Solution of generalized equilibrium problems and common fixed point of finite family of strict pseudocontractions with application

Thesis Overview

<p> </p><p>INTRODUCTION<br>The content of this thesis falls within the area of nonlinear operator theory. This area has attracted<br>attention of several researchers due to its wide range of application in dierent areas of pure and<br>applied sciences. The research documented in this thesis concentrated on the following topic:<br>Approximation of solution of generalized equilibrium problems and common xed point of nite<br>family of strict pseudocontractions.<br>1.1 Background of Study<br>In sciences, engineering, economics and in some other areas where there is a quantitative analysis,<br>we are greatly interested in describing how systems evolve in time, that is, in describing system’s<br>dynamics. We will restrict ourselves to one dimensional case for the purpose of illustration. We will<br>always write u = u(t), which is the state of the system. We think of the dependent variable u as<br>the state variable of a system that is varying with time t, which is the independent variable. Thus,<br>knowing u is virtually the same as knowing what state the system is, at time t. For example, u(t)<br>could be the number of patience admitted in a hospital, the quantity of data processed by CPU,<br>the concentration of a chemical substance such as sugar in the body, the number of immigrants<br>into a country, the current in an electrical circuit, the speed of a spacecraft, or the monthly sales<br>of an advertised item. Knowledge of u(t) for a given system tells us how the system changes with<br>respect to time. Often, we relate the state u(t) to its rates of change, as expressed by its derivatives<br>u<br>0<br>(t), u<br>00<br>(t); ; and so on. It is important to note that some of the dynamical system can be<br>described by the following model,<br>du<br>dt<br>+ Au = f(t; u(t)): (1.1)<br>Where A is an operator dened on some appropriate spaces. Equation (1.1) is called nonhomoge-<br>neous rst order ordinary dierential equation if f(t; u(t)) 6= 0, otherwise it is homogeneous rst<br>order ordinary dierential equation. Assuming that u(t) is a solution to equation (1.1) and suppose<br>that t0 is the initial reference time that we want to start studying the above model, we can always<br>use u(t) to make comparative analysis of the behaviour of the dynamical system between the time<br>t0 and t. If f(t; u(t)) = 0, then equation (1.1) becomes<br>du<br>dt<br>+ Au = 0: (1.2)<br>If we put A 0 in equation (1.1), then, equation (1.1) reduces to<br>du<br>dt<br>= f(t; u(t)): (1.3)<br>1<br>Picard proved that under some certain assumptions on f, its domain and co-domain, that problem<br>(1.3) is equivalent to problem of nding xed point of an operator T dened by<br>(Tu)(t) = (t)<br>= u0 +<br>Z t<br>t0<br>f(s; u(s))ds; (1.4)<br>where T is a self map dened on some appropriate innite dimensional function space and u0 =<br>u(t0). Though equation (1.3) looks simple, it happens that most times, we do not have exact<br>solution of equation (1.3) rather the numerical solution. This numerical solution corresponds to<br>the approximated xed point of some nonlinear operators. Furthermore, it is well known that at<br>equilibrium state, du<br>dt = 0, hence at equilibrium state, equation (1.2) becomes<br>Au = 0: (1.5)<br>Consequently, equation (1.2) reduces to problem of nding zero (zeros) of A which corresponds<br>(correspond) to problem of nding xed point of some operator T by dening A I ô€€€ T.