Approximation method for solving variational inequality with multiple set split feasibility problem in banach space

Thesis Overview

<p> General Introduction<br>In this chapter, we give a brief introduction of the subject matter and denitions<br>of some basic terms which will be used in our subsequent discussions.<br>1.1 Introduction<br>The Multiple sets split feasibility problem is to nd a point contained in the<br>intersection of a family of closed convex sets in one space so that its image<br>under a bonded linear transformation is contained in the intersection of a<br>family of closed convex sets in the image space. It generalizes the convex<br>feasibility problem and the two sets split feasibility problem. The problem<br>is formulated as<br>find x 2<br>n<br>i=1<br>Ci such that A(x) 2<br>m<br>t=1<br>Qt:<br>where A : X ! Y is a bounded linear operator, Ci X; i = 1; 2; 3; ; n<br>and Qt Y; t = 1; 2; 3; ;m are nonempty closed convex sets.<br>When n = m = 1, the problem reduce to the Split feasibiliry problem (SFP)<br>which is to nd<br>x 2 C such that A(x) 2 Q:<br>where C and Q are two nonempty closed convex subsets of X and Y respectively.<br>In Banach space, the multiple sets split feasibility problem is formulated<br>as nding an element x 2 X satisfying<br>x 2<br>n<br>i=1<br>Ci; A(x) 2<br>m<br>t=1<br>Qt:<br>2<br>3<br>where X and Y are two Banach spaces, m; n are two given integers, A :<br>X ! Y is a bounded linear operator, Ci; i = 1; 2; 3; ; n are closed convex<br>sets in X, and Qt; t = 1; 2; 3; ;m closed convex sets in Y .<br>The multiple sets split feasibility problem was rst introduced by Censor<br>and Elfving [9]. The problem arises in many practical elds such as<br>signal processing, image reconstruction [11], Intensity modulated radiation<br>therapy(IMRT)[10] and so on.<br>1.2 Preliminaries<br>Denition 1.2.1 A vector space over some eld say F is s nonempty set<br>E together with two binary operations of addition(+) and scalar multiplica-<br>tion(.) satisfying the following conditions for any v;w; z 2 E; ; 2 F:<br>1. v + w = w + v; the commutative law of addition,<br>2. (v + w) + z = v + (w + z); the associative law for addition,<br>3. There exists 0 2 E satisfying v + 0 = v; the existence of an additive<br>identity,<br>4. 8v 2 E there exists (ô€€€v) 2 E such that v+(-v) = 0; the existence of<br>an additive inverse,<br>5. (v + w) = v + w;<br>6. ( + ) v = v + v;<br>7. ( v) = () v;<br>8. 1 v = v.<br>Here, the scalar multiplication v is often written as v: The eld of scalars<br>will always be assumed to be either R or C and the vector space will be called<br>real or complex depending on whether the eld is R or C. A vector space is<br>also called a linear space.<br>Example 1.2.2 Space Rn. This is the Euclidean space, the underlying set<br>being the set of all nô€€€tuples of real numbers, written as x = (x1::::; xn), y =<br>(y1::::; yn), etc., and we now see that this is a real vector space with the two<br>algebraic operations dened in the usual fashion x+y = (x1+y1; :::; xn+yn)<br>and ax = (ax1; :::; axn), a 2 R.<br>Denition 1.2.3 The vectors fx1; x2; x3; g are said to form a basis for<br>E if they are linearly independent and E = spanfx1; x2; x3; g.