On j-fixed points of j-pseudocontractions with applications

Project Abstract

<p> Let E be a real normed space with dual space E and let A E ! 2E be any map. Let J E ! 2E be<br>the normalized duality map on E. A new class of mappings, J-pseudocontractive maps, is introduced<br>and the notion of J-fixed points is used to prove that T = (J ô€€€ A) is J-pseudocontractive if and<br>only if A is monotone. In the case that E is a uniformly convex and uniformly smooth real Banach<br>space with dual E, T E ! 2E is a bounded J-pseudocontractive map with a nonempty J-fixed<br>point set, and J ô€€€ T E ! 2E is maximal monotone, a sequence is constructed which converges<br>strongly to a J-fixed point of T. As an immediate consequence of this result, an analogue of a recent<br>important result of Chidume for bounded m-accretive maps is obtained in the case that A E ! 2E is<br>bounded maximal monotone, a result which complements the proximal point algorithm of Martinet and<br>Rockafellar. Furthermore, this analogue is applied to approximate solutions of Hammerstein integral<br>equations and is also applied to convex optimization problems.<br>viii <br></p>

Project Overview

<p> </p><p>INTRODUCTION<br>1.1 Background of study<br>The contributions of this thesis work fall within the general area of nonlinear functional analysis and<br>applications, in particular, nonlinear operator theory. We are interested in the solution or approximation<br>of solutions of nonlinear equations or inclusions (i.e., equations or inclusions defined by nonlinear operators)<br>in Banach spaces.<br>Problems in the area involve methods of fixed point theory and application of iterative algorithms to<br>approximate zeros or fixed points of nonlinear mappings. Research in the area is enormous due to varied<br>classification of Banach spaces, operators and topological assumptions on them (e.g., continuity, boundedness,<br>compactness, closedness e.t.c). The literature of the last four decades abounds with papers which<br>establish fixed point theorems for selfmaps or nonselfmaps satisfying a variety of contractive type conditions<br>on several ambient spaces. See figures 1.1, 1.2 and 3.1.<br>(1 &lt; p 2) Lp (2 p &lt; 1)<br>[<br>H<br>[<br>Rn<br>p-Uniformly convex q-Uniformly smooth<br>Uniformly convex Uniformly smooth<br>Reflexive Smooth<br>Unif. Gat. Diff. norm<br>Strictly convex<br>(a) Lattice of Banach spaces<br>k-Contractive maps<br>Nonexpansive maps<br>Strictly<br>pseudocontractive<br>Lipschitz<br>pseudocontractive<br>(b) Metric Fixed-point Operator lattice<br>Figure 1.1: Lattice of Spaces and Metric fixed-point Operator<br>1<br>k-Contractive maps<br>Nonexpansive maps<br>Strictly<br>pseudocontractive<br>Asymptotically<br>nonexpansve<br>Asymp. Nonexp. in the<br>Intermediate sense Pseudocontractive<br>Asymp. Strictly<br>pseudocontractive<br>Asymp. Strictly<br>pseudocontr. in the<br>Interm. Sense<br>Asymp.<br>pseudocontractive<br>Asymp. pseudocontr. in<br>the Interm. sense<br>Firmly<br>Quasi-nonexpansive<br>Quasi-nonexpansive<br>Demi-contractive<br>(a) Contractive-type self map Operator lattice<br>Figure 1.2: Lattice of Operators<br>Let H be a real inner product space. A map A : H ! 2H is called monotone if for each x; y 2 H,</p><p>ô€€€ ; x ô€€€ y</p><p>0 8 2 Ax; 2 Ay: (1.1.1)<br>Monotone mappings were first studied in Hilbert spaces by Zarantonello [120], Minty [84], Kaˇcurovskii<br>[64] and a host of other authors. Interest in such mappings stems mainly from their usefulness in applications.<br>1.1.1 Zeros of Monotone operators on Hilbert spaces<br>We consider the problem given by<br>Au 3 0 (1.1.2)<br>where A : H ! 2H is a monotone map on a Hilbert space. Problems of this kind find relevance in<br>several areas of applications. In particular, we have the following examples:<br>Convex optimisation problems<br>Let g : H ! R [ f1g be a proper convex function. The subdifferential of g, @g : H ! 