Iterative algorithms for single-valued and multi-valued nonexpansive-type mappings in real lebesgue spaces.

Thesis Overview

<p> General Introduction<br>Fixed Point Theory is concerned with solutions of the equation<br>x = Tx (1.0.1)<br>where T is a (possibly) nonlinear operator defined on a metric space. Any x<br>that solves (1.0.1) is called a fixed point of T and the collection of all such<br>elements is denoted by F(T). For a multi-valued mapping T : X ! 2X, a<br>fixed point of T is any x in X such that x 2 Tx:<br>Fixed Point Theory is inarguably the most powerful and effective tools<br>used in modern nonlinear analysis today. It is still an area of current intensive<br>research as it has vast applicability in establishing existence and uniqueness of<br>solutions of diverse mathematical models like solutions to optimization problems,<br>variational analysis, and ordinary differential equations. These models<br>represent various phenomena arising in different fields, such as steady state<br>temperature distribution, neutron transport theory, economic theories, chemical<br>equations, optimal control of systems, models for population, epidemics<br>and flow of fluids.<br>For example, given an initial value problem<br>dx(t)<br>dt = f(t; x(t));<br>x(t0) = x0:<br>(1.0.2)<br>This system is transformed into the functional equation<br>x(t) = x0 +<br>Z t<br>t0<br>f(s; x(s))ds:<br>1<br>To establish existence of solution to system (1.0.2), we consider the operator<br>T : X ! X(X = C([a; b])) defined by<br>Tx = x0 +<br>Z t<br>t0<br>f(s; x(s))ds:<br>Then finding a solution to the initial value problem (1.0.2) amounts to finding<br>a fixed point of T.<br>The existence(and uniqueness) of solution to equation (1.0.1), certainly, depends<br>on the geometry of the space and the nature of the mapping T. Existence<br>theorems are concerned with establishing sufficient conditions under<br>which the equation (1.0.1) will have a solution, but does not neccesarily show<br>how to find them. There are very many existence and uniqueness theorems in<br>the literature(see e.g. Kirk [67], Kato [62], Komura [68]).<br>Though existence theorems do not indicate how to construct a process starting<br>from a nonfixed point and convergent to a fixed point, they nevertheless<br>enhance understanding of conditions under which the existence of such fixed<br>points is guaranteed.<br>On the other hand, iterative methods of fixed points theory is concerned with<br>approximation or computation of sequences which converge to solutions of<br>(1.0.1). This is part of the problem that is being addressed in this thesis.<br>The pivot of the iterative methods of fixed point theory is the Banach contraction<br>mapping principle. It states that a self map T on a complete metric<br>space (X; d) satisfying<br>d(Tx; Ty) kd(x; y); 0 k &lt; 1; 8x; y 2 X; (1.0.3)<br>neccesarily has a unique fixed point and for any starting point x1, the sequence<br>fTnx1g converges strongly to that fixed point.<br>Many authors, see for example Alber [7], Boyd and Wong [25], have now investigated<br>more general conditions under which a mapping will have a unique fixed<br>point and also developed iterative sequences that converge to such fixed points.<br>If k = 1 in the inequality (1.0.3) above, the mapping T is tagged nonexpansive.<br>There are many examples that show that xn+1 = Tn(x) need not converge to<br>a fixed point of a nonexpansive mapping T, even if it has a unique fixed point.<br>We then need to impose additional conditions on T (and/or the space X) and<br>also modify the sequence Tn(x) to ensure convergence to a fixed point of T .<br>2<br>These notable iterative algorithms were introduced for nonexpansive mappings,<br>namely, the Krasnosel’skii sequence presented in [69] as: x1 2 X and<br>xn+1 =<br>1<br>2<br>(xn + Txn);<br>the Krasnoselskii-Mann algorithm given by: x1 2 X,<br>xn+1 = (1 ô€€€ )xn + Txn; 2 (0; 1);<br>the Halpern algorithm given in [59] as: u 2 X arbitrary and<br>xn+1 = nu + (1 ô€€€ n)Txn;<br>and the more general Mann sequence presented in [72] as<br>xn+1 = (1 ô€€€ n)xn + nTxn:<br>Diverse convergence theorems have been proved for these sequences, depending<br>on the smoothness of the underlying space and/or the compactness of the<br>mapping T:<br>Efforts to establish convergence theorems for nonexpansive mappings is likely<br>the most rewarding research venture in nonlinear analysis. It has helped in<br>the development of the geometry of Banach spaces and other related class of<br>mappings, namely, monotone and accretive operators.