Fixed Point Theory and Applications

, 2009:314581 | Cite as

Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme

Open Access
Research Article

Abstract

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.

Keywords

Nonexpansive Mapping Iterative Scheme Real Sequence Unique Fixed Point Smooth Banach Space 

1. Introduction

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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 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 [2]. The concept of Open image in new window -times reasonable expansive mapping in a complete metric space is defined in [3] and proven to possess a fixed point. In [5], the Open image in new window -stability of Picard's iteration is investigated with Open image in new window being a self-mapping of Open image in new window where ( Open image in new window ) is a complete metric space. The concept of Open image in new window -stability is set as follows: if a solution sequence converges to an existing fixed point of Open image in new window , 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, [9]. 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 Open image in new window of the Banach space Open image in new window . The final iteration process investigated in [6] consists of three forcing terms, namely, a contraction on Open image in new window , 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 Open image in new window . 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 Open image in new window plus a nonlinear term given by a continuous nondecreasing function, subject to an inequality-type constraint as proposed in [2], 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 Open image in new window of the Banach space Open image in new window . 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.

Theorem 2.1.

Consider the difference equation:
such that the error sequence Open image in new window is generated by

for all Open image in new window , where Open image in new window .

Assume that Open image in new window and Open image in new window are bounded real constants and Open image in new window ; for all Open image in new window . Then, the following properties hold.
  1. (i)
     
  2. (ii)
     
  3. (iii)

    Let Open image in new window and let Open image in new window a positive real sequence (i.e., all its elements are nonnegative real constants). Define Open image in new window if Open image in new window and Open image in new window if Open image in new window . Then, Open image in new window is a positive real sequence and Open image in new window is uniformly bounded if Open image in new window ; for all Open image in new window . If, furthermore, Open image in new window ; for all Open image in new window , then Open image in new window as Open image in new window .

     
  4. (iv)
     
  5. (v)

    (Corollary to Venter's theorem, [7]). Assume that Open image in new window for all Open image in new window , Open image in new window as Open image in new window and Open image in new window (what imply Open image in new window as Open image in new window and the sequence Open image in new window has only a finite set of unity values). Assume also that Open image in new window and Open image in new window is a nonnegative real sequence with Open image in new window . Then Open image in new window as Open image in new window .

     
  6. (vi)

    (Suzuki [8]; see also Saeidi [6]). Let Open image in new window be a sequence in Open image in new window with Open image in new window , and let Open image in new window and Open image in new window be bounded sequences. Then, Open image in new window .

     
  7. (vii)

    (Halpern [9]; see Hu [4]). Let Open image in new window be Open image in new window ; for all Open image in new window in (2.1) subject to Open image in new window , Open image in new window ; for all Open image in new window with Open image in new window being a nonexpansive self-mapping on Open image in new window . Thus, Open image in new window converges weakly to a fixed point of Open image in new window in the framework of Hilbert spaces endowed with the inner product Open image in new window , for all Open image in new window , if Open image in new window for any Open image in new window .

     
Proof.
  1. (i)
    Direct calculations with (2.1) lead to
     
Since Open image in new window is a convex parabola Open image in new window for all Open image in new window if real constants Open image in new window exist such that Open image in new window ; Open image in new window . The parabola zeros are Open image in new window so that Open image in new window if Open image in new window . If Open image in new window , then Open image in new window with Open image in new window . Thus, Open image in new window if Open image in new window , for all Open image in new window . If Open image in new window , then Open image in new window as Open image in new window . Property (ii) has been proven.
  1. (iii)

    If Open image in new window is positive then Open image in new window is positive from direct calculations through (2.1). The second part follows directly from Property (ii) by restricting Open image in new window for uniform boundedness of Open image in new window and Open image in new window for its asymptotic convergence to zero in the case of nonzero Open image in new window .