<br>We recall that if a function f is twice dierentiable at a point x i.e f00(x) exists and f00(x) 6= 0<br>and f0(x) = 0 then, x is an extremum point. This leads us to the following question: How do we<br>get the optimizer of a function whenever it exist without necessarily dierentiating f in the usual<br>sense? We have to note that some of the important operators involved in optimization problems<br>are not dierentiable in the usual sense. We give an example to illustrate our point. Consider the<br>map f : H ! R dened by<br>f(x) = kxk;<br>where H is a real Hilbert space. It is well known that f is not dierentiable at zero. However,<br>it is easy to see that zero is the minimizer. From the foregoing analysis, it is worthy to study<br>optimization problems.<br>Let us consider the problem of nding u 2 K such that<br>f(u; y) 0; 8 y 2 K; (1.6)<br>where K is a nonempty, closed and convex subset of real Hilbert space H and f : K K ! R; a<br>bifunction. We observe that it includes xed point problems and optimization problems as special<br>cases. Furthermore, if we consider a nonlinear operator A : K ô€€€! H and a problem of nding<br>x 2 K such that<br>f(u; y) = hAu; y ô€€€ ui 0; 8 y 2 K: (1.7)<br>We obtain another special case of equation (1.6)<br>If however, we consider the problem of nding u 2 K such that<br>f(u; y) + hAu; y ô€€€ ui 0; 8 y 2 K; (1.8)<br>then, we have a new problem which include problems (1.6) and (1.7) as special cases, we are going<br>to study problem (1.8) extensively in this thesis.<br>Problem (1.6) was introduced by Blum and Oettli (1994) and Noor and Oettli (1994). It has a<br>great impact and in uence in the development of several branches of Pure and Applied Sciences.<br>Motivated by the above example and forgoing analysis, We are interested in studying some iterative<br>algorithm for approximating the solution of equation (1.8) and common xed point of nite family<br>of strict pseudocontractions.<br>We present some preliminary results, denitions and some well known facts in Hilbert spaces,<br>understanding them plays a crucial role in comprehending the entire work. We shall therefore, im-<br>mediately turn to the preliminary section where most of the necessary denitions and explanation<br>of terms are displayed.<br>2<br>1.2 Preliminary<br>In this section, we give denitions of some crucial concepts that shall be needed in sequel.<br>Denition 1.1. Let T : D(T) H ! H be a map. then, T is said to be<br>(i) Asymptotically k-strictly pseudocontraction in the intermediate sense (Sahu, et ai., 2008)<br>with sequence f ng if there exists a constant k 2 [0; 1) and a sequence f ng [0;1) with<br>limn!1 n = 0 such that for all x; y 2 K and for all n 2 N;<br>lim sup<br>n!1<br>sup<br>x;y2K<br>(kTnx ô€€€ Tnyk ô€€€ (1 + n)kx ô€€€ yk2 ô€€€ kk(I ô€€€ Tn)x ô€€€ (I ô€€€ Tn)yk2) 0: (1.9)<br>(ii) k-Lipschitz if there exists k 0 such that for all x; y 2 D(T);<br>kTx ô€€€ Tyk kkx ô€€€ yk:<br>If k 2 [0; 1) in (ii); then T is called contraction and if k 2 [0; 1]; then the mapping T is called<br>nonexpansive.