Strong convergence of modified averaging iterative algorithm for asymptotically nonexpansive maps

Thesis Overview

<p> </p><p></p> <p>Introduction<br>1.1 General Introduction<br>The theory of xed points of nonlinear operators has found many powerful and important<br>applications in diverse elds such as Dierential Equations,Topology, Economics, Biology,<br>Chemistry, Engineering, Game Theory, Physics, Dynamics, Optimal Control, and Func-<br>tional Analysis . Iterative algorithms for approximating xed points of some nonlinear<br>operators belonging to certain classes of mappings that generalize nonexpansive mappings<br>and dened in appropriate Banach spaces have been ourishing area of research for many<br>mathematicians.The class of nonlinear mappings we studied in this work, is the class of<br>asymptotically nonexpansive mappings. This class of asymptotically nonexpansive map-<br>pings which has engaged the interest of many researchers(for example see [18] and the<br>references there in ) was rst introduced by Goebel and Kirk [10] in the year 1972.<br>Denition 1.1.1: Let K be a nonempty subset of a normed linear space E. A mapping<br>T : K ! K is said to be nonexpansive if<br>kTx ô€€€ Tyk kx ô€€€ yk; 8x; y 2 K:<br>Denition 1.1.2: Let K be a nonempty subset of a normed linear space E. A mapping<br>T : K ! K is called asymptotically nonexpansive, if there exists a sequence fkng; kn 2<br>1<br>[1;1) such that limn!1 kn = 1, and<br>kTnx ô€€€ Tnyk knkx ô€€€ yk<br>holds for each x; y 2 K and for each integer n 1. It is clear that every nonexpansive<br>mapping is asymptotically nonexpansive with kn = 1 8n 1.<br>The following example reveals that the class of asymptotically nonexpansive mappings<br>properly contains the class of nonexpansive mappings.<br>Example 1 (Goebel and Kirk [10]). Let B denote the unit ball in the Hilbert space l2<br>and let T be dened as follows:<br>T : (x1; x2; x3; :::) ! (0; x21<br>; a2x2; a3x3; :::):<br>where faig is a sequence of numbers such that 0 &lt; ai &lt; 1 and<br>Q1<br>i=2 ai =<br>1<br>2<br>.<br>Then, T is Lipschitz and kTx ô€€€ Tyk 2kx ô€€€ yk; 8x; y 2 B.<br>Moreover,<br>kTix ô€€€ Tiyk 2<br>Yi<br>j=2<br>ajkx ô€€€ yk 8i = 2; 3; :::<br>Thus,<br>lim<br>i!1<br>ki = lim<br>i!1<br>2<br>Yi<br>j=2<br>aj = 1<br>But let,<br>x = (<br>2<br>3<br>; 0; 0; :::)<br>and<br>y = (<br>1<br>2<br>; 0; 0; :::);<br>clearly x; y are in B.<br>kTx ô€€€ Tyk = j<br>4<br>9<br>ô€€€<br>1<br>4<br>j =<br>7<br>36<br>&gt;<br>1<br>6<br>= kx ô€€€ yk:<br>therefore T is not nonexpasnsive.<br>2<br>1.2 Some Banach Spaces and their Properties.<br>In this section we give the denition of some special Banach spaces as well as some of their<br>geometric properties.<br>Denition 1.2.1 A Banach space E is said to be uniformly convex if for any 2 (0; 2],<br>there exists = () &gt; 0 such that for all x; y 2 E with jjxjj 1, jjyjj 1 and jjxô€€€yjj &gt; ,<br>then jj 1<br>2 (x + y)jj 1 ô€€€ . Geometrically, a Banach space is uniformly convex if the unit<br>ball centred at the origin is uniformly round. Again, uniform convexity is a property of<br>the norm on E.<br>The modulus of convexity of E is dened by<br>E() = inf f1 ô€€€ jj<br>x + y<br>2<br>jj : jjxjj = 1 = jjyjj; jjx ô€€€ yjj g;<br>0 &lt; 2.<br>Denition 1.2.2 A Banach space E is said to be smooth if for all x 2 E, x 6= 0 with<br>jjxjj = 1, there exists f 2 E such that hx; fi = 1. Smoothness is a property of the norm.<br>In fact, E is smooth if and only if 8x; y 2 E, x 6= 0<br>lim<br>t!0<br>jjx + tyjj ô€€€ jjxjj<br>t<br>(1:1)<br>exists. The limit, when it exits, is of the form fx(y) with fx 2 E and is called the<br>Gateaux derivative of the norm in E. Thus T is smooth if and only if the norm is Gateaux<br>dierentiable.