ABSTRACT
The Kuratowski’s measure of noncompactness of bounded sets , k set contraction for k [0, 1), 1 set contraction and condensing mappings were studied. Some axiomatic properties of measure of noncompactness were used in determining a measure of noncompactness of a closed ball in an infinite dimensional Banach space and we used condensing property to solve some classes of nonlinear equations. The results confirmed the solvability of classes of non- linear equations considered.
TABLE OF CONTENTS
Title Page ii
Declaration iv
Certification v
Dedication vi
Acknowledgement vii
Abstract viii
CHAPTER ONE
1 INTRODUCTION 1
1.1 STATEMENT OF THE PROBLEM 6
1.2 AIM AND OBJECTIVES OF THE STUDY 6
1.3 MOTIVATION OF THE STUDY 7
1.4 SCOPE AND LIMITATION 7
1.5 DEFINITION OF TERMS 7
CHAPTER TWO
2 LITERATURE REVIEW 11
2.1 INTRODUCTION 11
2.2 REVIEW OF RELATED WORKS 11
CHAPTER THREE
3 MATERIALS AND METHODS 16
3.1 INTRODUCTION 16
3.2 KURATOWSKI PROPERTIES OF MEASURE OF NONCOM- PACTNESS 16
3.3 BALL PROPERTY IN INFINITE DIMENSIONAL BANACH SPACES 32
CHAPTER FOUR
4 ANALYSIS 39
4.1 RESULTS 39
CHAPTER FIVE
5 SUMMARY, CONCLUSION AND RECOMMENDATION 49
5.1 SUMMARY 49
5.2 CONCLUSION 50
5.3 RECOMMENDATION 50
REFERENCES 51
CHAPTER 1
INTRODUCTION
In this section, we present some notations and auxiliary results which will be utilized in the remaining part of the chapters. More facts here come from Ayerbe et al. (1997); Akhmerov et al. (1992).
We shall denote by R the set of real numbers. The half open interval [0, 1) ∈ R+ while N denotes the set of natural numbers. Further, assume that X is a real Banach space with norm ǁ.ǁX and zero element 0.
We will use the symbol ǁ.ǁ instead of ǁ.ǁX if it does not lead to misunder- standing. The symbol B(x, r) shall denote the closed ball B(x, r) centered at x with radius r, while ∂B(x, r) shall denote the boundary of B(x, r) and we will write Br to denote the ball B(0, r). Let BX = {x ∈ X : ǁxǁ ≤ 1} and let SX = {x ∈ X : ǁxǁ = 1}. Let B , D be subsets of X. We denote by D the closure of D and c0(B) the closed convex envelop of B. Next, let us denote by B(X), the family of all nonempty and bounded subsets of X.
In 1939, Kuratowski introduced a measure of noncompactness of bounded sets in a metric space which is now called the Kuratowski measure of noncompact- ness. Measure of noncompactness has found applications in fixed point theory, differential equations, functional equations, integral or intrgro-differential equations, etc. If B is a bounded set of a metric space, the Kuratowski measure of noncompactness of B or α − measure of B is defined by
α(B) = inf {ϵ > 0 : B can be covered by finitely many sets of diameter ≤ ϵ}.
Darbo (1959) used this measure to generalized the classical Schauders fixed point theorem to a class of operators called k-set-contractive operators which satisfy the condition α(T (A)) ≤ kα(A) for some k ∈ [0, 1).