<br>(iii) k-strictly pseudocontractive mapping if there exists a constant k 2 [0; 1) such that for all<br>x; y 2 D(T):<br>kTx ô€€€ Tyk2 kx ô€€€ yk2 + kkx ô€€€ Tx ô€€€ (y ô€€€ Ty)k2:<br>(iv) rmly nonexpansive if for all x; y 2 D(T);<br>kTx ô€€€ Tyk2 hTx ô€€€ Ty; x ô€€€ yi :<br>(v) monotone if for all x; y 2 D(T); hTx ô€€€ Ty; x ô€€€ yi 0:<br>(vi) -inverse strongly monotone if there exists &gt; 0 such that for all x; y 2 D(T);<br>hTx ô€€€ Ty; x ô€€€ yi kTx ô€€€ Tyk2:<br>Furthermore, a point x 2 D(T) is called xed of T if Tx = x.<br>Remark 1.2. (i) It has been shown by Marino and Xu (2007) that the class of strict pseu-<br>docontractions are Lipschitz with Lipschitz constant 1+k<br>1ô€€€k . Therefore, the class of strict<br>Pseudocontractions is a subclass of uniformly continuous mappings, as well as a subclass of<br>Lipschitz pseudocontractive mappings.<br>(ii) It is easy to see that every nonexpansive map is 0-strictly pseudocontraction. Hence, the<br>class of strict pseudocontractions contains the class of nonexpansive maps. We, however,<br>emphasize that the converse is false. In fact, we have the following example.<br>Example 1.3. Let H be a real Hilbert space and let T : H ! H be dened by<br>T(x) = ô€€€2x<br>It is not dicult to see that T is not nonexpansive map. We argue as follow to show that T is<br>strictly pseudocontraction. First, we observe that for any x; y 2 H;<br>kTx ô€€€ Tyk2 = 4kx ô€€€ yk2 = (1 + 3)kx ô€€€ yk2<br>=</p><p>1 +</p><p>3<br>9</p><p>(9)</p><p>kx ô€€€ yk2<br>= kx ô€€€ yk2 +<br>3<br>9<br>k3(x ô€€€ y)k2<br>= kx ô€€€ yk2 +<br>1<br>3<br>k(1 + 2)x ô€€€ (1 + 2)y)k2<br>3<br>= kx ô€€€ yk2 +<br>1<br>3<br>k(1 ô€€€ (ô€€€2))x ô€€€ (1 ô€€€ (ô€€€2))y)k2<br>= kx ô€€€ yk2 +<br>1<br>3<br>k(I ô€€€ T)x ô€€€ (I ô€€€ T)yk2<br>kx ô€€€ yk2 + kk(I ô€€€ T)x ô€€€ (I ô€€€ T)yk2; 8 k 2</p><p>1<br>3<br>; 1</p><p>:<br>Denition 1.4. The generalized mixed equilibrium problems (abbreviated GMEP) for operators<br>f; ; B is a problem of nding u 2 K such that<br>f(u; y) + (y) ô€€€ (u) + hBu; y ô€€€ ui 0; 8 y 2 K; (1.10)<br>where K is nonempty, closed and convex subset of a real Hilbert space H, f is a real valued<br>bifunction with domain K K, is a proper extended real valued function with domain K, that<br>is, : K ! R [ f+1g and B an operator dened from K to H. The solution set of (1.10) is<br>denoted by<br>GMEP(f;;B) := fu 2 K : f(u; y) + (y) ô€€€ (u) + hBu; y ô€€€ ui 0; 8 y 2 K:<br>It is easy to see that u 2 GMEP(F;;B) implies that<br>u 2 D() := fu 2 H : (u) &lt; +1g:<br>If 0 B in (1.10), then, inequality (1.10) reduces to the Classical equilibrium problem<br>(abbreviated EP(f)), that is, the problem of nding u 2 K such that<br>f(u; y) 0; 8 y 2 K: (1.11)<br>Solution set of (1.11) is denoted by<br>EP(f) := fu 2 K : f(u; y) 0; 8 y 2 Kg:<br>If 0 f in (1.10), then (1.10) reduces to the Classical variational inequality problem<br>GMEP(0; 0;B), that is, the problem of nding u 2 K such that<br>hBu; y ô€€€ ui 0; 8 y 2 K: (1.12)<br>Solution set of (1.12) is denoted by<br>V:I(B;K) = fu 