Contributions to operator theory and applications

Thesis Overview

<p> GENERAL INTRODUCTION AND<br>PRELIMINARIES<br>This thesis is divided into two parts. The contributions of the first part fall within the general<br>area of operator theory while the second part is concerned with applications to mathematical<br>modelling of communicable diseases, in particular, malaria models. For the first part, we shall,<br>‘ in particular, devote attention to existence and approximation methods for finding solutions of<br>operator equations or fixed points of certain nonlinear mappings defined in a subset of a Banach<br>space. The classes of mappings st~di&amp;l’k’cl;ide: the K-positive definite operators, the suppressive<br>operators and certain accretive-type operators.<br>It is well known that many physically significant problems can be modelled in terms of an<br>I I<br>initial value problem of the form<br>where A is an accetive-type operator defined in an appropriate Banach space. It is clear that if<br>the solution of equation (1.0) is independent oft, then Au = 0 and the solutions of this equation<br>correspond to the equilibrium of the system (1.0). Consequently, considerable research<br>efforts have been devoted within the past half a century to finding techniques for the determination<br>of zeros of accretive-type operators (see, for e.g., [17, 18, 19, 22, 52, 54, 581). The study of<br>operator equat.ions is partly linked with fixed point theory for; u is a fixed point of the operator<br>A if and only if u is the solution of the operator equation ( I – A)u = 0, where I is the identity<br>operator. The classical importance and application of fixed point theory can be seen largely in<br>the theory of ordinary differential equations. The existence or construction of a solution to a<br>differential equation is often reduced to the existence or location of a fixed point for an operator<br>defined on a subset of a space of functions. In this thesis we shall.employ fixed point techniques<br>where appropriate.<br>Direct and iterative methods for finding solutions of operator equations or fixed points of an<br>operator defined in an appropriate Banach space have been studied by many authors. These studies<br>have given rise to the development of results and techniques which are now widely available<br>in the literature.<br>Petryshyn [54] considered the operator equation .4u = f , in a Hilbert space, when A is Kpositive<br>definite. Let HI be a dense subspace of a Hilbert space, H. An operator T with domain<br>‘ D(T) 2 HI is called continuously HI invertzble if the range of T, R(T), with T considered as<br>an operator restricted to HI is dense in H and T has a bounded inverse on R(T). Let H be a<br>complex and separable HilberC space%ndaA ‘be a linear unbounded operator defined on a dense<br>domain D(A) in H with the property that there exist a continuously D(A)-invertible closed<br>linear operator K with D(A) c D(K), and a constant c &gt; 0 such that<br>(1.0) (Au, Ku) 12 c(. ~.. Ku~(~u ,E D(A),<br>then A is called I&lt;-positive definite (Kpd) (see e.g., Petryshyn [54]). If K = I (the identity operator)<br>inequality (1.0) reduces to (Au, u) 2 cJu(a~n~d ,in this case, A is called positive definite.<br>If in addition c = 0, A is a positive operator (or accretive operator). Positive definite operators<br>have been studied by several authors (see e.g.[15, 17, 18, 19, 28, 38, 581). It is clear that the<br>class of K-pd operators contains among others, the class of positive definite operators, and also<br>contains the class of invertible operators (when K = A) as its subclasses. Furthermore, Petryshyn<br>[54] remarked that for a proper choice of K, the ordinary differential operators of odd order, the<br>weakly elliptic partial differential operators of odd order, are members of the class of K-pd operators.<br>Moreover, if the operators are bounded, the class of Kpd operators forms a subclass of<br>symmetrizable operators studied by Reid [58].<br>In [54], Petryshyri proved the following theorem.<br>Theorem P If A is a Kpd operator and D(A) = D(K), then there exists a constant a! &gt; 0 such<br>that for all u E D(K),<br>ll.4ull &lt; allKuII.<br>Furthermore, the operator A is closed, R(A) = H and the equation Au = f , f E H, has a<br>unique solution.