<br>4<br>Denition 1.2.4 A vector space E is said to be nite dimensional if the<br>number of vectors in a basis of E is nite.<br>Note that if E is not nite dimensional, it is said to be innite dimensional.<br>Example 1.2.5 In analysis, innite dimensional vector spaces are of greater<br>interest than nite dimensional ones. For instance, C[a; b] and l2 are innite<br>dimensional, whereas Rn and Ck are nite dimensional for some n; k 2 N.<br>Denition 1.2.6 A normed space E is a vector space with a norm dened<br>on it, here a norm on a (real or complex) vector space E is a real-valued<br>function on E whose value at an x 2 E is denoted by kxk and which satises<br>the following properties, for x; y 2 E and 2 R<br>1. kxk 0;<br>2. kxk = 0 i x = 0;<br>3. kxk = jjkxk;<br>4. kx + yk kxk + kyk;<br>Denition 1.2.7 A sequence fxng in a normed linear space X is<br>(i) convergent to x 2 X if given &gt; 0, there exists N 2 N such that<br>kxn ô€€€ xk &lt; whenever n N<br>(ii) said to be Cauchy if given &gt; 0; there exists N 2 N such that<br>Remark 1.2.8 Every convergent sequence is Cauchy but the converse is not<br>necessarily true.<br>Denition 1.2.9 A space X is said to be complete if every Cauchy sequence<br>in X converges to an element of X.<br>Denition 1.2.10 A Banach space is a complete normed space (complete<br>in the metric dened by the norm).<br>Example 1.2.11 The space lp is a Banach space with norm given by<br>kxk = (<br>1X<br>j=1<br>jxjp)<br>1<br>p<br>Denition 1.2.12 An inner product space (E; h; i) is a vector space E<br>together with an inner product h; i : E E ! C such that for all vectors x,<br>y, z and scalar a we have<br>1. hx + y; zi = hx; zi + hy; zi;<br>5<br>2. hx; yi = hx; yi;<br>3. hx; yi = hy; xi;<br>4. hx; xi 0 and hx; xi = 0 i x = 0;<br>A norm on E can also be dene as<br>1. kxk2 = hx; xi, 8x 2 E<br>2. x and y are orthogonal if hx; yi = 0<br>Inner product space generalizes notion of dot product of nite dimensional<br>spaces.<br>Denition 1.2.13 A Hilbert space is a complete inner product space.<br>In a Banach space E, beside the strong convergence dened by the norm,<br>i.e., fxng E converges strongly to a if and only if limn!1 kxn ô€€€ ak = 0,<br>we shall consider the weak convergence, corresponding to the weak topology<br>in E. We say that fxng E converges weakly to a if for any f 2 E<br>hxn; fi ! ha; fi as n ! 1.<br>Remark 1.2.14 Any weakly convergent sequence fxng in a Banach space<br>is bounded.<br>Denition 1.2.15 Let E be a Banach space. Consider the following map<br>J : E ! E dened for each x 2 E, by<br>J(x) = x 2 E<br>where<br>x : E ! R<br>is given by<br>x(f) = hf; xi; for each f 2 E:<br>Clearly J is linear, bounded and one-to-one. The mapping J dened above<br>is called the canonical map(or canonical embedding) of E onto E.<br>Denition 1.2.16 Let E be a normed linear space and J be the canonical<br>embedding of E onto E. If J is onto, then E is called re exive.<br>Proposition 1.2.17 1. In re exive Banach space each bounded sequence<br>has a weakly convergent subsequence.<br>2. The spaces Lp and lp, p &gt; 1, are re exive.<br>6<br>3. The spaces L1 and l1 are non-re exive.<br>Denition 1.2.18 A Banach space E is said to be strictly convex if kx+yk<br>2 &lt;<br>1 for all x; y 2 U; where U = fz 2 E : kzk = 1g with x 6= y.