<br>Denition 1.2.3 A Banach space is uniformly smooth if and only if the limit (1.1) exists<br>uniformly on the set<br>U = f(x; y) 2 E E : jjxjj = jjyjj = 1g:<br>Denition 1.2.4 A Banach space E is said to be an Opial space (see for example [1],<br>[16]) if for each sequence fxng1 n=1 in E which converges weakly to a point x 2 E<br>lim inf jjxn ô€€€ xjj &lt; lim inf jjxn ô€€€ yjj;<br>3<br>for all y 2 E, y 6= x. It is known (see [17]) that every Hilbert space and every lp(1 &lt; p &lt; 1)<br>space enjoy the property. Also in [9], D Van showed that any separable Banach space can<br>be equivalently re-normed so that it satises Opial condition. Indeed, for any normed<br>space E the existence of a weakly sequentially continuous duality map implies that E is<br>an Opial’s space, but the converse implication does not hold. Notably, the Lebesgue space<br>Lp is not an Opial space for p 6= 2.<br>Denition 1.2.5 A function : [0;1) ! [0;1) is said to be a guage function if is<br>continuous and strictly increasing with (0) = 0 and lim<br>t!1<br>(t) = 1.<br>Denition 1.2.6 Let E be a real Banach space. Let E denote the topological dual of E<br>and 2E be the collection of all subsets of E. Let : [0;1) ! [0;1) be a gauge function.<br>The mapping J:E ! 2E dened by<br>Jx = ff 2 E : hx; fi = kxkkfk; kfk = (kxk)g:<br>is called duality map with gauge function , where h:; :i denotes the generalized duality<br>paring between E and E. We note that if 1 &lt; q &lt; 1, then (t) = tqô€€€1 is a gauge<br>function. The duality mapping Jq : E ! 2E with gauge (t) = tqô€€€1 dened for each<br>x 2 E by<br>Jx = ff 2 E : hx; fi = kxkkfk; kfk = kxkqô€€€1g:<br>is called the generalised duality mapping. If q = 2, we obtain<br>J2 := J : E ! 2E<br>;<br>dened for all x 2 E by<br>J2x := J(x) = ff 2 E : hx; fi = kxk2 = kfk2g:<br>J2 is known as the normalised duality map. It is well known ( see for instance [2; 8]) that<br>for 1 &lt; q &lt; 1; Jq(x) = kxkqô€€€2J(x); for x 2 X; x 6= 0: The following theorem has been<br>proved for uniformly convex Banach space.<br>4<br>Theorem 1.2.7(Xu, [30]) Let p &gt; 1 and r &gt; 0 be two xed real numbers.Then a Banach<br>space X is uniformly convex if and only if there exists a continuous, strictly increasing<br>and convex function<br>g : R+ ! R+; g(0) = 0;<br>such that for all x; y 2 Br and 0 1,<br>kx + (1 ô€€€ )ykp kxkp + (1 ô€€€ )kykp ô€€€Wp()g(kx ô€€€ yk): (1:3)<br>where Wp() := p(1 ô€€€ ) + (1 ô€€€ )p and Br := fx 2 X : kxk rg:<br>The next result below establishes the existence of a xed point for asymptotically nonex-<br>pansive mapping in a nonempty, closed, convex and bounded subset of a uniformly convex<br>Banach space.<br>Theorem 1.2.8([10] Theorem 1) Let K be a nonemtpty, closed, convex and bounded<br>subset of a uniformly convex Banach space X, let F : K ! K be an asymptotically<br>nonexpansive mapping,then F has a xed point.<br>Proof For each x 2 K and r &gt; 0, Let S(x; r) denote the spherical ball centred at x with<br>radius r. Let y 2 K be xed, and let the set Ry consist of those numbers for which there<br>exists an integer k such that<br>K (1<br>i=kS(Fiy; )) 6= ;:<br>If d is the diameter of K then d 2 Ry, so Ry 6= ;. Let 0 = g:l:b:Ry, and for each &gt; 0,<br>dene C = [1k<br>=1(1i<br>=kS(Fiy; + )). Thus for each &gt; 0 the set C K are nonempty<br>and convex, so re exivity of X implies that<br>C = &gt;0( C K) 6= ;:<br>Note that for x 2 C and &gt; 0 there exists an integer N such that if i N; kx ô€€€ Fiyk<br>0 + .