Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme
© M. De la Sen. 2009
Received: 18 February 2009
Accepted: 27 April 2009
Published: 4 June 2009
A generalization of Halpern's iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern's iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern's iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.
Fixed point theory is a powerful tool for investigating the convergence of the solutions of iterative discrete processes or that of the solutions of differential equations to fixed points in appropriate convex compact subsets of complete metric spaces or Banach spaces, in general, [1–12]. A key point is that the equations under study are driven by contractive maps or at least by asymptotically nonexpansive maps. By that reason, the fixed point formalism is useful in stability theory to investigate the asymptotic convergence of the solution to stable attractors which are stable equilibrium points. The uniqueness of the fixed point is not required in the most general context although it can be sometimes suitable provided that only one such a point exists in some given problem. Therefore, the theory is useful for stability problems subject to multiple stable equilibrium points. Compared to Lyapunov's stability theory, it may be a more powerful tool in cases when searching a Lyapunov functional is a difficult task or when there exist multiple equilibrium points, [1, 12]. Furthermore, it is not easy to obtain the value of the equilibrium points from that of the Lyapunov functional in the case that the last one is very involved. A generalization of the contraction principle in metric spaces by using continuous nondecreasing functions subject to an inequality-type constraint has been performed in . The concept of -times reasonable expansive mapping in a complete metric space is defined in  and proven to possess a fixed point. In , the -stability of Picard's iteration is investigated with being a self-mapping of where ( ) is a complete metric space. The concept of -stability is set as follows: if a solution sequence converges to an existing fixed point of , then the error in terms of distance of any two consecutive values of any solution generated by Picard's iteration converges asymptotically to zero. On the other hand, an important effort has been devoted to the investigation of Halpern's iteration scheme and many associate extensions during the last decades (see, e.g., [4, 6, 9, 10]). Basic Halpern's iteration is driven by an external sequence plus a contractive mapping whose two associate coefficient sequences sum unity for all samples, . Recent extensions of Halpern's iteration to viscosity iterations have been proposed in [4, 6]. In the first reference, a viscosity-type term is added as extraforcing term to the basic external sequence of Halpern's scheme. In the second one, the external driving term is replaced with two ones, namely, a viscosity-type term plus an asymptotically nonexpansive mapping taking values on a left reversible semigroup of asymptotically nonexpansive Lipschitzian mappings on a compact convex subset of the Banach space . The final iteration process investigated in  consists of three forcing terms, namely, a contraction on , an asymptotically nonexpansive Lipschitzian mapping taking values in a left reversible semigroup of mappings from a subset of that of bounded functions on its dual. It is proven that the solution converges to a unique common fixed point of all the set asymptotic nonexpansive mappings for any initial conditions on . The objective of this paper is to investigate further generalizations for Halpern's iteration process via fixed point theory by using two more driving terms, namely, an external one taking values on plus a nonlinear term given by a continuous nondecreasing function, subject to an inequality-type constraint as proposed in , whose argument is the distance between pairs of points of sequences in certain complete metric space which are not necessarily directly related to the sequence solution taking values in the subset of the Banach space . Another generalization point is that the sample-by-sample sum of the scalar coefficient sequences of all the driving terms is not necessarily unity but it converges asymptotically to unity.
2. Stability and Boundedness Properties of a Viscosity-Type Difference Equation
In this section a real difference equation scheme is investigated from a stability point of view by also discussing the existence of stable limiting finite points. The structure of such an iterative scheme supplies the structural basis for the general viscosity iterative scheme later discussed formally in Section 4 in the light of contractive and asymptotically nonexpansive mappings in compact convex subsets of Banach spaces. The following well-known iterative scheme is investigated for an iterative scheme which generates real sequences.
The real sequences , , and are uniformly bounded if if and if ; for all . If, furthermore, if and , if , with if and only if ; for all , then the sequences , , and converge asymptotically to the zero equilibrium point as and is monotonically decreasing.
Let and let a positive real sequence (i.e., all its elements are nonnegative real constants). Define if and if . Then, is a positive real sequence and is uniformly bounded if ; for all . If, furthermore, ; for all , then as .
(Corollary to Venter's theorem, ). Assume that for all , as and (what imply as and the sequence has only a finite set of unity values). Assume also that and is a nonnegative real sequence with . Then as .
(Halpern ; see Hu ). Let be ; for all in (2.1) subject to , ; for all with being a nonexpansive self-mapping on . Thus, converges weakly to a fixed point of in the framework of Hilbert spaces endowed with the inner product , for all , if for any .