<br>In the case that K is bounded and A is closed, F. E. Browder[l9] otained a result similar to<br>the second part of Theorem P.<br>In chapter two of this thesis, we extend the definition of a Kpd operator to real Banach spaces,<br>X . In particular, if X is a real – separable Ban,ach space with a strictly convex dual, we prove that ,, . a*,. +. ‘,&gt;<br>the equation Au = f , f E X, where A is a Kpd operator with the same domain as K has a unique<br>solution. Furthermore, if X = Lp (or lp), p &gt; 2, and is separable, we construct an iteration<br>process which converges strongly to this solution.<br>(1<br>In chapter three, we contiriue with the study of Kpd operators. We prove convergence of a<br>simplified iteration sequence to the solution of a Kpd operator in real uniformly smooth Banach<br>spaces. We extend the notion of a Kpd operator to F’rCchet differentiable operators, and in this<br>case, prove convergence of an iteration scheme to the unique solution of a Kpd operator equation<br>in arbitrary Banach space. This result is also derived directly from the inverse function theorem.<br>Let E be a real normed linear space with dual E*. We denote by J the normalized duality mapping<br>from E to 2E* defined by<br>Jx = {f E E* : (x, f) = = /l f 112),<br>where (., .) denotes the generalized duality pairing. It is well known that if E* is strictly convex<br>then J is single-valued and if E* is uniformly convex thcn J is unifornlly continuous on bounded<br>subsets of E (see e.g., [6]). We shall denote the single-valued duality mapping by j.<br>Let K be a subset of a real Banach space E. A map T : K -+ K is called a strict contraction<br>if t,here exists k E [0, 1) such that llTx – Tyll 5 kllx – yll, and it is called nonexpansive if, for<br>arbitrary x, y E K, /ITx – Tyl/ &lt; 1/17: – yl 1. The map T is called pseudocontractive if, for each<br>x, y E K, there exists j(x – y) E J(x – y) such that<br>Let I&lt; be a nonempty convex subset of a real normed linear space E. For strict contraction<br>mappings, nonexpansive and pseudocontractive self-mappings T of K with a fixed point in K,<br>three well known iterative methods,+thewlebmted Picard method, the Mann iteration method (see,<br>for example [48]) and the Ishikawa iteration method (see, for example [41]) have been successfully<br>employed to approximate such fixed points. If, however the domain of T, D(T), is a proper subset<br>of E (and this is the case in several applications), and T maps K into El these iteration methods<br>may not be well defined. Under this situation, for Hilbert spaces, this problem has been overcome<br>by the introduction of the metric projection (see, for e.g. [20, 261). The advantage of this is<br>that if K is a nonempty closed convex subset of a Hilbert space H and PK : H -+ K is the<br>metric projection of H onto K, then PK is nonexpansive. This fact which is central in most of the<br>proofs actually characterizes Hilbert spaces and consequently, it is not available in general Banach<br>spaces. In this connection, Alber [I] recently introduced a generalized projection operator IIK in<br>an arbitrary Banach space E which is an analogue of the metric projection in Hilbert spaces.<br>We recall the definition of the metric projection operator PK : E -+ K, K C E. The operator<br>PK : E -+ K is called a metric projection operator if it assigns to each x E E its nearest point<br>2 E K i.e, fi is the solution for the minimization problem<br>Now let E be a real normed linear space and K C E. Consider the operator IIK : E -+ K defined<br>by<br>IIlcz = 2; fi : V(x, 2) = inf V(x, 0,<br>CEK<br>where<br>Observe that in a Hilbert space H, (1.1)r educes to V(x,() = llx – (I[$,x E H, &lt; E K .<br>Existence and uniqueness of’ the operator IIK follow from the properties of the functional<br>V(x,() and the strict monotonicity of the mapping J (see for e.g. [4]).I n a Hilbert space, ITK<br>becomes PK .<br>Using the Lyapunov functional defined in (1.1), Alber and Guerre-Delabriere [2] introduced<br>the following classes of non-self mappings.<br>., ,? . 4 “1, 0. ‘&gt;a ‘<br>Let K be a nonempty subset of a Banach space E. A map T : K -+ E is called strongly<br>suppressive on I&lt; if there exists 0 &lt; q &lt; 1 such that for all x, y E K,<br>T is called weakly suppressive of class Ca(t) if there exists a continuous and nondecreasing function<br>Q,(t)d efined on !J?