2H, is defined<br>for each x 2 H by<br>@g(x) =</p><p>x 2 H : g(y) ô€€€ g(x)</p><p>y ô€€€ x; x<br>8 y 2 H</p><p>:<br>It is easy to check that @g is a monotone operator on H, and that 0 2 @g(u) if and only if u is a minimizer<br>of g. Setting @g A, it follows that solving the inclusion 0 2 Au, in this case, is solving for a minimizer<br>of g.<br>2<br>In particular, as an example of the above, where g(x) = jxj, the subdifferential of g at zero, @g(0) =<br>[ô€€€1; 1], which trivially contains zero. Hence, zero is the minimizer of g.<br>Equilibrium problem of dynamical systems: Evolution equation<br>The equation 0 2 Au when A is a monotone map from a real Hilbert space to itself also appears in<br>evolution systems. Consider the evolution equation for a single-valued operator,<br>du<br>dt<br>+ Au = 0<br>where A is a monotone map from a real Hilbert space to itself. At equilibrium state, du<br>dt = 0 so that<br>Au = 0, whose solutions correspond to the equilibrium state of the dynamical system.<br>In particular, consider the following diffusion equation<br>8&lt;<br>:<br>@u<br>@t (t; x) = 4u(t; x) + g(u(t; x)); t 0; x 2<br>;<br>u(t; x) = 0; t 0; x 2 @<br>;<br>u(0; x) = u0(x); u0 2 L2(<br>);<br>(1.1.3)<br>where<br>is an open subset of Rn.<br>By simple transformation i.e., by setting v(t) = u(t; :); where v : [0;1) ô€€€! L2(<br>) is defined by<br>v(t)(x) = u(t; x) and f(‘)(x) = g(‘(x)); such that f : L2(<br>) ô€€€! L2(<br>); we see that equation (1.1.3)<br>is equivalent to</p><p>v0(t) = Av(t) + f(v(t)); t 0;<br>v(0) = u0;<br>(1.1.4)<br>where A is a nonlinear monotone-type mapping defined on L2(<br>).<br>Setting f to be identically zero, at an equilibrium state (i.e., when the system becomes independent of<br>time) we see that equation (1.1.4) reduces to<br>Au = 0: (1.1.5)<br>Thus, approximating zeros of equation (1.1.5) is equivalent to the approximation of solutions of the<br>diffusion equation (1.1.3) at equilibrium state.<br>Hammerstein integral equations<br>Definition 1.1.1. Let<br>Rn be bounded. Let k :</p><p>! R and f :<br>R ! R be measurable<br>real-valued functions. An integral equation (generally nonlinear) of Hammerstein-type has the form<br>u(x) +<br>Z</p><p>k(x; y)f(y; u(y))dy = w(x); (1.1.6)<br>where the unknown function u and inhomogeneous function w lie in a Banach space E of measurable<br>real-valued functions.<br>By simple transformation (1.1.6) can put in the abstract form<br>u + KFu = 0; (1.1.7)<br>Interest in Hammerstein integral equations stems mainly from the fact that several problems that arise in<br>differential equations, for instance, elliptic boundary value problems whose linear part possesses Green’s<br>function can, as a rule, be transformed into the form (1.1.6) (see e.g., Pascali and Sburian [88], p. 164).<br>3<br>1.1.2 Extension of Hilbert space Monotonicity to arbitrary normed spaces<br>We recall that for a Hilbert space H, H = H. So, in the definition of a monotone operator in a Hilbert<br>space, the map A : H ! H could have been A : H ! H. Thus, the notion of monotone mappings has<br>been extended to real normed spaces. We now briefly examine two well-studied extensions of Hilbert<br>space monotonicity to arbitrary normed spaces, say , E.<br>A<br>A : E ! E A : E ! E<br>Accretive Monotone<br>Figure 1.3: Extension of Hilbert space monotonicity<br>Accretive-type mappings<br>Let E be a real normed space with dual space E. A map<br>J : E ! 2E defined by<br>Jx :=</p><p>x 2 E :</p><p>x; x<br>= kxk:kxk; kxk = kxk</p><p>is called the normalized duality map on E. We denote Jô€€€1 by J<br>A map A : D(A) ! 2E is called accretive if for each x; y 2 D(A), there exists j(x ô€€€ y) 2 J(x ô€€€ y)<br>such that</p><p>ô€€€ ; j(x ô€€€ y)</p><p>0 8 2 Ax; 2 Ay: (1.1.8)<br>Roughly speaking, accretive mappings acting in a space E are generalizations of non-decreasing realvalued<br>functions. More precisely, A is said to be accretive if for all x1; x2 2 D(A), y1 2 Ax1, y2 2 Ax2<br>and 0,<br>kx1 ô€€€ x2k kx1 ô€€€ x2 + (y1 ô€€€ y2)k:<br>A is called maximal accretive if, in addition, the graph of A is not properly contained in the graph of any<br>other accretive operator. It is m-accretive if and only if A is accretive and R(I + tA) = E for all t &gt; 0.