2 K : hBu; y ô€€€ ui 0 ; 8 y 2 Kg:<br>If B 0 f in (1.10), then (1.10) reduces to the following minimization problem: nd u 2 K<br>such that<br>(y) (u); 8 y 2 K: (1.13)<br>Solution set of (1.13) is denoted by Argmin(), where<br>Argmin() := fu 2 K : (y) (u); 8 y 2 Kg:<br>If B 0 in (1.10), then (1.10) reduces to the mixed equilibrium problem (abbreviated MEP(f;; 0),<br>that is, the problem of nding u 2 K such that<br>f(u; y) + (y) ô€€€ (u)+ 0; 8 y 2 K: (1.14)<br>4<br>Solution set of (1.14) is denoted by<br>MEP(f; ) := fu 2 K : f(u; y) + (y) ô€€€ (u)+ 0; 8 y 2 Kg:<br>If 0 in (1.10), then (1.10) reduces to the Generalized equilibrium problem, that is,<br>the problem of nding u 2 K such that<br>f(u; y) + hBu; y ô€€€ ui 0; 8 y 2 K: (1.15)<br>Solution set of (1.15) is denoted by<br>GEP(f;B) := fu 2 K : f(u; y) + hBu; y ô€€€ ui 0; 8 y 2 Kg:<br>If f 0 in (1.10), then (1.10) reduces to the Generalized variational inequality problems,<br>that is, the problem of nding u 2 K such that<br>(u) ô€€€ (y) + hBu; y ô€€€ ui 0; 8 y 2 K: (1.16)<br>Solution set of (1.16) is denoted by<br>GV I(;B;K) := fu 2 K : (u) ô€€€ (y) + hBu; y ô€€€ ui 0; 8 y 2 Kg:<br>From the forgoing discussion so far, we observe that (1.10) solves three dierent types of prob-<br>lems simultaneously i.e., it solves problem of optimization, variational inequality and equilibrium<br>problems.<br>Throughout this thesis, we assume that our bifunction f, satises the following conditions,<br>namely:<br>A1 f(x; x) = 0; 8 x 2 K;<br>A2 f is monotone in the sense that<br>f(x; y) + f(y; x) 0; 8 x; y 2 K;<br>A3 f is hemi-continuous, that is,<br>lim sup<br>t!0+<br>f(tz + (1 ô€€€ t)x; y) f(x; y); 8 x; y; z 2 K;<br>A4 The function f(x; <img alt="" src="https://s.w.org/images/core/emoji/11/svg/1f642.svg">&nbsp;is convex and lower semicontinous, 8 x 2 K. Though the following<br>denition is well known, we still present it here for clarity sake.<br>Denition 1.5. Let E be a real vector space.The map<br>1. k:k : E ! [0;1) satisfying the following conditions:<br>(i) kxk 0; 8 x 2 E and kxk = 0 if and if x = 0,<br>(ii) For any 2 R, kxk = jjkxk; 8 x 2 E,<br>(iii) kx + yk kxk + kyk; 8 x; y 2 E,<br>is called a norm on E and the pair (E; k:k) is called a normed vector space.<br>2. h:; :i : E E ! R satisfying the following conditions:<br>(i) hx; xi 0; 8 x; y 2 E and hx; xi = 0 if and only if x = 0,<br>(ii) symmetricity, that is, hx; yi = hx; yi ; 8 x; y 2 E,<br>5<br>(iii) bilinear, that is, linear in both rst and second argument.<br>is called real inner product on E and the pair (E; h:; :i) is called a real inner product<br>space.<br>Remark 1.6. If (E; h:; :i) is an inner product space and we consider the map k:k : E ! R dened<br>by kxk =<br>p<br>hx; xi. One can easily verify that k:k is a norm on E. It is called the norm induced by<br>the inner product.<br>From now onward, we will always assume that:<br>(i) H is a real Hilbert space.<br>(ii) K is nonempty, closed and convex subset of H.<br>(iii) h:; :i is an inner product associated with H.<br>(iv) k:k is the norm induced by the inner product.