+ such that Q, is positive on !J?+{o), Q,(O) = 0, lim Q,(t)= +m and Vx,y E K ,<br>t++m<br>The map T is called n,onextensive if<br>It is easy to see that each weakly suppressive mapping is nonextensive, and each strongly suppressive<br>opcrator is weakly suppresive with @it) = (1 – q)t. In Hilbert spaces, nonextensive<br>I operators are nonexpansive and vice versa and strongly suppressive operators coincide with strict 1<br>contractions.<br>Assuming the existence of fixed point’s for the above classes of operators, Alber and Guerre-<br>I Delabriere [4] proved convergence theorems with the help of generalized projection maps. They<br>1 also introduced the following classes of operators. 1<br>A mapping T with domain D(T) and range R(T) in E is called weakly contractive (see e.g.,<br>I [2, 3, 41 ) if there exists a contiiiuous and nondecreasing function : [0, m) := %+ + Xf such<br>I that @ is positive on W+ {0), Q(0) = 0, lirn,, @it) = co and for x, y E D(T) there exists 1<br>j(x – y) E J(x – y) such that<br>(1.5) IITx -Tyll I llx – yll – @(I12 – 911).<br>It is called d-weakly contractive if for all x, y E D(T) there exist j(x – y) E J(x – y ) and @(t) as<br>I above such that<br>‘i’<br>The d-weakly contractive operators include several important classes of nonlinear operators. 1<br>In particular, they include the weakly contractive operators.<br>11 . .<br>In [4], Alber and Guerre-Delabriere proved, interalia, the following theorem.<br>Theorem AG Let T : G + H be a d-weakly contractive map, G a closed convex bounded subset<br>of a Hilbert space H and suppose that a fixed point x* E int(G) of T exists. Then the sequence<br>{x,) defined by<br>XI E G; zn+l := PG(xIL- ailI(xn- Tx,)), R = 1,2,. ..,<br>6<br>where PG is the metric projection onto the set G, {a,) is a sequence of positive numbers such<br>that Cy a, = co and lim,,+a a, = 0 converges strongly to s*. Moreover, there exist a constant<br>C &gt; 0 and a bounded sequence {x,,) c {x,), 1=1,2, … such that<br>Furthermore,<br>For a Banach space El the modulus of smoothness of E is the function p~ : [0, co) -+ [0, co)<br>defined by<br>E is said to be uniformly smooth if lim = 0. Typical examples of such spaces are the<br>7+0+<br>Lebesgue Lp, the sequence lpl and the Sobolev Wpm spaces, 1 &lt; p &lt; oo.<br>It is known (see, e.g. [38, 39, 631) that pE(0) = 0. Moreover, hE(r) = r-lpE(r) is continuous,<br>nondecreasing and hE (0) = 0. IN<br>In chapter four of this thesis, we extend Theorem AG in various directions. In particular,<br>we prove that Theorem AG remains true in real uniformly smooth Banach spaces and without<br>the boundedness condition imposed on the domain. Furthermore, we prove a related convergent<br>theorem in our general setting when the fixed point x* of T exists but is not necessarily in the interior<br>of G. Finally we prove a convergent theorem for approximating a fixed point of a uniformly<br>continuous d-weakly contractive and bounded self-map T of G with F(T) # 8, in arbitrary real<br>Banach spaces.<br>In 1972, Goebel and Kirk [39] introduced a class of mappings generalizing the class of nonexpansive<br>operators. Let K be a nonempty subset of a normed space E. A mapping T : K + K is<br>called asymptotically nonexpansive if there exists a sequence {Ic,,), Ic, 2 1, such that limn,, Ic, =<br>1, and I ITnx – TnyJI 5 Icnl lx – yll for each x, y in K and for each integer n 2 1.<br>Later in 1993, Bruck et a1 introduced and studied another class of asymptotic nonexpansive maps.<br>A mapping T : K + K is called asympiotically nonexpansive i n the intermediate sense (see e.g.,<br>Bruck et a1 [22]) provided T is uniformly continuous and<br>lim sup sup (IIT7’x – Tnyll – IIx – yII) 5 0.<br>n,tm i :,,ycK 1<br>Asymptotic pseudocontractive operators have also been introduced and studied, first by Schu (see<br>e.g., [59]) and then by a host of other authors, as a generalization of asymptotic nonexpansive<br>maps. T : K + K is called asymptotic all?^ pseudocontractive if there exists a sequence {Ic,),<br>Ic, 2 1, lim Ic, = 1 such that<br>for each x, y E K.<br>. ,?. 4 “7. .*. , ,&gt; ‘<br>It is easy to see that asymptotically pseudocontractive maps include the asymptotic nonexpansive<br>ones. These classes of maps have been studied by various authors.