). Under what condition can be asserted that L¹(μ,G) is proximinal in L¹(μ,X). These questions are addressed in the section (1) of chapter (4). The answer depends on the S-property. Some interesting results have been achieved. Among of which it is shown that if G has the S-property then L∞ (μ,G) has the S-property . lt is also proved that if G has the S-property then Lᶲ (μ P⁻¹ɢ (0))= P⁻¹Lᶲ (μ,G) (0) I ask our God to be our assistant to continue our efforts so as to achieve the hopes and desires of all scholars in mathematics.