<br>A mapping M : X ! X is called ô€€€strongly monotone if<br>hx ô€€€ y;Mx ô€€€Myi kx ô€€€ yk2; 8x; y 2 X;<br>and A : X ! X is called ô€€€strongly accretive if<br>hAx ô€€€ Ay; j(x ô€€€ y)i kx ô€€€ yk2; 8x; y 2 X;<br>where h:; :i is the duality pairing between X and X; j(xô€€€y) 2 J(xô€€€y) where<br>J is the normalized duality mapping. When = 0, these mappings are called<br>monotone and accretive, respectively. If X is Hilbert space, these two notions<br>agree and they are simply refered to as monotone.<br>Accretive mappings have properties that are similar to those of monotone mappings.<br>However, the use of the strongly nonlinear mapping J make the study<br>of such mappings difficult. In a sense, the duality mapping on a Banach space<br>has all the properties of the Banach space that makes it differ from a Hilbert<br>space and the space can be characterized, almost, exclusive by the mapping.<br>3<br>These two ideas have proved to be very useful in many areas of interest. The<br>idea of accretive operators appear very often in partial differential equation,<br>in the existence theory of nonlinear evolution equations. On the other hand,<br>the idea of monotone operators appear in optimization theory and that, in<br>particular, include the increasingly important set-valued mapping called the<br>subdifferential. Given a convex, lower semicontinous function f, the subdifferential<br>is @f : X ! 2X given by<br>@f(x) := fx 2 X : f(y) ô€€€ f(x) hy ô€€€ x; xi; 8y 2 Xg:<br>The subdifferential is a monotone mapping and it is well known that 0 2 @f(x)<br>if and only if f(x) = inf<br>x2X<br>f(x). This motivates the study of the more general<br>problem of finding a zero, i.e x such that 0 2 Ax, of a monotone operator A.<br>The question on the existence of zeros is studied under the concept of maximal<br>monotone operators. A monotne mapping A is maximal monotone if the graph<br>G(A) is a maximal element when graphs of monotone operators in X X are<br>partially ordered by set inclusion. In that case, for any (x; y) 2 X X, the<br>inequality<br>hy1 ô€€€ y2; x1 ô€€€ x2i 0; 8×2 2 D(A); y2 2 Ax2<br>implies y1 2 Ax2: Maximal accretive mappings are defined accordingly.<br>The accretive operators are intimately connected with an important generalization<br>of nonexpansive mappings called the pseudocontractive mappings. A<br>mapping is pseudocontractive in the terminology of Browder and Petryshyn<br>[23] if for x; y in X, and for all r &gt; 0,<br>kx ô€€€ yk k(x ô€€€ y) + r[(x ô€€€ Tx) ô€€€ (y ô€€€ Ty)]k; :<br>By a result of Kato [62], this is equivalent to<br>h(I ô€€€ T)x ô€€€ (I ô€€€ T)y; j(x ô€€€ y)i 0:<br>Thus, a mapping T is pseudocontractive if and only if the complementary operator<br>A := I ô€€€ T is accretive. Moreover, the zeros of A coincides with the<br>fixed points of T.<br>Another interesting relationship is that the resolvent of an accretive mapping<br>A always exists(i.e I +A is invertible ) and it is nonexpansive. The resolvent<br>of A is a set valued mapping J : X ! 2X defined by<br>J(x) = (I + A)ô€€€1x; &gt; 0:<br>4<br>In this case, Aô€€€1(0) = Fix(J). More precisely, the mapping J is in fact<br>firmly nonexpansive, i.e<br>kJ(x) ô€€€ J(y)k2 hx ô€€€ y; J(x) ô€€€ J(y)i; 8x; y 2 X:<br>The existence and approximation algorithms for zeros of maximal monotone<br>operators are usually formulated in relation with the corresponding problem<br>for fixed points of firmly nonexpansive mappings. This makes the study of<br>firmly nonexpansive, and the more general pseudocontractive mappings, an<br>important tool for monotone operators and the theory of optimization.<br>The metric projection operator has become a veritable tool in dealing with variational<br>inequalities problem by iterative-projection method in Hilbert spaces.