<br>Motivated by .Goebel and Kirk [39], Bruck eta1 [22] and Schu [59] , we introduce and study,<br>I1<br>in chapter five, the class of asym,ptotically d-weakly contractive maps.<br>A map T with domain D(T) and range R(T) in E will be called asymptotically d-weakly contractive<br>if there exists a continuous and nondecreasing function @ : [0, m) := R+ + R+ such<br>that @ is positive on R+{O), g(0) = 0, limt,,@(t) = m and for x, y E D(T), there exists<br>j(x – y) E J(x – y) such that<br>where {k,) is a real sequence such that k, 2 1, lim,,,k, = 1 and P is a retraction from R(T)<br>to D(T).<br>Clearly, if T is a self-map, then condition (*) reduces to the following one:<br>With the help of the d-weakly contractive maps, we prove a version of Theorem AG in real<br>uniformly smooth Banach spaces. A corollary of our theorem extends Theorem AG from Hilbert<br>spaces to real uniformly smooth Banach spaces. Furthermore, the boundednes requirement on the<br>domain in Theorem AG is dispensed with in our more general setting. A related result deals with<br>the approximation of fixed points of uniformly continuous asymptotically d-weakly contractive<br>self-maps with nonempty fixed point sets, in arbitrary real Banach spaces.<br>Part two<br>Malaria is a mosquito-borne infection caused by protozoa of the genus plasmodium. Four<br>species of the parasite, namely: P. falciparuin, P. vivax, P. ovale, and P. malariae infect humans.<br>Malaria remains the most important of the tropical diseases, being widespread throughout the<br>tropics, but also occurring in many temperate regions.<br>The parasites are transmitted by the bite of infected female mosquitoes of the genus Anopheles.<br>Mosquitoes become infected by feeding on the blood of infected people, and the parasites<br>then undergo another phase of reproduction in the infected mosquito. Clinical symptoms such as<br>fever, pains, and sweats may develop a few days after an infected mosquito bite.<br>In many parts of Africa, where malaria has long been highly endemic, people are infected so<br>frequently that they develop a degree of acquired immunity, and may become asymptomatic carriers<br>of the infection [36]. Treatment and control have become more difficult in recent years with<br>the spread of drug resistant strains of malaria parasites [14, 36, 621. Drugs such as chloroquine,<br>nivaquine, quinine, and fansidar are used for treatment. More recent and more powerful drugs<br>‘include mefloquine, and halofant,rine.<br>It is estimated that 267 milli6~’@%pk $re presently infected, with 107 million clinical cases<br>annually; the number of countries affected is put at 103 [62].<br>The biology of the four species of plasmodium,is generally similar and consists of two discrete<br>phases-sexual and asexual . The parasite &amp;rates to the liver where it remains latent for several<br>days while replicating. The latent period is followed by penetration of red blood cells and asexual<br>replication within them. Asexual parasites in the blood, after surviving some developmental<br>period, give rise to sexual stages called gametocytes. Gametocytes can remain in the blood for<br>more than two years [36].<br>The emergence of drug-resistant strains of malaria parasite has become a significant health<br>problem. Recent pronouncernents by the World Health Organisation indicate the availability of<br>strains that are resistant to virtually all known drugs.<br>Among the four species of plasmodium, P. falciparum causes the most serious illness and it is<br>the most widespread in the tropics. This paper therefore focuses on the dynamics of P. falciparum<br>malaria, although the analysis is similar for all forms of malaria infections.<br>An early fundamental model in the art of mathematical modelling of malaria, due to Ross-<br>Macdonald describes the basic features of the interaction between infected humans (y) and infected<br>mosquito vectors (9). The model is defined as follows:<br>where a = bpg, p, r., p are some constants defined as follows (see for e.g. [12]):<br>N is the size of the human population;<br>M is the size of the femtle mosquito population;<br>. ,. , . *1. ,t’ ,<br>$$ is the number of female mosquitoes per human host;<br>/3 is the rate of biting on man by a single mosquito (number of bites per unit time);<br>b is the proportion of infected bites on ,man that produce an infection;<br>r is the per capita rate of recovery for humans (: is the average duration of infection in the<br>human host) ;<br>p is the per capita mortality rate for mosquitoes (iis t he average lifetime of a mosquito).