More generally, let X be a Banach Space. A map T : X −→ X is called con- densing whenever α(T (A)) < α(A) for α(A) > 0 for every bounded subset A of X with α(A) > 0. We shall study various conditions on T which guarantee surjectivity (the ontoness property) of the map I − T ,where I is the identity mapping on a Banach space X. We shall study the various properties of measure of noncompactness, k-set-contraction defined by Kuratowski (1939), condensing maps studied by Massatt (1978), Massatt (1983). This work is a study of results obtained by Petryshyn (1972). Also if T : X −→ X is 1-set-contraction with (I − T )(B(0, r)) close for each r > 0 and if (rp) and (mp) = (m(rp)) are sequences of positive numbers such that
(i) ǁT (x) − γxǁ ≥ mp for all x ∈ Brp (0) and γ > 1,
then I − T is surjective provided that (mp) → ∞ as p → ∞. It will be shown that if j : X −→ 2X∗ is a normalized duality map and T : X −→ X is k-set-contractive for some k ≥ 0, then I + λT is surjective for each λ in Σ = {λ ∈ [0, 1] : λk < 1}
(ii) If ⟨T (x), J(x)⟩ ≥ ⟨T (0), J(x)⟩ for x ∈ X,
and I + λT is surjective for all λ ≥ 0 and if T is accretive, that is if
(iii) ⟨T (x) − T (y), j(x − y)⟩ ≥ 0 for x, y ∈ X, where j(x − y) ∈ J(x − y). As a special case, we shall study Browder’s theorem for Lipschitzian accretive map, and also study the generalized result of Kirk (1970), to certain continuous maps T : Br(0) −→ X which includes k-set- contractions and which need be neither Lipchitzian nor satisfy the boundary condition T (∂Br(0)) ⊂ Br(0).
Definition 1.0.1 Suppose that X is a metric space, and if A, C ∈ X and h∗(A, C) = sup{d(a, C) : a ∈ A}, h∗(C, A) = sup{d(c, A) : c ∈ C}, then
h(A, C) = max{h∗(A, C), h∗(C, A)} is called Hausdorff distance between sets A and C
Remark 1.0.1 Given ϵ > 0, let Aє = {x ∈ X : d(x, A) < ϵ} and Cє =
{x ∈ X : d(x, C) < ϵ} (the open ϵ-expansion of A and C respectively). Then from the above definition, we have h∗(A, C) = inf{ϵ > 0 : A ⊆ Cє} , h∗(C, A) = inf{ϵ > 0 : C ⊆ Aє} and h(A, C) = inf{ϵ > 0 : A ⊆ Cє, C ⊆ Aє} . Directly from the above definition we have the following properties hold for any A, C, D
(1) h(A, A) = 0
(2) h(A, C) = h(C, A)
(3) h(A, C) ≤ h(A, D) + h(D, C)
Property (2) is called Hausdorff symmetric distance.
Example 1.0.1 Let X = R, A = [0, 1] and B = [2, 4]. Then supa∈A d(a, B) = 2, supb∈B d(b, A) = 3 and h(A, B) = max{supa∈A d(a, B), supb∈B d(b, A)} = max{2, 3} = 3
Note that d(a, B) = infb∈B d(a, b)
Proposition 1.0.1 Suppose that (X, dX) is a bounded metric space and Pf (X) denotes the collection of nonempty closed subsets of X. Then
(a) For any A, B ∈ Pf (X), we set h(A, B) = sup |dist(x, A) − dist(x, B)|. So h is a metric on Pf (X). It is called Hausdroff metric.
(b) For any A, B ∈ Pf (X), we set h¯(A, B) = max{supa∈A dist(a, B), supb∈B dist(b, B)}. So h(A, B) = h¯(A, B) ∀A, B ∈ Pf (X).
(c) For any A, B ∈ Pf (X), we set h¯(A, B) = inf{ϵ > 0 : A ⊆ BєandB ⊆ Bє}, where Bє = {x ∈ X : dist(x, B) ≤ ϵ} for all B ∈ Pf (X). So h¯(A, B) = h¯(A, B) for all A, B ∈ Pf (X).
We now write out the various properties or concepts which we will use in this work and prove some of the known results. We state Proposition (1.1) in Ayerbe et al. (1997) as follows, their proofs will be done in Chapter Three. Let (X, d) be a complete metric space and B be the family of bounded subsets of X.
A map α : B −→ [0, +∞) is called a measure of noncompactness defined on X if it satisfies the following properties.
(a) Regularity: α(B) = 0 iff B is a precompact set.
(b) Invariance under closure: α(B) = α(B) for all B ∈ B(X).
(c) Maximum property: α(B1 ∪ B2) = max{α(B1), α(B2)} for all B1, B2 ∈ B(X).
From these axioms,
Theorem 1.0.2 If X is a Banach space the following properties holds:
(1) Monotonicity: B1 ⊂ B =⇒ α(B1) ≤ α(B), B1, B ∈ B(X).