<br>Variational inequality problem V IP(A;C) involving an accretive operator A<br>and a convex set C can be proved to be equivalent to the fixed point problem<br>involving the nonexpansive mapping<br>T = PC(I ô€€€ A)<br>for arbitrary positive number . Conversely, given a differentiable functional<br>f, the V IP(rf;C) is simply the optimality condition for the minimization<br>problem<br>min<br>x2C<br>f(x):<br>Metric projection operators in Hilbert spaces are accretive and nonexpansive<br>and gives absolutely best approximations of any element of the closed convex<br>set. However, in the Banach space setting, this operator no longer possess<br>most of those properties that made them so effective in Hilbert spaces.<br>To study monotone-type mappings and the related pseudocontractive mappings<br>in Banach spaces, some analogues of the Hilbert space type projection<br>operators were introduced. These mappings are natural extentions of the classical<br>projection operators to Banach spaces. They have also helped in the<br>approximation of monotone operator in Banach spaces.<br>In the last five years or so, intensive effort are invested in developing feasible<br>iterative algorithm for approximating fixed points of multivalued pseudocontractive<br>type mappings and/or, correspondingly, zeros of monotone mappings<br>in Hilbert spaces and in the general Banach spaces. In each case, attempts<br>are made to recover Hilbert space type identities for these mappings. Most<br>of the study aim to derive a generalization of the multi-valued nonexpansive<br>mapping introduced in the classical work of Nadler [80]. Such method depends<br>heavily on the characterisation of the Hausdorf distance defined on closed and<br>bounded sets. The generalizations of existing ideas, on the other hand, should<br>5<br>be due to the generalization of some properties of the Hausdorff distance.<br>In this thesis, we first establish some new characterizations of the Hausdorf<br>metric and use the ideas thereby to define some more general class of multivalued<br>pseudocontractive mappings and prove convergent theorems for the<br>class of mappings defined. Attempts would be made to apply some of the<br>ideas obtained to real problems of interest. An example in this regard include<br>applications to split equality fixed point problems, introduced by Moudafi and<br>Al-Shemas[79] in (2013), which is formulated as finding a point x in a convex<br>set C and y in a convex set Q such that their images Ax and By under some<br>linear transformations A and B satisfy Ax = By. It serves as an inverse problem<br>model in which constraints are imposed on the solutions in the domain of<br>a linear mapping as well as in its range.<br>This thesis gives new insight and direction in the study of a general class of<br>multivalued pseudocontractive mappings. It also studies a new method for<br>finding a common solution of a monotone operator and family of a general<br>class of nonexpansive mappings in some classical Banach spaces using the idea<br>of generalized projections.<br>The rest of the thesis is organized as follows. Chapter 2 introduces some notions<br>and recalls some basic definitions and ideas which are the bedrocks for<br>the formulation of our theorems and for effective reading of the subsequent<br>chapters.Detailed literature review involving multi-valued nonexpansive and<br>pseudocontractive-type mappings are presented. In Chapter 3, convergence<br>of a coupled iterative algorithm to a solution of some split equality problem<br>is presented. Chapter 4, deals with some contributions to convergence theorems<br>for a general class of multivalued striclty pseudocontractive mappings<br>and Chapter 5 deals with the extension to finite and countable family. Chapter<br>6 is devoted to convergence theorems for a class of multivalued Lipschitz<br>pseudocontractive mappings. We finally present in Chapter 7, an iterative algorithm<br>for common element of zeros of a monotone mapping and fixed points<br>of a general class of nonexpansive mappings in real Banach spaces.<br>6 <br></p>