<br>In this simple model, the total population of both humans and vectors is assumed fixed, so that the<br>dynamical variables (y and q) are the proportion infected in each population. The first equation<br>describes changes in the proportion of humans infected. New infections are acquired at a rate<br>that depends on the following factors:<br>(i) the number of mosquito bites per person per unit time (pg)<br>(ii) the probability that the biting mosquito is infected (q)<br>(iii) the probability that a bitten human is uninfected (1 – y)<br>(iv) the probability that an uninfected person thus bitten will actually become infected (b).<br>Infections are lost by infected people returning to the uninfected class, at a characteristic recovery<br>rate r. Similarly, the second equation describes changes in the proportion of mosquitoes infected.<br>Population changes are determined by the following factors:<br>(i) the number of bites per mosquito per unit time (p)<br>(ii) the probability that the biting mosquito is uninfected (1 – q)<br>(iii) the probability that the bitten human is infected (y).<br>The loss term (pq) arises from the death of infected mosquitoes. The loss terms for the infected<br>humans and infected vectors both involve death and recovery. For human hosts the recovery rate<br>is typically faster than the death*r.~t&amp;,”whgrefaosr vectors the opposite is the case. The origin is<br>a local asymptotic equilibrium for this model if pr &gt; [email protected] Thus infection dies out if the product<br>of the death rates (p and r) for the two populations is large in the sense that pr &gt; [email protected] Thus the<br>above formulation is a sensible approximation.<br>I<br>However, this model is highly simplified. The model simply assumes that an infected individual<br>either recovers to join the susceptible group N(l – y) or dies. It fails to distinguish between<br>the various infected categories of human and vector hosts. Thus it cannot describe accurately the<br>recent trend in malaria infection.<br>This basic model has been studied, modified and generalized in different directions by various<br>authors (see for e.g., [12, 13, 14, 471). Aron and May [12] extended this model by introducing<br>another population group x- the latent infected humans (infected but not yet infectious). They<br>conjectured that if the incubation interval in the mosquito has duration T the second equation in<br>the basic model could be replaced by the two dynamical systems:<br>where the circumflex denotes evaluation at time T in the past: % = y(t – T); etc. Here, the two<br>categories of mosquito (uninfected and infected-and-infectious) are now replaced by three categories:<br>a proportion, 1 – q – z, that are uninfected; a proportion q that are infected and infectious;<br>and a third, new proportion z that are latent (infected but not yet infectious).<br>Bailey [13] considered two interacting populations-human hosts and mosquito vectors with<br>each group consisting of three subgroups, viz susceptibles, infectives, and isolated (recovered and<br>immune). These are designated by x, y, and z respectively for the human populations and x’,<br>y’, and z’ respectively for the vector populations. It follows that the number of new infections<br>occurring in the human population in time interval 6t is /3xy16t, where /3 is the infection rate.<br>Since the converse arrangement -is”YeQdired’to hold for vectors, namely that susceptible vectors<br>are infected by human infectives, the number of new infections occurring in susceptible vectors<br>in time interval 6t is therefore given by /3’x1y6t. If in addition the overall removal rates for the<br>two populations are assumed to 11e y and y’, respectively, then the numbers of removals occurring<br>in time 6t are yybt and y’y’6t for humans and .vectors. respectively. The system of differential<br>equations for the dynamic process involved is given as<br>X = -PxY’, X’ = -plxly + y’y’,<br>Y = Pxy1-YY7 Ij’ = /3’x’y – y’y1,<br>2 = YY 7<br>where p, p’, y, y’ are some constants. Due to its relatively short life-cycle, isolation by immunity<br>is negligible in the mosquito vectors, and hence the only isolation process is by death which is<br>assumed to occur equally in all groups. Hence i’ r 0 (see [14], pp. 61-68).