Equation (2.1) under the form
with and being a nonexpansive self-mapping on under the weak or strong convergence conditions of Theorem 2.1(vii) is known as Halpern's iteration , which is a particular case of the generalized viscosity iterative scheme studied in the subsequent sections. Theorem 2.1(vi) extends stability Venter's theorem which is useful in recursive stochastic estimation theory when investigating the asymptotic expectation of the norm-squared parametrical estimation error . Note that the stability result of this section has been derived by using discrete Lyapunov's stability theorem with Lyapunov's sequence what guarantees global asymptotic stability to the zero equilibrium point if it is strictly monotonically decreasing on and to global stability (stated essentially in terms of uniform boundedness of the sequence ) if it is monotonically decreasing on . The links between Lyapunov's stability and fixed point theory are clear (see, e.g., [1, 2]). However, fixed point theory is a more powerful tool in the case of uncertain problems since it copes more easily with the existence of multiple stable equilibrium points and with nonlinear mappings. Note that the results of Theorem 2.1 may be further formalized in the context of fixed point theory by defining a complete metric space , respectively, for the particular results being applicable to a positive system under nonnegative initial conditions, with the Euclidean metrics defined by .
3. Some Definitions and Background as Preparatory Tools for Section 4
The four subsequent definitions are then used in the results established and proven in Section 4.
It is possible to define a partial preordering relation " " by ; for all for any semigroup . Thus, , for some existing and , such that if is left reversible. The semigroup is said to be left-amenable if it has a left-invariant mean and it is then left reversible, [6, 13].
A representation of a left reversible semigroup as Lipschitzian mappings on , a nonempty weakly compact convex subset of , with Lipschitz constants is said to be a nonexpansive (resp., asymptotically nonexpansive, ) semigroup on if it holds the uniform Lipschitzian condition (resp., ) on the Lipschitz constants.
A representation of a left reversible semigroup as Lipschitzian mappings on with Lipschitz constants is said to be a contractive (resp., asymptotically contractive) semigroup on if it holds the uniform Lipschitzian condition (resp., ) on the Lipschitz constants.
The iteration process (3.1) is subject to a forcing term generated by a set of Lipschitzian mappings where is a sequence of means on , with the subset (defined in Definition 3.5 below) containing unity, where is the Banach space of all bounded functions on endowed with the supremum norm, such that where is the dual of .
Some particular characterizations of sequences of means to be invoked later on in the results of Section 4 are now given in the definitions which follow.
Parallel definitions follow for right-invariant and strongly right-amenable sequences of means. is said to be left (resp., right)-amenable if it has a left (resp., right)-invariant mean. A general viscosity iteration process considered in  is the following:
(ii) is a representation of a left reversible semigroup with identity being asymptotically nonexpansive, on a compact convex subset of a smooth Banach space, with respect to a left-regular sequence of means defined on an appropriate invariant subspace of ;
It has been proven that the solution of the sequence converges strongly to a unique common fixed point of the representation which is the solution of a variational inequality . The viscosity iteration process (3.1) generalizes that proposed in  for and and also that proposed in [14, 15] with , and ; for all . Halpern's iteration is obtained by replacing and in (3.1) by using the formalism of Hilbert spaces, for all (see, e.g., [4, 9, 10]). There has been proven the weak convergence of the sequence to a fixed point of for any given if for , also proven to converge strongly to one such a point if and as , and . On the other hand, note that if , , and with , for all , then the resulting particular iteration process (3.1) becomes the difference equation (2.1) discussed in Theorem 2.1 from a stability point of view provided that the boundedness of the solution is ensured on some convex compact set ; for all .
4. Boundedness and Convergence Properties of a More General Difference Equation
The viscosity iteration process (3.1) is generalized in this section by including two more forcing terms not being directly related to the solution sequence. One of them being dependent on a nondecreasing distance-valued function related to a complete metric space while the other forcing term is governed by an external sequence . Furthermore the sum of the four terms of the scalar sequences , , and and at each sample is not necessarily unity but it is asymptotically convergent to unity.
The following generalized viscosity iterative scheme, which is a more general difference equation than (3.1), is considered in the sequel
for all for a sequence of given finite numbers with (if , then the corresponding sum is dropped off) which can be rewritten as (2.1) if ; for all (except possibly for a finite number of values of the sequence what implies ) by defining the sequence
The iterative scheme is subject to the following assumptions.
is single valued.
Note that Assumption 1( ) is stronger than the conditions imposed on the sequence in Theorem 2.1 for (2.1). However, the whole viscosity iteration is much more general than the iterative equation (2.1). Three generalizations compared to existing schemes of this class are that an extracoefficient sequence is added to the set of usual coefficient sequences and that the exact constraint for the sum of coefficients being unity for all is replaced by a limit-type constraint as while during the transient such a constraint can exceed unity or be below unity at each sample (see Assumption 1( ). Another generalization is the inclusion of a nonnegative term with generalized contractive mapping involving another iterative scheme evolving on another, and in general distinct, complete metric space (see Assumptions 1( ) and 1( ). Some boundedness and convergence properties of the iterative process (4.1) are formulated and proven in the subsequent result.
The difference iterative scheme (4.1) and equivalently the difference equation (2.1) subject to (4.2) possess the following properties under Assumption 1.
