# Project Materials

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

ABSTR

In this , an 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.

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 s . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Review 12
2.1 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 ple 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 , 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 if every Cauchy sequence in X converges to an element of X.

Definition 1.2.10 A Banach space is a normed space ( 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 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 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.

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