<br>Let the initial states at t = 0 be (x,, yo, 0) and (xi, yb, 0). It is clear from the above equations<br>that for a proper epidemic outbreak to occur, y, &gt; 0 and yo’ &gt; 0. That is<br>Let N and N’ be the populations of humans and vectors respectively, so that N = x + y and<br>N’ = x’ + y’. It follows that if both y, and yL are small, then x, z N and xb z N’. Thus<br>Eliminating y, we have<br>Thus for an epidemic build-up to occur we must have the condition<br>NN’ &gt; -YY’ PP’<br>This model, like the first, also describes the basic interaction between the infected human host<br>population and the mosquito vec.t,o;,~,~pul,ationb ut with an additional group in each population.<br>,?<br>Proposition PS [55] Consider the system<br>with f (., .) and g(.) continuous throughout a compact subset E of R~D.e fine P(E) as the projection<br>of E onto the x2 axis. Assume<br>I. E is positively invariant for the system,<br>and<br>2. x2 = i2 is (m equilibrium point that is globally asymptotically stable on a subset A of P(E) .<br>Then every trajectory startzng in -4′ = {(TIx, 2) : xz E A) tends asymptotically to a point of the<br>form (21, :t2), where li-l is an, eqzdibrium of XI = f (zl, &amp;).<br>Consider the system<br>where wij, N;, gi are constants and wij &gt; 0, Ni &gt; 0: gi &gt; 0, yi 2 0. Let y be the vector whose<br>components are yi, i=l, … ,n; A the matrix of linear terms, and Q(y) the vector of quadratic<br>terms in (1.7). Then the vector form of the system is<br>Lajmanovich and Yorke (see [44]) proved that solutions to system (1.8) are globally asymptotically<br>stable with limiting value dktermined by the stability modulus s(A) of the matrix A. This is<br>defined as the maximum real part of the eigenvalues of A, i.e.,<br>s(A) = max{Re X : X an, eigenvalue of A).<br>Precisely, the following theorem w.,a s I? g~ped,,byL.a jmanovich and Yorke.<br>THEOREM LY The solutions to system (1.8) approach the origin if s(A) 5 0 and approach a<br>unique positive equilibrium. 6 if s(A) &gt; 0, provided yi(0) &gt; 0 for some i. Furthermore in this case<br>0 &lt; &amp;(0) &lt; IVi for’ each i=l, 2, … ,n. ,, . ..<br>Throughout this part, we shall adopt the following notations where applicable.<br>N denotes the total human population;<br>x denotes susceptibles (the number of people that are uninfected);<br>y denotes infectives (the number of people that are severely infected);<br>15<br>v denotes infected vectors (the population of vectors that can transmit the disease).<br>Where y and E’ occur simultaneously, we designate y as the number of individuals infected<br>with parasites that are sensitive to drugs and Y as the number of individuals that are infected<br>with the resistant parasites with or without the sensitive parasites. Also where applicable, v and<br>V denote respectively the population of vectors carrying sensitive strains and resistant strains,<br>with or without the sensitive strains of the parasite.<br>In chapter six, models describing recent trend in malaria infection are formulated. The first<br>model deals with single strain infection while the second incorporates resistant strains. Each of<br>the models results in a system of nonlinear ordinary dzfferential equations. A version of Proposition<br>PS is proved in R4. Our major theorem, Theorem 3 gives conditions for the existence of<br>different endemic states. Since major health efforts should be geared towards the elimination<br>of resistant parasites (and hence resistant infection) our findings and conclusions enable us to<br>forecast the trend in resistant infection.<br>In chapter seven, we formulate models treating both sensitive and resistant infections in single<br>and multigroup populations. We consider a case of a homogenous interacting population groups.<br>Theorem LY plays a major role i.n. , .t .h .e analvsis of our models. Results which forecast the trend .L .V ,<br>of resistant infection are established.<br>Finally in chapter eight we consider the more general multigroup population in Rn. Local<br>asymptotic equilibrium are obtained in R?,. A. general extension theorem is proved. The main<br>tool in this direction is the Poincar4-Bendisson theorem <br></p>