Approximation Method for Solving Variational Inequality with Multiple Set Split Feasibility Problem in Banach Space

Approximation Method for Solving Variational Inequality with Multiple Set Split Feasibility Problem in Banach Space

ABSTRACT

In this thesis, an iterative algorithm for approximating the solutions of a variational inequality problem for a strongly accretive, L-Lipschitz map and solutions of a multiple sets split feasibility problem is studied in a uniformly convex and 2-uniformly smooth real Banach space under the assumption that the duality map is weakly sequentially continuous. A strong convergence theorem is proved.

TABLE OF CONTENTS

Acknowledgment i
Certification ii
Approval iii
Abstract v
Dedication vi
1 General Introduction 2
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 10
1.4 Significance of the Study . . . . . . . . . . . . . . . . . . . . . 11
1.5 Aim and Objectives . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Scope and Limitations . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Literature Review 12
2.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Strong convergence theorem for solving variational inequality with multiple set split feasibility problem 16
3.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Summary and Conclusion 32
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.4 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . 32

CHAPTER ONE

General Introduction

In this chapter, we give a brief introduction of the subject matter and definitions of some basic terms which will be used in our subsequent discussions.

1.1 Introduction

The Multiple sets split feasibility problem is to and a point contained in the intersection of a family of closed convex sets in one space so that its image under a bonded linear transformation is contained in the intersection of a
family of closed convex sets in the image space. It generalizes the convex feasibility problem and the two sets split feasibility problem. The problem is formulated as
find x 2
n
i=1
Ci such that A(x) 2
m
t=1
Qt:
where A : X ! Y is a bounded linear operator, Ci X; i = 1; 2; 3; ; n
and Qt Y; t = 1; 2; 3; ;m are nonempty closed convex sets.
When n = m = 1, the problem reduce to the Split feasibility problem (SFP) which is to nd
x 2 C such that A(x) 2 Q:

where C and Q are two nonempty closed convex subsets of X and Y respectively.

In Banach space, the multiple sets split feasibility problem is formulated as ending an element x 2 X satisfying
x 2
n
i=1
Ci; A(x) 2
m
t=1
Qt:
2
3
where X and Y are two Banach spaces, m; n are two given integers, A:
X ! Y is a bounded linear operator, Ci; i = 1; 2; 3; ; n are closed convex

sets in X, and Qt; t = 1; 2; 3; ;m closed convex sets in Y.

The multiple sets split feasibility problem was rst introduced by Censor and Elfving [9]. The problem arises in many practical elds such as signal processing, image reconstruction [11], Intensity modulated radiation therapy (IMRT)[10] and so on.

1.2 Preliminaries Definition

1.2.1 A vector space over some eld say F is s nonempty set E together with two binary operations of addition(+) and scalar multiplication(.) satisfying the following conditions for any v;w; z 2 E; ; 2 F:
1. v + w = w + v; the commutative law of addition,
2. (v + w) + z = v + (w + z); the associative law for addition,
3. There exists 0 2 E satisfying v + 0 = v; the existence of an additive identity,
4. 8v 2 E there exists (􀀀v) 2 E such that v+(-v) = 0; the existence of an additive inverse,
5. (v + w) = v + w;
6. ( + ) v = v + v;
7. ( v) = () v;
8. 1 v = v.
Here, the scalar multiplication v is often written as v: The eld of scalars will always be assumed to be either R or C and the vector space will be called real or complex depending on whether the eld is R or C. A vector space is
also called a linear space.

Example 1.2.2 Space Rn. This is the Euclidean space, the underlying set being the set of all n􀀀tuples of real numbers, written as x = (x1::::; xn), y = (y1::::; yn), etc., and we now see that this is a real vector space with the two algebraic operations dened in the usual fashion x+y = (x1+y1; :::; xn+yn) and ax = (ax1; :::; axn), a 2 R.
Definition 1.2.3 The vectors fx1; x2; x3; g are said to form a basis for E if they are linearly independent and E = spanfx1; x2; x3; g.

Definition 1.2.4 A vector space E is said to be nite dimensional if the number of vectors in a basis of E is nite.
Note that if E is not nite dimensional, it is said to be indefinite dimensional.

Example 1.2.5 In analysis, indefinite dimensional vector spaces are of greater interest than nite dimensional ones. For instance, C[a; b] and l2 are indefinite dimensional, whereas Rn and Ck are nite dimensional for some n; k 2 N.

Definition 1.2.6 A normed space E is a vector space with a norm dened on it, here a norm on a (real or complex) vector space E is a real-valued function on E whose value at an x 2 E is denoted by kxk and which satisfies the following properties, for x; y 2 E and 2 R
1. kxk 0;
2. kxk = 0 i x = 0;
3. kxk = jjkxk;
4. kx + yk kxk + kyk;
Definition 1.2.7 A sequence fxng in a normed linear space X is (i) convergent to x 2 X if given > 0, there exists N 2 N such that kxn 􀀀 xk < whenever n N (ii) said to be Cauchy if given > 0; there exists N 2 N such that

Remark 1.2.8 Every convergent sequence is Cauchy but the converse is not necessarily true.

Definition 1.2.9 A space X is said to be complete if every Cauchy sequence in X converges to an element of X.

Definition 1.2.10 A Banach space is a complete normed space (complete in the metric dened by the norm).

Example 1.2.11 The space lp is a Banach space with norm given by
kxk = (
1X
j=1
jxjp)
1
p

Definition 1.2.12 An inner product space (E; h; i) is a vector space E together with an inner product h; i : E E ! C such that for all vectors x, y, z and scalar a we have
1. hx + y; zi = hx; zi + hy; zi;
5
2. hx; yi = hx; yi;
3. hx; yi = hy; xi;
4. hx; xi 0 and hx; xi = 0 i x = 0;
A norm on E can also be dene as
1. kxk2 = hx; xi, 8x 2 E
2. x and y are orthogonal if hx; yi = 0

Inner product space generalizes notion of dot product of nite dimensional spaces.