<br>Denition 1.2.19 A Banach space E is said to be smooth, if for every 0 6=<br>x 2 E there exists a unique x 2 E such that kxk = 1 and hx; xi = kxk<br>i.e., there exists a unique supporting hyperplane for the ball around the origin<br>with radius kxk at x.<br>Denition 1.2.20 The modulus of convexity of a normed space E is the<br>function E : (0; 2] ! [0; 1] dened by<br>E() = inff1 ô€€€ k<br>1<br>2<br>(x + y)k; kxk = kyk = 1; kx ô€€€ yk = g:<br>Denition 1.2.21 The modulus of smoothness of a normed space E is the<br>fuction E : [0;1) ! [0;1) dened by<br>E(r) =<br>1<br>2<br>supfkx + yk + kx ô€€€ yk ô€€€ 2 : kxk = 1; kyk rg:<br>Denition 1.2.22 A Banach space E is said to be uniformly convex, if for<br>any 2 (0; 2] there exists a = () &gt; 0; such that for any x; y 2 E with kxk =<br>kyk = 1 and kx ô€€€ yk then kx+y<br>2 k 1 ô€€€ :<br>Remark 1.2.23 1. Every uniformly convex space is re exive<br>2. E is uniformly convex i E() &gt; 0:8 2 (0; 2]<br>Denition 1.2.24 A Banach space E is said to be uniformly smooth, if<br>lim<br>r!0<br>(<br>E(r)<br>r<br>) = 0:<br>where E(r) is the modulus of smoothness.<br>Remark 1.2.25 1. E is continuous, convex and nondecreasing with E(0) =<br>0 and E(r) r<br>2. The function r 7! E(r)<br>r is nondecreasing and fullls E(r)<br>r &gt; 0 for all<br>r &gt; 0:<br>Denition 1.2.26 Let q &gt; 1 be a real number. A normed space E is said<br>to be q-uniformly smooth if there is a constant d &gt; 0 such that<br>E(r) dq:<br>When 1 &lt; q 2; E is said to be 2-uniformly smooth.<br>7<br>Denition 1.2.27 A mapping A : E1 ! E2 is said to be bounded and linear<br>if there exists real numbers c; and such that for x; y 2 E1,<br>kAxk ckxk<br>and<br>A(x + y) = Ax + Ay:<br>Denition 1.2.28 Let E1 and E2 be two re exive, strictly convex and smooth<br>Banach spaces. The mapping A : E1 ô€€€! E2 is called a bounded linear op-<br>erator endowered with the operator norm kAk = supkxk1 kAxk. The dual<br>operator A : E<br>2 ô€€€! E<br>1 dened by hAy; xi = hy; Axi8x 2 E1; y 2 E<br>2 is<br>called the adjoint operator of A. The adjoint operator A has the property.<br>kAk = kAk<br>Denition 1.2.29 A continuous strictly incresing function g : R+ ô€€€! R+<br>such that g(0) = 0 and limt!1 g(t) = 1 is called a guage function.<br>Denition 1.2.30 The ganeralized daulity map J : E ô€€€! 2E<br>with respect<br>to the guage function is dened by<br>J(x) = fx 2 E; hx; xi = kxkkxk; kxk = (kxk)g:<br>For p &gt; 1; if (t) = tpô€€€1; then Jp : E ô€€€! 2E<br>dened by<br>Jp(x) = fx 2 E; hx; xi = kxkkxk; kxk = (kxk) = kxkpô€€€1g:<br>is also called the generalized duality map.<br>In particular, if p = 2 then<br>J2x := Jx = ff 2 E : hx; fi = kxk2 = kfk2g<br>is called the normalized duality mapping<br>Proposition 1.2.31 The duality map of a Banach space E has the follow-<br>ing properties;<br>1. It is homogeneous<br>2. It is additive i E is a Hilbert space.<br>3. It is single-valued i E is smooth.<br>4. It is surjective i E is re exive.<br>5. It is injective or strictly monotone i E is strictly convex<br>8<br>6. It is norm to weak* uniformly continous on bounded subsets of E if E<br>is smooth<br>7. If E is Hilbert, J and Jô€€€1 are identity.