(2) Minimum property: α(B1 ∩ B2) ≤ min{α(B1), α(B2)} for all B1, B2 ∈ B(X).
(3) Non-singularity: if B is a finite set, then α(B) = 0.
(4) Kuratowski property: If {Bn}n is a decreasing sequence of non-empty closed and bounded subsets of X and limn→∞ α(Bn) = 0, then the intersection B∞ = T∞n=1(Bn) is nonempty and compact.
(5) Semi-homogeneity: α(λB) = |λ|α(B) for any λ ∈ F and B ∈ B(X).
(6) Algebraic semi-additivity: α(B1 + B2) ≤ α(B1) + α(B2) for all B1, B2 ∈ B(X).
(7) Invariance under translations: α(x0 + B) = α(B) for any x0 ∈ X, B ∈ B(X).
(8) Lipschitzianity: |α(B1) − α(B2)| ≤ Lαρ(B1, B2), where ρ denotes the Hausdroff semimetric, ρ(B1, B2) = inf {ϵ > 0 : B2 ⊂ B1+ϵB(0, 1), B1 ⊂ B2 + ϵB(0, 1)}.
(9) Continuity: for every B ∈ B(X), and for all ϵ > 0, there is a δ > 0
such that |α(B) − α(B1)| < ϵ for all B1 satisfying ρ(B1, B) < δ.
(10) Invariance under passage to the convex envelop: α(c0(B)) = α(B) ∀B ∈ B(X).
Proposition 1.0.3 (a) Let (Xi, di), i = 1, 2, 3. be metric spaces. Assume that f : X1 −→ X2 is a k1 − set − contraction and g : X2 −→ X3 is a k2 − set − contraction. Then gf is a k1k2-set-contraction.
(b) Let (X, d) be a metric space and Y a Banach space. Assume that f : X −→ Y is a k1 − set − contraction and g : X −→ Y is a k2-set- contraction, then (f + g) is (k1 + k2)-set-contraction.
Here we give a non-trivial k-set-contraction map.
Theorem 1.0.4 Let X be a Banach space, D ⊂ X and T and S are two operators defined from D into X such that T is compact and S is k-set contractive, that is, there is k ∈ [0, 1) such that ǁS(x) − S(y)ǁ ≤ kǁx − yǁ ∀x, y ∈ D. Then T + S is a k-set contractive .
1.1 STATEMENT OF THE PROBLEM
We will in the course of this work, state the following problem extracted from Petryshyn (1972) which we shall prove and apply the result in solving some various classes of non-linear equations in Banach space X.
Suppose that E is a bounded open subset of X and T : E −→ X is a 1-
set-contraction of E into X such that either one of the following conditions holds.
(i) There exists x0 ∈ E such that T (x) − x0 = λ(x − x0) for some x ∈ ∂E, and λ ≤ 1
(ii) E is convex and T (∂E) ⊂ E , where ∂E is boundary of E and (I −T )(E) is a closed set in X. Then T has a fixed point in E.
1.2 AIM AND OBJECTIVES OF THE STUDY
The aim of this thesis is to study some results in the work of Petryshyn (1972).Specifically, the following are the objectives of this work:
(1) To study the properties of Kuratowski measure of noncompactness, k- set-contractions and condensing maps; and
(2) To apply the properties in (1) above in solving various classes of non- linear equation in a Banach space X.
1.3 MOTIVATION OF THE STUDY
Various classes of nonlinear operators such as Darbor’s k-set-contraction ,condensing operators, (every k-set-contraction is condensing and the converses false) and 1-set contraction has important application in verifying the existence of solutions of certain nonlinear operator equation in Banach spaces X.
1.4 SCOPE AND LIMITATION
We shall study the axiomatic properties of Kuratowski measure of noncom- pactness in metric spaces and Banach spaces X. Also we shall show Dardo’s k-set-contraction,1-set-contraction and condensing mappings. Moreover, we also investigate some results in finite and infinite dimensional Banach spaces. We also use these properties and Darbo’s theorem of k-set-contractions and condensing maps in solving some nonlinear equations, i.e (I − T )(X) = X. We limit this work to real Banach space X.