<br>5<br>Now let x 2 C and suppose the sequence fFnxg does not converge to x (i.e., suppose<br>Fx 6= x). Then there exists &gt; 0 and a subsequence fFnixg of fFnxg such that<br>kFnix ô€€€ xk ; i = 1; 2; ::::<br>For m &gt; n,<br>kFnx ô€€€ Fmxk knkx ô€€€ Fmô€€€nxk;<br>where kn is the Lipschitz constant for Fn obtained from the denition of asymptotic<br>nonexpansiveness. Assume 0 &gt; 0 and choose &gt; 0 so that (1 ô€€€ (<br>0+))(0 + ) &lt; 0.<br>Select n so that kxô€€€Fnxk and also that kn(0 +</p><p>2<br>) 0 +. If N n is suciently<br>large, then m &gt; N implies<br>kx ô€€€ Fmô€€€nyk 0 +</p><p>2<br>and we have<br>kFnx ô€€€ Fmyk knkx ô€€€ Fmô€€€nyk 0 + ;<br>kx ô€€€ Fmyk 0 + :<br>Thus by uniform convexity of X, if m &gt; n,<br>k(<br>x + Fnx<br>2<br>) ô€€€ Fmyk (1 ô€€€ (</p><p>0 +<br>))(0 + ) &lt; 0;<br>and this contradicts the denition of 0. Hence we conclude 0 = 0 or Fx = x. But 0 = 0<br>implies fFnyg is a Cauchy sequence yielding Fny ! x = Fx as n ! 1: Therefore the set<br>consists of a single point which is a xed point under F.<br>Kirk, Yanez and Shin [13] improved the result of Goebel and Kirk [10]. They proved that<br>if a re exive Banach space E has the property that each of its closed, bounded and convex<br>subset has the xed point property for nonexpansive maps, then it will also have the<br>xed point property for asymptotically nonexpansive mapping which has a nonexpansive<br>iterate.<br>Theorem 1.2.9 Let H be a real Hilbert space and C a nonempty closed and convex<br>subset of H. Let T : C ! C be an asymptotically non expansive map. Then Fix(T) =<br>fx 2 C : Tx = xg is closed and convex.<br>6<br>Proof:<br>Convexity.<br>For any x; y 2 Fix(T) and 2 (0; 1). Let z := (1 ô€€€ )x + y. Then,<br>kTnz ô€€€ zk2 = kTnz ô€€€ [(1 ô€€€ )x + y]k2<br>= k(1 ô€€€ )(Tnz ô€€€ x) + (Tnz ô€€€ y)k2<br>= (1 ô€€€ )kTnz ô€€€ xk2 + kTnz ô€€€ yk2 ô€€€ (1 ô€€€ )kx ô€€€ yk2<br>(1 ô€€€ )k2n<br>kz ô€€€ xk2 + k2n<br>kz ô€€€ yk2 ô€€€ (1 ô€€€ )kx ô€€€ yk2<br>= (1 ô€€€ )k2n<br>k(1 ô€€€ )x + y ô€€€ xk2 + k2n<br>k(1 ô€€€ )x + y ô€€€ yk2 ô€€€ (1 ô€€€ )kx ô€€€ yk2<br>= 2(1 ô€€€ )k2n<br>kx ô€€€ yk2 + k2n<br>k(1 ô€€€ )x + (1 ô€€€ )yk2 ô€€€ (1 ô€€€ )kx ô€€€ yk2<br>= 2(1 ô€€€ )k2n<br>kx ô€€€ yk2 + (1 ô€€€ )2k2n<br>kx ô€€€ yk2 ô€€€ (1 ô€€€ )kx ô€€€ yk2<br>= (1 ô€€€ )[k2n<br>+ (1 ô€€€ )k2n<br>ô€€€ 1]kx ô€€€ yk2<br>= (1 ô€€€ )(k2n<br>ô€€€ 1)kx ô€€€ yk2 ! 0:<br>Therefore,<br>kTnz ô€€€ zk ! 0:<br>Now,<br>0 kTz ô€€€ zk<br>kTz ô€€€ Tnzk + kTnz ô€€€ zk<br>k1kz ô€€€ Tnô€€€1zk + kTnz ô€€€ zk ! 0:<br>Hence<br>Tz = z:<br>We now show that Fix(T) is closed. Let fxng Fix(T) be arbitrary and let xn ! x as<br>n ! 1, we show that x is in Fix(T):<br>Tx = T lim<br>n!1<br>xn = lim<br>n!1<br>Txn = lim<br>n!1<br>xn = x:<br>7<br>Two other denitions of asymptotically nonexpansive maps has also appeared in the lit-<br>erature. One of the denitions which is weaker than Denition 1.1.2 was introduced by<br>Kirk[12] and requires that<br>lim sup<br>n!1<br>sup<br>y2K<br>(kTnx ô€€€ Tnyk ô€€€ kx ô€€€ yk) 0:<br>for every x 2 K and that TN be continuous for some integer N &gt; 1:<br>The other denition which has appeared require that<br>lim sup<br>n!1<br>(kTnx ô€€€ Tnyk ô€€€ kx ô€€€ yk) 0 8x; y 2 K:<br>This, however, has been shown to be unsatisfactory from the point of view of xed<br>point theory. Tingly [24] constructed an example of a closed convex set K in a Hilbert<br>space and a continuous map T : K ! K which actually satises the following condition<br>limn!1 kTnx ô€€€ Tnyk = 0 8x; y 2 K but has no xed point.