Iterative Approximation of Equilibrium Points of Evolution Equations

Iterative Approximation of Equilibrium Points of Evolution Equations

Abstract

Suppose that E is a real Banach space .which is both uniformly convex and q-uniformly smooth and that T is a Lipschitz pseudocontractive self-mapping of a closed convex and bounded subset K of E. Suppose F(T) denotes the set of fixed points of T and U denotes the sunny nonexpansive retraction of K onto F(T) and w any point of K, it is proved that the sequence {Xn} n=0∞generated from an arbitrary x0 € K by Xn+1 = βnw +(1+βn) 1/n+1 ∑3=0n{(1-a3)1+a3T}Xn, (where I denotes the identity operator on E and {an}∞n=0, and {βn}∞n=0 are real sequences in (0,1], satisfying certain conditions) converges strongly to Uw. This result i s similar to, and in some sence, is an improvement on the theorems of Chidume (Proc. Amer. Math. Soc. 129(8) (2001) .2245-2251) and Ishikawa (Proc. Amer. Math. Soc. 44(l) (l974), 147-150). Furthermore, suppose that E is an arbitrary real normed linear space and A : E + 2E is a uniformly continuous and uniformly quasi-accretive multi-valued map with nonempty closed values such that the range of (I – A) is bounded and the inclusion f € Ax has a solution x* € E for an arbitrary but fixed f € E. Then it is proved that the sequence { xn)∞n=0 generated from an arbitrary xo € E by Xn+1= (1-cn)Xn+cnCn1 Cn € (1-A)xn Ɐ n ≥0 (where {cn)∞n=0 is a, real sequence in [ O , l ) satisfying certain conditions) converges strongly to x*. Moreover, suppose E is an arbitrary real normed linear space and T : D(T) c E → E is locally Lipschitzian and uniformly hemicontractive map with open domain D(T) and a fixed point x* € D(T). Then there exists a neighbourhood B of x* such that the sequence { xn, ) ∞n=0 generated from a3 arbitrary x0 € B c D(T) by xn+1=(1-cn)xn+cnTxn,Ɐ n ≥ 0 (where {cn}∞n=0 is a real sequence in [O,1) satisfying certain conditions) remains in B and converges strongly to x*. These results are improvements on the results of Alber and Delabriere (Operator Theory, Advances and Applications 98 (1997) ,7-22), Bruck (Bull. Amer. Math. Soc. 79(1973),1259-1262), Chidume and Moore (J. Math. Anal. Appl 245(l) (2000) ,142-160) and OsiIike (Nonlinear Analysis 36 (1) (1999) ,I-9). Finally, if E is a real Banach space and T : E → E a map with F(T) := {x € E : Tx = x}≠0 and satisfying the accretive-type condition (x – Tx, j(x – x*)) ≥גּ║x- Tx║2 for all x € E, x* € F(T) and גּ > 0, then a necessary and sufficient condition for the convergence of the sequence {xn} ∞n=0=, generated from an arbitrary x0 € E by xn+l = (1 – cn)xn + c,Txn, Ɐ n ≥ 0 . (where {cn}∞n=0 is a real sequence in [0, I) satisfying certain conditions) to a fixed point of T is established. This result extends the results of Maruster (Proc. Amer. Math. Soc.66 (1977), 69-73) and Chidume (J. Nigerian Math. Soc. 3(1984),57-62) and resolves a question raised by Chidume (J. Nigerian Math. Soc. 3(1984),57-62).

Chapter One

Introduction

The contributions in this thesis fall within the general area of Nonlinear Functional Anal-ysis, an area that has witnessed an increasing amount of research interest in recent years. 1%devote attenticn t o an important topic in this area: iterative approximation of equilib-rium points of evolution equations. Sevcral physically significant problems can be modeled in terms of an initial value problem of the form where A is an operator satisfying certain conditions in appropriate (Banach) spaces, f is an arbitrary’but fixed point in the space and x is any point in the space. When x is independent of time t, (1-1) becomes Ax = f . A solution of the equation Ax = f is precisely the equilibrium point of the system (1.1). The question of whether or not there exists an element x* in the space under consideration which satisfies (solves) the equation (1.1) is the question of existence. Whether or not there exists only one or many solutions is the question of uniqucnrss. Several results addrcssing thcsc two questions concerning various classes of operator equations (as well as inclusions and inequalities) have been established (see [4]-[9],[15],[36],[49][53),[57],-and [72]). When a solution is guaranteed to exist, the issoe of how t o actually obtain such a solution is often a problem. The nature of the solution is such that it is ofteri not easy to obtain it explicitly. In such situatichs we natura1Iy turn to approximation. Consider, for instmce, the problem of finding the zero of a real nonlinear function. We generally rely on such iterative methods as Newtoq, Newton-Raphson, Regula Falsi, chords/secant, etc. Moreover, certain differential equa-tions do not have a closed form solution, they can only bc solved numerically using various linear, single and multi step methods such as Runge-Kutta, Adarns-Moulton, Bashforth, Xystrom, Euler, Sirnpson, etc methods. These methods consist in taking an initial guess at the solution and then progressively obtaining better approximations to the solution; thus ope obtains a sequence of approximants which one expects to converge to the actual solution. This is the question of iterative approximation or construction of a solution. In this thesis we are interested in generating a sequence of approximants that will converge to equilibrium points of the system (1.1) whcn the opcrator A is of thc accretive-type. This method (cdled the method of iterative or successive approximation) of generating a sequence of approximants that will converge to the rcquired solution has bccn known and used for a very long time; in fact, it is reported to havc bccn uscd as far back as 323BC.