<br>In a normed space, “m-accretive” implies “maximal accretive” . The converse assertion need not be<br>true. The first counterexample was constructed in lp by B.D. Calvert (1970). Moreover, A. Cernes<br>(1974) showed that even if both E and E are uniformly convex, but E is not a Hilbert space, then there<br>are maximal accretive mappings which are not m-accretive. However, it was proved by G. Minty (1962)<br>that in Hilbert spaces, the notions of ”m-accretive” and ”maximal accretive” are equivalent (see e.g.,<br>[80]) In a Hilbert space, the normalized duality map is the identity map, and so, in this case, inequality<br>(1.1.8) and inequality (1.1.1) coincide. Hence, accretivity is one extension of Hilbert space monotonicity<br>to general normed spaces.<br>Monotone-type mappings in arbitrary normed spaces<br>Let E be a real normed space with dual E. A map A : E ! 2E is called monotone if for each x; y 2 E,<br>the following inequality holds:</p><p>ô€€€ ; x ô€€€ y</p><p>0 8 2 Ax; 2 Ay: (1.1.9)<br>4<br>It is called maximal monotone if, in addition, the graph of A is not properly contained in the graph of<br>any other monotone operator. Also, A is m-monotone if and only if it is monotone and R(J +tA) = E<br>for all t &gt; 0. When E is a strictly convex Banach space with a Fr´echet differentiable norm, a maximal<br>monotone operator from E into E is m-monotone (see e.g., Kido [71]).<br>It is obvious that monotonicity of a map defined from a normed space to its dual is another extension of<br>Hilbert space monotonicity to general normed spaces.<br>The extension of the monotonicity condition from a Banach space into its dual has been<br>the starting point for the development of nonlinear functional analysis…: The monotone<br>mappings appear in a rather wide variety of contexts, since they can be found in many<br>functional equations. Many of them appear also in calculus of variations, as subdifferential<br>of convex functions (Pascali and Sburian [88], p. 101).<br>1.1.3 Application of Fixed Point Techniques<br>The theory of fixed point proves to be one of the most useful tools of modern mathematics. This comes<br>from earlier development of the theory and the fact that most important nonlinear problems in applications<br>can be transformed to fixed point problems.<br>Definition 1.1.2. Let X be a non-empty set and f be a self-map on X. A fixed point of f is a point<br>x 2 X such that f(x) = x. If f is a multivalued then a point p in X is called a fixed point of f if p 2 fp.<br>Theorems concerning the existence and properties of fixed points are known as fixed point theorems.<br>Several fixed point theorems include the Banach contraction mapping principle, Brouwer fixed point<br>theorem, Schauder fixed point theorem and a host of others (see e.g., Asati et al. [5], Khamsi [67], Smith<br>[107], Lee [74]).<br>Let E be a real normed space and A : E ! E be an accretive operator. Assume that Au = 0 has<br>a solution. Browder [14] introduced an operator T : E ! E by T = I ô€€€ A and called the map T,<br>pseudo-contractive. It is clear that zeros of A correspond to fixed points of T (i.e., Au = 0 if and<br>only if Tu = u). The class of pseudocontractive maps properly contains the class of nonexpansive maps<br>which are a generalisation of contraction maps. A map T : E ! E is called nonexpansive if for each<br>x; y 2 E, the inequality kTx ô€€€ Tyk kx ô€€€ yk is true.<br>Several existence theorems have been proved for the equation Au = 0; where A is of the monotone-type<br>(or accretive-type) (see e.g., Brezis [11], Browder [14], Deimling [50], Pascali and Sburian [88], e.t.c.).<br>Likewise, several results have appeared in the literature for approximating zeros of accretive-type (or<br>fixed points of pseudo-contractive) mappings in certain Banach spaces (see e.g., Chidume et al.[20],<br>Takahashi [113], Bruck [17], and host of other authors).