<br>1.3 Iterative Algorithms for Asymptotically Nonex-<br>pansive Mappings<br>1.3.1 Modied Mann Iterative Algorithm<br>The averaging iteration process,<br>xn+1 = (1 ô€€€ n)xn + nTnxn; n 1;<br>where T : K ! K is asymptotically nonexpansive in the sense of denition 1.1.2, K a<br>closed, convex and bounded subset of a Hilbert space was introduced by Schu[23].<br>In [21] Schu used the modied Mann iteration method,<br>xn+1 = (1 ô€€€ n)xn + nTnxn; n 1<br>and proved the following theorem.<br>8<br>Theorem(1.3.1): Let H be a Hilbert space, K a nonempty closed convex and bounded<br>subset of H. Let T : K ! K be a completely continuous asymptotically nonexpansive map<br>with sequence fkng1 n=1 with kn 2 [1;1) for all n 1; limn!1 kn = 1 and<br>P1<br>n=1 (k2n<br>ô€€€ 1) &lt;<br>1. Let fng1 n=1 be a sequence in [0,1] satifying the condition &lt; n &lt; 1 ô€€€ for some<br>&gt; 0. Then the sequence fxng generated from an arbitrary x1 2 K by<br>xn+1 = (1 ô€€€ n) xn + nTnxn; n 1;<br>converges strongly to a xed point of T.<br>1.3.2 Iterative method of Schu<br>In this subsection, consider algorithm for approximating xed points of asymptotically<br>nonexpansive mappings which deals with almost xed points,<br>xn := nTnxn<br>of an asymptotically nonexpansive mappings T. Schu [24] proved the convergence of this<br>sequence fxng to some xed point of T under additional assumption that T is uniformly<br>asymptotically regular and (I ô€€€ T) is demiclosed. These assumptions had actually been<br>made by Vijayaraju[27] to ensure the existence of a xed point of T By strengthening<br>the asymptotic regularity of T, Schu established the convergence of an explicit iteration<br>scheme,<br>zn+1 := nTnzn<br>to some xed point of T.<br>1.3.3 Halpern-type process<br>One of the most useful results concerning algorithms for approximating xed points of non-<br>expansive mappings in real uniformly smooth Banach spaces is the celebrated convergence<br>theorem of Riech[29] who proved that the implicit sequence fxng dened as,<br>xn =<br>1<br>n<br>u + (1 ô€€€<br>1<br>n<br>)Txn:<br>9<br>converges strongly to a xed point of T. Several authors have tried to obtain a result<br>analogous to that of Reich [19] for asymptotically nonexpansive mappings. Suppose K<br>is a nonempty bounded closed convex subset of a real uniformly smooth Banach space E<br>and T : K ! K is an asymptotically nonexpansive mapping with sequence kn 1 for all<br>n 1. Fix u 2 K and dene, for each integer n 1, the contraction mapping Sn : K ! K<br>by, (see [6]),<br>Sn(x) = (1 ô€€€<br>tn<br>kn<br>)u +<br>tn<br>kn<br>Tnx;<br>where ftng [0; 1) is any sequence such that tn ! 1.Then by the Banach Contraction<br>Mapping Principle, there is a unique point xn xed by Sn, i.e. there is xn such that<br>xn = (1 ô€€€<br>tn<br>kn<br>)u +<br>tn<br>kn<br>Tnxn:<br>For some existing results on asymptotically nonexpansive maps, an interested reader<br>should see [4,7,14,18,20,22] and the references there in.<br>1.4 Organization of Thesis<br>We have introduced in this chapter (Chapter One), various iteration methods for asymp-<br>totically nonexpansive maps and some existing results on them. We also studied some of<br>the important Banach spaces which are encountered in this work and their properties.<br>In Chapter Two (the Preliminaries), we presented most of the classical results on the se-<br>quences of real numbers encountered in Operator Theory. We also looked at projection<br>maps and some other results vital to this work.<br>In Chapter Three, we study certain averaging iterative algorithm for approximating the<br>xed point of asymptotically nonexpansive mappings introduced by Goebel and Kirk [10].<br>10</p> <br><p></p>