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

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Approximation Method for with Multiple Set Split Feasibility Problem in Banach

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 S

Acknowledgment i
Certification ii
Approval iii
Abstract v
Dedication vi
1 General Introduction 2
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 of the Problem . . . . . . . . . . . . . . . . . . . . 10
1.4 Significance of the Study . . . . . . . . . . . . . . . . . . . . . 11
1.5 Aim and Objectives . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Scope and s . . . . . . . . . . . . . . . . . . . . . . 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 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 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 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 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 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 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 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 , 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.

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