Definition 1.2.13 A Hilbert space is a complete inner product space.

In a Banach space E, beside the strong convergence dened by the norm, i.e., fxng E converges strongly to a if and only if limn!1 kxn 􀀀 ak = 0, we shall consider the weak convergence, corresponding to the weak topology
in E. We say that fxng E converges weakly to a if for any f 2 E
hxn; fi ! ha; fi as n ! 1.

Remark 1.2.14 Any weakly convergent sequence fxng in a Banach space is bounded.

Definition 1.2.15 Let E be a Banach space. Consider the following map
J : E ! E dened for each x 2 E, by
J(x) = x 2 E
where
x : E ! R
is given by
x(f) = hf; xi; for each f 2 E:

Clearly J is linear, bounded and one-to-one. The mapping J dened above is called the canonical map(or canonical embedding) of E onto E.

Definition 1.2.16 Let E be a normed linear space and J be the canonical embedding of E onto E. If J is onto, then E is called re exive.

Proposition 1.2.17 1. In re exive Banach space each bounded sequence
has a weakly convergent subsequence.
2. The spaces Lp and lp, p > 1, are re exive.
6
3. The spaces L1 and l1 are non-re exive.

Definition 1.2.18 A Banach space E is said to be strictly convex if kx+yk
2 < 1 for all x; y 2 U; where U = fz 2 E : kzk = 1g with x 6= y. Definition 1.2.19 A Banach space E is said to be smooth, if for every 0 6= x 2 E there exists a unique x 2 E such that kxk = 1 and hx; xi = kxk i.e., there exists a unique supporting hyperplane for the ball around the origin with radius kxk at x.

Definition 1.2.20 The modulus of convexity of a normed space E is the function E : (0; 2] ! [0; 1] dened by E() = inff1 􀀀 k 1 2 (x + y)k; kxk = kyk = 1; kx 􀀀 yk = g: Definition 1.2.21 The modulus of smoothness of a normed space E is the fuction E : [0;1) ! [0;1) dened by E(r) = 1 2 supfkx + yk + kx 􀀀 yk 􀀀 2 : kxk = 1; kyk rg: Definition 1.2.22 A Banach space E is said to be uniformly convex, if for any 2 (0; 2] there exists a = () > 0; such that for any x; y 2 E with kxk =
kyk = 1 and kx 􀀀 yk then kx+y
2 k 1 􀀀 :
Remark 1.2.23 1. Every uniformly convex space is re exive
2. E is uniformly convex i E() > 0:8 2 (0; 2]
Definition 1.2.24 A Banach space E is said to be uniformly smooth, if
lim
r!0
(
E(r)
r
) = 0:
where E(r) is the modulus of smoothness.
Remark 1.2.25 1. E is continuous, convex and nondecreasing with E(0) =
0 and E(r) r
2. The function r 7! E(r)
r is nondecreasing and full ls E(r)
r > 0 for all
r > 0:
Definition 1.2.26 Let q > 1 be a real number. A normed space E is said
to be q-uniformly smooth if there is a constant d > 0 such that
E(r) dq:
When 1 < q 2; E is said to be 2-uniformly smooth. 7 Definition 1.2.27 A mapping A : E1 ! E2 is said to be bounded and linear if there exists real numbers c; and such that for x; y 2 E1, kAxk ckxk and A(x + y) = Ax + Ay: Definition 1.2.28 Let E1 and E2 be two re exive, strictly convex and smooth Banach spaces.

The mapping A : E1 􀀀! E2 is called a bounded linear operator endowed with the operator norm kAk = supkxk1 kAxk. The dual operator A : E 2 􀀀! E 1 dened by hAy; xi = hy; Axi8x 2 E1; y 2 E 2 is called the adjoint operator of A. The adjoint operator A has the property. kAk = kAk Definition 1.2.29

A continuous strictly increasing function g : R+ 􀀀! R+ such that g(0) = 0 and limit!1 g(t) = 1 is called a gauge function. Definition 1.2.30 The ganeralized duality map J : E 􀀀! 2E with respect to the guage function is dened by J(x) = fx 2 E; hx; xi = kxkkxk; kxk = (kxk)g: For p > 1; if (t) = tp􀀀1; then Jp : E 􀀀! 2E
dened by

Jp(x) = fx 2 E; hx; xi = kxkkxk; kxk = (kxk) = kxkp􀀀1g:
is also called the generalized duality map.
In particular, if p = 2 then
J2x := Jx = ff 2 E : hx; fi = kxk2 = kfk2g
is called the normalized duality mapping
Proposition 1.2.31 The duality map of a Banach space E has the follow-
ing properties;

1. It is homogeneous
2. It is additive i E is a Hilbert space.
3. It is single-valued i E is smooth.
4. It is surjective i E is re exive.
5. It is injective or strictly monotone i E is strictly convex
6. It is norm to weak* uniformly continuous on bounded subsets of E if E is smooth
7. If E is Hilbert, J and J􀀀1 are identity.

If E is re exive, strictly convex and smooth, then J is bijective. In this case
the inverse J􀀀1 : E 􀀀! E is given by J􀀀1 = J with J being the duality
mapping of E.
Definition 1.2.32 The duality mapping Jp
E is said to be weakly sequentially
continuous if for each xn ! x weakly, we have Jp
E(xn) ! Jp
E(x) weakly.