<br>If E is re exive, strictly convex and smooth, then J is bijective. In this case<br>the inverse Jô€€€1 : E ô€€€! E is given by Jô€€€1 = J with J being the duality<br>mapping of E.<br>Denition 1.2.32 The duality mapping Jp<br>E is said to be weakly sequentially<br>continuous if for each xn ! x weakly, we have Jp<br>E(xn) ! Jp<br>E(x) weakly.<br>Denition 1.2.33 A mapping PC : E ! C is said to be a projection of x<br>onto C if for all y 2 C there exists a unique element PC(x) 2 C such that<br>kx ô€€€ PC(x)k = miny2C kx ô€€€ yk. Moreover, if Jp is the duality mapping of<br>E, then x0 2 C is the projection of x onto C i<br>hJp(x0 ô€€€ x); y ô€€€ x0i 0 8y 2 C:<br>Denition 1.2.34 A mapping T : C ! C is said to be nonexpansive if<br>kTx ô€€€ Tyk kx ô€€€ yk for any x; y 2 C:<br>Denition 1.2.35 A mapping T : E ! E is said to be accretive if<br>hTx ô€€€ Ty; j(x ô€€€ y)i 0 8x; y 2 E:<br>Denition 1.2.36 A mapping T : E ! E is said to be strongly accretive if<br>there exists &gt; 0 such that<br>hTx ô€€€ Ty; j(x ô€€€ y)i kx ô€€€ yk2 8x; y 2 E:<br>Remark 1.2.37 If E := H a Hilbert space, then a strongly accretive map<br>T is strongly monotone.<br>Denition 1.2.38 A mapping T : E ! E is said to be L-Lipschitz contin-<br>uous on E if there exists L &gt; 0 such that<br>kTx ô€€€ Tyk Lkx ô€€€ yk 8 x; y 2 E:<br>Denition 1.2.39 Let C be a nonempty, closed and convex subset of E.<br>The problem of nding x 2 C such that<br>hj(x ô€€€ x); Txi 0<br>is called a variational inequality problem(VI), where T : E ! E is strongly<br>accretive and Lô€€€Lipschitz continuous.<br>9<br>Lemma 1.2.40 [8] Let E be q- uniformly smooth Banach space. Then,<br>there exists a constant dq &gt; 0, such that<br>kx + ykq kxkq + qhy; jxi + dqkykq:<br>Lemma 1.2.41 [25] Let E be 2- uniformly smooth Banach space with best<br>smoothness constant k &gt; 0: Then,<br>kx + yk2 kxk2 + 2hy; jxi + 2kkyk2:<br>Lemma 1.2.42 Let C be a nonempty, closed and convex subset of a smooth,<br>strictly convex and re exive Banach space E and let (x; z) 2 E C. Then ,<br>z = PCx iff hy ô€€€ z; j(x ô€€€ z)i 0 8y 2 C:<br>Lemma 1.2.43 [4] Let E be a normed linear space. Then, the following<br>inequality hold:<br>kx + yk2 kxk2 + 2hy; j(x + y)i for x; y 2 E; j(x + y) 2 J(x + y):<br>Lemma 1.2.44 ([19]see also [3]) Let E be a uniformly convex Banach space<br>with modulus of convexity () of order q; q 2, then there exists &gt; 0<br>such that the following inequality hold:<br>kPCx ô€€€ xkq kx ô€€€ ykq ô€€€ kPCx ô€€€ ykq 8 y 2 C:<br>Lemma 1.2.45 Let E be a uniformly smooth Banach space with best smooth-<br>ness constant k satisfying 0 &lt; k &lt; p1<br>2<br>: Suppose T : E ! E is strongly<br>accretive and L- Lipschitz continuous on E , 0 &lt; &lt; 1, 0 1 ô€€€ and<br>0 &lt; &lt; 2<br>L2 : Then,<br>k(1ô€€€)xô€€€Txô€€€[(1ô€€€)y ô€€€Ty]k (1ô€€€ ô€€€ )kxô€€€yk 8 x; y 2 E;<br>where<br>= 1 ô€€€<br>p<br>1 ô€€€ (2 ô€€€ L2) 2 (0; 1]:<br>Proof<br>By L-Lipschitz contuinity and strongly accretivity of F, we have<br>k(x ô€€€ y) ô€€€ (Tx ô€€€ Ty)k2 = k(Tx ô€€€ Ty) ô€€€ (x ô€€€ y)k2<br>2kTx ô€€€ Tyk2 ô€€€ 2hTx ô€€€ Ty; j(x ô€€€ y)i<br>+ 2k2kx ô€€€ yk2<br>2kTx ô€€€ Tyk2 ô€€€ 2hTx ô€€€ Ty; j(x ô€€€ y)i<br>+ kx ô€€€ yk2<br>2L2kx ô€€€ yk2 ô€€€ 2kx ô€€€ yk2 + kx ô€€€ yk2<br>= (1 ô€€€ 2 + 2L2)kx ô€€€ yk2<br>10<br>Thus<br>k(x ô€€€ y) ô€€€ (Tx ô€€€ Ty)k<br>p<br>1 ô€€€ 2 + 2L2kx ô€€€ yk:j (1.2.1)<br>Now, using (1.2.1)<br>k(1 ô€€€ )x ô€€€ Tx ô€€€ [(1 ô€€€ )y ô€€€ Ty]k<br>= k(1 ô€€€ )(x ô€€€ y) ô€€€ (Tx ô€€€ Ty)k<br>= k(1 ô€€€ )x ô€€€ y) ô€€€ [(x ô€€€ y) ô€€€ (Tx ô€€€ Ty)]k<br>(1 ô€€€ ô€€€ )kx ô€€€ y)k + k(x ô€€€ y) ô€€€ (Tx ô€€€ Ty)k<br>(1 ô€€€ ô€€€ )kx ô€€€ y)k +<br>p<br>1 ô€€€ 2 + 2L2kx ô€€€ yk<br>= (1 ô€€€ ô€€€ )kx ô€€€ y)k<br>where<br>= 1 ô€€€<br>p<br>1 ô€€€ (2 ô€€€ L2):<br>This completes the proof.