**Format****Pages****Chapters**

# 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 ntuples 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) = tp1; then Jp : E ! 2E

dened by

Jp(x) = fx 2 E; hx; xi = kxkkxk; kxk = (kxk) = kxkp1g:

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 J1 are identity.

If E is re exive, strictly convex and smooth, then J is bijective. In this case

the inverse J1 : E ! E is given by J1 = 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.