1.5 DEFINITION OF TERMS
Here, we define some terms that will be use in the proceeding chapters.
Definition 1.5.1 We say that set G ⊂ R is open, if for every x ∈ G, ∃ r > 0 such that B(x, r) ⊂ G.
Definition 1.5.2 Let Y be a topological space. A collection Σ of subsets of Y is said to cover Y if the union of all sets in Σ equals Y i.e, A∈Σ A = Y . If Σ is collection of an open subset of X, then we call Σ an open covering of Y .
Definition 1.5.3 A topological space Y is said to be compact if every open covering Σ of Y can be reduced to a finite subcovering.
Definition 1.5.4 Let T : X −→ X be a continuous mapping in a Banach space X. T is called a k-set-contraction if for all A ⊂ X with A bounded, T (A) is bounded, then ∃k ∈ [0, 1) s α(T (A)) ≤ kα(A).
Definition 1.5.5 Let T : X −→ X be a continuous mapping in a Banach space X, then T is called condensing densifying whenever α(A) > 0 and α(T (A)) < α(A).
Definition 1.5.6 Let T : X −→ X be a continuous mapping in a Banach space X, T is called 1- set contraction whenever α(T (A)) ≤ α(A), for all bounded subsets A of X.
Definition 1.5.7 Let T : X −→ X be a continuous mapping in a Banach space X, then T is called semiclosed 1-set-contraction if T is 1-set-contraction and I − T is closed.
Definition 1.5.8 If B is a bounded set of a metric space, let α(B)=inf {ϵ > 0 : B can be covered by finitely many sets of diameter ≤ ϵ}.Then α(B) is called Kuratowski measure of noncompactness of B.
Definition 1.5.9 Given a map T : X −→ X, every solution of the equation x = T (x) is called a fixed point of T. Let Y be a topological space and A ⊂ Y , then ∂A shall denote the boundary of A and which is given by A \ int(A) where int(A)=Y \ (Y \ A). Thus ∂A = A ∩ Y \ A.
Definition 1.5.10 Let (X, τX) and (Y, τY ) be topological spaces, then
(a)A map f : X −→ Y is said to be continuous, if for every U ∈ τY , f −1(U ) ∈ τX. (b) A map f : X −→ Y is said to be open, if for each V ∈ τX, f (V ) ∈ τY .
Definition 1.5.11 A metric (a distance function) on a nonempty set X is a mapping
d : X × X −→ R+ satisfying the following conditions ∀x, y, z ∈ X.
(1) d(x, y) ≥ 0
(2) d(x, y) = 0 iff x = y
(3) d(x, y) = d(y, x)
(4) d(x, y) ≤ d(x, z) + d(z, y).
Then the pair (X, d) is called a metric space.
Remark 1.5.1 If the condition (3) does not satisfy, the pair (X, d) is called qusimetric space.
Definition 1.5.12 Let X be an arbitrary real Banach space and let J denote the normalized duality mapping from X to 2X∗ given by J(x) = {f ∈ X∗ :
⟨x, f ⟩ = ǁxǁ2; ǁxǁ2 = ǁf ǁ2}, where X∗ denotes the dual space of X and ⟨., .⟩
denotes the generalized duality paring.
Definition 1.5.13 A map T : X −→ X is said to be accerative if for x, y ∈ X then ⟨T (x) − T (y), j(x − y)⟩ ≥ 0, where j(x − y) ∈ J(x − y).
Definition 1.5.14 A map A : X −→ X is said to be strongly accretive if there exists β > 0 such that ⟨A(x) − A(y), J(x − y)⟩ ≥ βǁx − yǁ2 for all x, y ∈ X.
Definition 1.5.15 The space (X, d) is called complete if each Cauchy se- quence {xn} ⊂ X is convergent to an element x∗ ∈ X i.e
limn−→∞ d(xn, x∗) = 0.
Definition 1.5.16 Let (X, d) be any metric space. The sequence {xn} of points of X is said to converge to a point x of X, if for each ϵ > 0 there exists a positive integer m, such that d(xn, x) < ϵ ∀ n ≥ m. i.e; d(xn, x) → 0 as n → ∞.
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