<br>(v) F(T) = fx 2 D(T) : Tx = xg:<br>Denition 1.7. Let fxng be a sequence in H. Then, fxng is said to converge to x 2 H<br>(i) strongly, if 8 &gt; 0; 9 n 2 N such that 8 n n; kxn ô€€€ xk &lt; ;<br>(ii) weakly, if 8 f 2 H, the sequence ff(xn)gn1 converges to f(x) in R with the usual topology.<br>Denition 1.8. A net (or generalized sequence ) in H indexed by A := [0; 1] is an operator from<br>A to H. It is denoted by fxg2A:<br>Denition 1.9. (i) Let fxg2A be a net in H, fxg2A converges to a vector x as ! 0 if<br>fxg2A lies eventually in every neighbourhood of x. i.e 8 V 2 Nbh(x); 9 b 2 A such that<br>b ) x 2 V:<br>(ii) A point x 2 H is a cluster point of the net fxg2A if fxg2A frequently lies in every<br>neighbourhood of x. i.e 8 V 2 Nbh(x); 8 b 2 A; 9 2 A such that b and x 2 V:<br>Denition 1.10. Let f : H ! R [ f1g and and x0 2 H, where H is a real Hilbert as we have<br>pointed out before. Then, f is lower semicontinuous at x0 if, for every net (x)2A H such that<br>x ! x0 as ! 0+, Then, f(x0) lim inf<br>!0+<br>f(x)<br>1.2.1 Some Facts in Hilbert Spaces<br>(i) Given a nonempty, closed and convex subset K of H, let PK : H ! K be the projection<br>operator. It is well known that for arbitrary vector x 2 H, z = PKx if and only if<br>hx ô€€€ z; y ô€€€ zi 0; 8 y 2 K: (1.17)<br>The following identities are also well known in Hilbert spaces:<br>(ii) for any t 2 [0; 1] and for any x; y 2 H;<br>ktx + (1 ô€€€ t)yk2 = tkxk2 + (1 ô€€€ t)kyk2 ô€€€ t(1 ô€€€ t)kx ô€€€ yk2: (1.18)<br>(iii) for any x; y 2 H<br>kx ô€€€ yk2 = kxk2 + kyk2 ô€€€ 2 hx; yi : (1.19)<br>(iv) It is also well known that given any vector y 2 H; there exists fy 2 H such that<br>fy(x) = hx; yi ; 8 x 2 H: (1.20)<br>Where H denotes the dual space of H, i.e the set of all bounded linear operators from H to R.<br>Remark 1.11. It is easy to see using equation (1.20) that xn * x if and only if for any y 2 H;<br>hxn; yi ! hx; yi.<br>6<br>1.3 Statement of Problem<br>Several Authors have published articles on how to approximate the solution of generalized equi-<br>librium problems and common xed points of nite family of strict pseudocontractions<br>For example, Marino and Xu (2007) proved that: Given a self mapping T from a nonempty, closed<br>and convex subset K of a real Hilbert space H , the sequence fxng dened recursively by the<br>formula<br>xn+1 = nxn + (1 ô€€€ n)Txn; n 0; (1.21)<br>converges weakly to a xed point of T. Where the initial guess x0 2 K is arbitrary, and fng is<br>a real control sequence in the interval (0; 1): They proved the above result under the additional<br>hypothesis that<br>(i) T is k-strictly pseudocontraction that admits at least a xed point,<br>(ii) k &lt; n &lt; 1; for all n 1 and<br>1X<br>n=0<br>(n ô€€€ k)(1 ô€€€ n) = 1:<br>Hu and Cai (2011) proved the following theorem for class of asymptotically pseudocontractive<br>mapping in the intermediate sense:<br>Theorem 1.12. (Hu and Cai, 2011 ) Let C be a nonempty, closed and convex subset of a real<br>Hilbert space H and N 1 be an integer, f : C C ! R be a bifunction satisfying A1 – A4 and A<br>be an -inverse strongly monotone mapping of C into H. Let, for each 1 i N; Ti : C ! C be<br>a uniformly continuous ki -strictly asymptotically pseudocontractive mapping in the intermediate<br>sense for some 0 ki &lt; 1 with sequences f ng [0;1) such that<br>1X<br>n=1<br>n;i &lt; 1 and