<br>Let E be a real normed space and T := I ô€€€ A : E ! E a pseudocontractive mapping. If K is a<br>nonempty convex subset of E and F(T) := fx 2 K : Tx = xg 6= ;, the following recursion formula<br>has been used to approximate fixed points of T, x0 2 K,<br>xn+1 = (1 ô€€€ n)xn + nTxn; n 0;<br>where fng is a real sequence satisfying appropriate conditions. The most general iterative scheme for<br>bounded pseudocontractive maps seems to be that obtained from the following<br>Theorem 1.1.3 (C. E. Chidume [23]). Let E be a uniformly smooth real Banach space with modulus of<br>smoothness E, and let A : E ! 2E be a multi-valued bounded mô€€€accretive operator with D(A) = E<br>such that the inclusion 0 2 Au has a solution. For arbitrary x1 2 E, define a sequence fxng by,<br>xn+1 = xn ô€€€ nun ô€€€ nn(xn ô€€€ x1); un 2 Axn; n 1;<br>5<br>where fng and fng are sequences in (0; 1) satisfying the following conditions:<br>(i) limn!1 n = 0; fng is decreasing; (ii)<br>P<br>nn = 1;<br>P<br>E(nM1) &lt; 1, for some<br>constantM1 &gt; 0; (iii) limn!1<br>h<br>nô€€€1<br>n<br>ô€€€1<br>i<br>nn<br>= 0. There exists a constant 0 &gt; 0 such that E(n)<br>n<br>0n.<br>Then, the sequence fxng converges strongly to a zero of A.<br>1.2 Statement of Problem<br>In studying the inclusion (1.1.2) on real Banach spaces more general than Hilbert spaces when A is<br>of accretive-type mapping, several iterative algorithms have been constructed and results obtained for<br>approximating solutions of problems of the equation (see e.g., the following monographs: Berinde [9],<br>Browder [14], Chidume [22], Reich [90], and the references contained in them). Consequently, this has<br>generated interests and the question asked if similar results for the case of monotone-type mappings in<br>arbitrary Banach spaces can be obtained, where A maps a space into its dual.<br>Regrettably, the pursuit of analogous results has only been greeted with very little progress and seemingly<br>unpropitious prospects as the success for the accretive-type case doesn’t quite easily carry over to<br>the case of monotone-type mappings. The difficulty, for the most part, seems to be that all efforts made<br>to apply directly known geometric properties of Banach spaces proved abortive; also developing and<br>understanding concepts with applying knowledge of the structure and geometry of the dual space, existence<br>and uniqueness theorems for monotone-type mappings in arbitrary Banach spaces, weak topology<br>and relevant tools of functional analysis, and other notions of operator theory were rather too slow for<br>the ambitious researcher. Also, defining the iterative sequence to make sense posed a challenge.<br>Furthermore, the technique of converting the inclusion (1.1.2) into a fixed point problem of defining the<br>map T := I ô€€€ A is not applicable since, in this case when A is monotone, A maps E into E and such<br>T is never well-defined as the identity map does not make sense.<br>1.3 Aim and Objectives of Study<br>The aim of this work is to contribute to the efforts being made to approximate solutions of inclusion<br>(1.1.2) where A is of monotone-type. We consider the problem of solving zeros of nonlinear equations<br>of maximal monotone-type mappings with no continuity assumption. We proceed thus.<br>1. We introduce, as far as we know, a class of mappings called J-Pseudocontractive mappings and<br>study the concept of J-fixed points. We establish the relationship between monotone mappings<br>and J-pseudocontractive mappings and between J-fixed points and zeros of operators.<br>2. We construct an iterative algorithm which converges to a J-fixed point of a J-Pseudocontractive<br>mappings and hence, by extension, to a zero of a monotone mapping.<br>3. We apply our results to:<br>Zeros of maximal monotone mappings. ( This corresponds, as noted earlier, to the equilibrium<br>state of some dynamical system)<br>Proximal point algorithm<br>Solutions Hammerstein integral equations<br>Convex minimization problems<br>6</p> <br><p></p>