(i) . Also, and for any norm defined on the smooth Banach space and there exists a nonempty bounded compact convex set such that the solution of (4.2) is permanent in , for all and some sufficiently large finite with .
with and which always holds for sufficiently large finite since as . It has been proven by complete induction that the first part of Property (i) holds with the set being built such that for the given initial condition . For a set of initial conditions with any set convex and bounded, a common set might be defined for any initial condition of (4.1) in with a redefinition of the constant as . The second part of Property (i) follows for any norm on from the property of equivalence of norms. Furthermore, the real sequences and converge strongly to a finite limit in since they are uniformly bounded so that Property (ii) has also been proven. Property (iii) follows directly from (4.1) and Property (ii). Property (iv. ) follows since is a nonnegative -vector sequence provided that if what follows from simple inspection of (4.1). Properties (iv. )-(iv. ) follow directly from separating nonnegative positive and nonpositive terms in the right-hand side of the expression in Property (iii).
The convergence properties of Theorem 4.1(ii) are now related to the limits being fixed points of the asymptotically nonexpansive semigroup which is the representation as Lipschitzian mappings on of a left reversible semigroup with identity.
Let be the set of fixed points of the asymptotically nonexpansive semigroup on . Then, the common strong limit of the sequences and in Theorem 4.1(ii) is a fixed point of located in and, thus, a stable equilibrium point of the iterative scheme (4.1) provided that , and then , is sufficiently large.
Property (ii) follows directly from Theorem 4.1(iii)-(iv).
Note that the boundedness property of Theorem 4.1(i) does not require explicitly the condition of Assumption 1( ) that is asymptotically nonexpansive. On the other hand, neither Theorem 4.1 nor Theorem 4.2 requires Assumption 1( ).
Definition 4.4 (see ).
It is known that if is weakly compact, is a mean on (see Definition 3.5), and is in for each , then there is a unique such that for each . Also, if is smooth, that is, the duality mapping of is single valued then a retraction of onto is sunny and nonexpansive if and only if , for all [6, 11].
Note that Theorem 4.2 proves the convergence to a fixed point in , with being constructively proven to be nonempty by first building a sufficiently large convex compact so that the solution of the iterative scheme (4.1) is always bounded on . Note also that Theorems 4.1 and 4.2 need not the assumption of being a left-invariant -stable subspace of containing " " and to be a left-invariant mean on , although it is assumed to be strongly left regular so that it fulfils ; for all (Assumption 1( ), see Definition 3.6. However, the convergence to a unique fixed point in the set is not proven under those less stringent assumptions. Note also that Assumption 1( ) required by Theorem 4.1 and also by Theorem 4.2 as a result is one of the two properties associated with the -stability of .
The results of Theorems 4.1 and 4.2 with further considerations by using Definitions 4.4 and 4.5 allow to obtain the convergence to a unique fixed point under more stringent conditions for the semigroup of self-mappings , as follows.
If Assumption 1 hold and, furthermore, is a left-invariant -stable subspace of then the sequence , generated by (4.1), converges strongly to a unique ; for all , for all , for all which is the unique solution of the variational inequality . Equivalently, where is the unique sunny nonexpansive retraction of onto .
The proof follows under similar tools as those used in  since is a nonempy sunny nonexpansive retract of which is unique since is nonexpansive for all .
The iterative scheme (4.27) keeps the applicable parts of Assumptions 1( )–1( ), 1( ) for the nonidentically zero parameterizing sequences , and . Assumptions 1( ) and 1( ) are modified with the replacements , , and .
The following properties hold under Assumption 2.
(i) ; for all . Also, and for any norm defined on the smooth Banach space and there exists a nonempty bounded compact convex set such that the solution of (4.2) is permanent in , for all and some sufficiently large finite with .
(iv)Assume that the nonempty convex subset of the smooth Banach space , which contains the sequence of means on , is such that each element ; for all , for some so that (4.1) is a positive viscosity iteration scheme (4.27). Then,
Note that the results of Section 4 generalize those of Section 2 since the iterative process (4.1) possesses simultaneously a nonlinear contraction and a nonexpansive mapping plus terms associated to driving terms combining both external driving forces plus the contribution of a nonlinear function evaluating distances over, in general, distinct metric spaces than that generating the solution of the iteration process. Therefore, the results about fixed points in Theorem 2.1(vi)-(vii) are directly included in Theorem 4.1.
Venter's theorem can be used for the convergence to the equilibrium points of the solutions of the generalized iterative schemes (4.1) and (4.27), provided they are positive, as follows.
Then, the sets of fixed points of the positive iteration schemes (4.1) and (4.27) contain a common stable equilibrium point which is a unique solution to the variational equations of Theorems 4.8 and 4.9; that is, and that .
Outline of Proof
The following result is obvious since if the representation is nonexpansive, contractive or asymptotically contractive (Definitions 3.3 and 3.4), then it is also asymptotically nonexpansive as a result.
The author is very grateful to the Spanish Ministry of Education by its partial support of this work through Grant DPI2006-00714. He is also grateful to the Basque Government by its support through Grants GIC07143-IT-269-07 and SAIOTEK S-PE08UN15. The author is also grateful to the reviewers for their interesting comments which helped him to improve the final version of the manuscript.
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