fcn;ig [0;1)<br>such that limn!1 cn;i = 0: Let k = maxfki : 1 i Ng; n = maxf n;i : 1 i Ng and<br>cn = maxfcn;i : 1 i Ng. Assume that F := Ni<br>=1F(Ti) EP is nonempty. Let fxng and fung<br>be sequences generated initially by arbitrary element x1 2 C and then by<br>8&gt;&lt;<br>&gt;:<br>f(un; y) + hAxn; y ô€€€ uni + 1<br>rn<br>hy ô€€€ un; un ô€€€ xni 0; 8 y 2 K;<br>zn = (1 ô€€€ n)un + nTk(n)<br>i(n) un;<br>xn+1 = nun + (1 ô€€€ n)zn;<br>(1.22)<br>where fng; fng and frng satisfy the following conditions:<br>(i) 0 &lt; a n 1; fng (0; 1);<br>(ii) 0 &lt; n 1 ô€€€ k ô€€€ &lt; 1; fng (0; 1);<br>(iii)<br>1X<br>n=1<br>ncn &lt; 1;<br>(iv) 0 &lt; b rn c 2.<br>Then, the sequences fxng and fung converge weakly to an element of F.<br>Huang and Ma (2014) proved the following theorem by slightly adjusting scheme (1.12) and con-<br>sidering the class of strict pseudocontractions. They obtained the following theorem:<br>7<br>Theorem 1.13. (Huang and Ma, 2014) Let K be a nonempty, closed and convex subset of a real<br>Hilbert space H. Let T : C ! H be a -inverse-strongly monotone mapping. Let F be a bifunction<br>from C C to R satisfying conditions (A1)ô€€€(A4). Let S : C ! C be a k-strict pseudocontraction.<br>Assume that F := EP(F; T)<br>T<br>F(S) is not empty. Let fng; fng; f ng and fng be sequences<br>in (0; 1). Let frng be a sequence in (0; 2), and let feng be a bounded sequence in C. Let fxng be<br>a sequence generated in the following manner:<br>8&gt;&lt;<br>&gt;:<br>x 2 C;<br>F(un; u) + hTxn; u ô€€€ uni + 1<br>rn<br>hu ô€€€ un; un ô€€€ xni 0 8 u 2 K;<br>xn+1 = nxn + n(nun + (1 ô€€€ n)Sun) + nen n 1;<br>(1.23)<br>Assume that the sequences fng; fng; f ng nd fng; frng satisfy the following restrictions: 0 &lt;<br>a n a<br>0<br>&lt; 1, 0 k n b &lt; 1, 0 &lt; c rn d &lt; 2 and<br>1X<br>n=1<br>n &lt; 1: Then, the sequence<br>fxng converges weakly to some point x 2 F, where x = limn!1 PF xn:<br>The problem is that all their results concluded weak convergence which seems to be less useful in<br>applications compare to strong convergence. In this thesis, we studied the above problem and we<br>constructed iterative algorithm by modifying the operators used in scheme (1.12) as Huang and<br>Ma did, drop the error term introduced in scheme (1.13) and use a modied Halpern scheme which<br>seems better than Mann’s scheme in several ways to study the convergence analysis of the new<br>problem.<br>1.4 Motivations<br>Our motivation arises from application point of view, the work of Huang and Ma (2014) and that<br>of Hu and Cai (2011) precisely theorems (1.12) and (1.13), respectively.<br>1.5 Objectives<br>Our objectives are the following :<br>(i) to introduce a scheme that will have computational advantage over the existing ones.<br>(ii) to prove strong convergence theorem using our scheme which seems to be more useful in<br>application.<br>1.6 Limitations<br>We proved our result in Hilbert space setting, so we are faced with the challenge of whether our<br>result is valid in more general Banach spaces. It is dicult in practice to get operators that<br>are inverse strongly monotone.This calls for further research on how to relax the inverse strongly<br>monotone condition.</p> <br><p></p>