<br>Lemma 1.2.46 [2] Let fang be a sequence of nonnegative real numbers.<br>Suppose that for any integer m, there exists an integer p such that p m<br>and ap ap+1. Let n0 be an integer such that an0 an0+1 and dene for<br>all integer n n0 by<br>(n) = maxfk 2 N : n0 k n; ak ak+1g:<br>Then f (n)gnn0 is a nondecreasing sequence satisfying limn!1 (n) = 1<br>and the following inequalities hold true:<br>a(n) a(n)+1 an a(n)+1 8n n0:<br>Lemma 1.2.47 [24] Assume fang is a sequence of nonnegative real num-<br>bers satisfying the condition<br>an+1 (1 ô€€€ n)an + nn; 8 n n0:<br>where fng is a sequence in (0; 1) and fng is a sequence in R such that<br>(4i)<br>P1<br>n=0 n = 1;<br>(ii) lim supn!1 n 0:<br>Then limn!1 an = 0:<br>1.3 Statement of the Problem<br>The split feasiblity problem (SFP) in nite-dimensional Hilbert spaces was<br>rst introduced by Censor et al.[9] for modeling inverse problems which<br>arise from phase retrievals and in medical image reconstruction. Since then,<br>11<br>various algorithms have been introduced and studied for solving SFP and<br>MSSFP<br>Recently, Anh[1] proposed a parallel method for solving the variational<br>inequality with the MSSFP and proved a strong convergence of the iterative<br>process in frame work of a Hilbert space. Now the problem is to establish<br>new convergence theorems that hold in more general Banach space than<br>Hilbert space.<br>1.4 Signicance of the Study<br>When a multiple set split feasibility problem exists in a setting of a Banach<br>space more general than Hilbert, the result of Anh[1] cannot be used to get<br>solutions. While our result in this thesis can be applied to such problem to<br>some extent.<br>1.5 Aim and Objectives<br>The aim of this work is to present a new theorem for solution of some nonlinear<br>operator problem. The aim is achieved through the following objective<br>To establish some existance results of solutions and the convergence of an<br>iterative process for solving variational inequalities with multiple sets split<br>feasibility problem in a uniformly convex and 2-uniformly smooth Banach<br>spaces<br>1.6 Scope and Limitations<br>The theorems considered in this research work hold in a uniformly convex<br>and 2-uniformly smooth Banach space whose duality mapping is weakly<br>sequentially continuous. It involves solution of a variational inequality problem.<br>The work is limited to Banach spaces with weakly sequentially continuous<br>duality maps which excludes some Banach spaces such as Lp spaces and<br>sobolev spaces.<br>1.7 Methodology<br>Various well known computational techniques, theorems and results in the<br>literature related to approximations of solutions of the multiple sets split<br>feasibility problems are used. <br></p>