Definition 1.1.

Let (E,p) be a metric space and let T : S + E be a map. A point x E E is called a ,fixed point of T if Tx = x. A map T : E + E is called a strict contraction if there exists k E LO, 1) such that Vx, y E E the following inequality holds:

The map T is called Lipschitz if (1.2) is satisfied with k 2 0 (i.e., k could be greater than 1). If (1.2) is satisfied with k = 1 then T is calIcd rtonezprtsive.

A result of Picard [72] which was given precise abstract formulation by Banach [5] and

Caccioppoli [15] in what is now generally known as the contraction mapping principle states that if (E,p) is a complete metric space and T : E -+ E is a strict contraction then T has a unique fixed point and the sequence ( x , ) ~ = ,of approximants generated from an arbi tary xo E E by

xn+l = T.T, = TnxO (1.3)

converges strongly to this unique fixed point. This scheme (1.3) has been applied to solve several physically significant problems (see [18], [%3], [60] and [61]). The beauty of the iteration process (1.3) lies in its simplicity and fast convergence rate (it normally converges as fast a.;a geometric progression with common ratio ,u E (O,1)). Because of the wide accepta,bility of this result of Picard [72] efforts were made to extend it to larger classes of mappings, starting with the nonexpansive mappings, (even in slightly restricted spaces). However, it was discovered that, this result fails, if, for example, one takes S = {T E R2 : 1 1 ~ 1 51~ 1) and T : S -+ S a rotation about the origin of coordinates through an angle of q, counterclockwise. Then, Vx E S, T x = %a, SO that JITx- Tyll = 11s- yll,

V x, y F_ K. Clearly Tz*== z* if and only if z* = (0,O). For the initial guess zo = (1, O ) ,

= Tz,, yields n sequence which clusters at eight points on S = (x E R2 : llxll < 1).

Krasnoselskii [XI studied the above problem and established t,he strong convergence of a sequence {xn)r=, of iterates obtained from an arbitrary xo E E by the averaging process to the unique fixed point of the nonexpanskrc map T. Possibly, inspired by this, Schaeffcr [791 introduced the following more general iteration process. For an arbitrary xi-, t E and some X E (U,1) the sequence (xn)?& is defined by

The iterative sequence (1.5) has been studied by various researchers for the purpose of constructmg solutions of various classes of operator equations and fixed points of various nonlinear maps in various normed linear spaces (see [18], 1231, [60] and 1611). Observe that if one def nes the operator then T and Tx have the same set of fixed points. Hence one may replace (1.5) by the following Picard-Iike process

xn+l = TAx,= q x 0 . (1-7)

Observe further, that setting X = $ in (1.5) yields (1.4), the Krasnoselskii process. Thus

both procosses -Ire actually Picard-like proccsses and generally converge, whco they do, at least as fast as a geometric progression. However, the basic advantage of these later averaging processes (1.4) and (1.5) is that for proper choices of A, TAcan, under fairly general conditions, be made a contraction operator (see [I91 and [60]).

If the constant X in the Schaeffer process (1.5) is replaced by a variable real sequence { c ~ } c~ =[O, ~I), then we obtain an iteration process which is similar to the one introduced by Mann [56]. Let IC be a convex nonempty subset of E and let T be a self-map on IC. Then the sequence {x,)?=, iteratively generated from an arbitrary so E K by

where {cn}?=, is a real sequence satisfying the following conditions (i) co = 1, 0 5 c, < 1 M V n > 1 (ii) lim c, = 0 and (iii) C c, = oo is called the Mann iteration process and is
n+ ~c n = O

being applied ir, the iterative approximation of solutions to various classes of nonlinear equations involving various types of nonlinear maps (see [I]-[3],[19]-[33],[37]-[43],[54]-[56], [59]-[65],[68]-[71],[70]-[go]). In applications, it is a t times convenient th replace thc
OC)
conditions (ii) and (iii) with c i < oo and C c,(l – %) = w respectively. n>O n=O

Ishikawa [48] studied the class of nonlinear maps for which the Mann scheme s e e m 4
not to converge and introduced a new iteration process thus: Let (/3,L)r==obe
real sequences satisfying the following conditions: (i) 0 5 a, 5 Pn < 1 V n >_ 0, (ii)
lim 13, – M
0 and (iii) C anPn = cm. Then the sequence {xn)F=, iteratively generated
n-+m n=O
from an arbitr:t.ry xo E K by

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