     
  2. (iv)
    If Open image in new window ; for all Open image in new window and Open image in new window , then from recursive evaluation of (2.1):
     

If Open image in new window and Open image in new window , for all Open image in new window for some Open image in new window , then Open image in new window , for all Open image in new window ; thus, Open image in new window is monotonically strictly decreasing so that it converges asymptotically to zero.

Equation (2.1) under the form

with Open image in new window and Open image in new window being a nonexpansive self-mapping on Open image in new window under the weak or strong convergence conditions of Theorem 2.1(vii) is known as Halpern's iteration [4], 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 [7]. Note that the stability result of this section has been derived by using discrete Lyapunov's stability theorem with Lyapunov's sequence Open image in new window what guarantees global asymptotic stability to the zero equilibrium point if it is strictly monotonically decreasing on Open image in new window and to global stability (stated essentially in terms of uniform boundedness of the sequence Open image in new window ) if it is monotonically decreasing on Open image in new window . 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 Open image in new window , respectively, Open image in new window for the particular results being applicable to a positive system under nonnegative initial conditions, with the Euclidean metrics defined by Open image in new window .

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.

Definition 3.1.

Open image in new window is a left reversible semigroup if Open image in new window ; for all Open image in new window .

It is possible to define a partial preordering relation " Open image in new window " by Open image in new window ; for all Open image in new window for any semigroup Open image in new window . Thus, Open image in new window , for some existing Open image in new window and Open image in new window , such that Open image in new window if Open image in new window is left reversible. The semigroup Open image in new window is said to be left-amenable if it has a left-invariant mean and it is then left reversible, [6, 13].

Definition 3.2 (see [6, 13]).

Open image in new window is said to be a representation of a left reversible semigroup Open image in new window as Lipschitzian mappings on Open image in new window if Open image in new window is a Lipschitzian mapping on Open image in new window with Lipschitz constant Open image in new window and, furthermore, Open image in new window ; for all Open image in new window .

The representation Open image in new window may be nonexpansive, asymptotically nonexpansive, contractive and asymptotically contractive according to Definitions 3.3 and 3.4 which follow.

Definition 3.3.

A representation Open image in new window of a left reversible semigroup Open image in new window as Lipschitzian mappings on Open image in new window , a nonempty weakly compact convex subset of Open image in new window , with Lipschitz constants Open image in new window is said to be a nonexpansive (resp., asymptotically nonexpansive, [6]) semigroup on Open image in new window if it holds the uniform Lipschitzian condition Open image in new window (resp., Open image in new window ) on the Lipschitz constants.

Definition 3.4.

A representation Open image in new window of a left reversible semigroup Open image in new window as Lipschitzian mappings on Open image in new window with Lipschitz constants Open image in new window is said to be a contractive (resp., asymptotically contractive) semigroup on Open image in new window if it holds the uniform Lipschitzian condition Open image in new window (resp., Open image in new window ) on the Lipschitz constants.

The iteration process (3.1) is subject to a forcing term generated by a set of Lipschitzian mappings Open image in new window where Open image in new window is a sequence of means on Open image in new window , with the subset Open image in new window (defined in Definition 3.5 below) containing unity, where Open image in new window is the Banach space of all bounded functions on Open image in new window endowed with the supremum norm, such that Open image in new window where Open image in new window is the dual of Open image in new window .

Definition 3.5.

The real sequence Open image in new window is a sequence of means on Open image in new window if Open image in new window .

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.

Definition 3.6.

The sequence of means Open image in new window on Open image in new window is

(1)left invariant if Open image in new window ; for all Open image in new window , for all Open image in new window , for all Open image in new window in Open image in new window for Open image in new window ;

(2)strongly left regular if Open image in new window , for all Open image in new window , where Open image in new window is the adjoint operator of Open image in new window defined by Open image in new window ; for all Open image in new window , for all Open image in new window .

Parallel definitions follow for right-invariant and strongly right-amenable sequences of means. Open image in new window is said to be left (resp., right)-amenable if it has a left (resp., right)-invariant mean. A general viscosity iteration process considered in [6] is the following:

where

(i)the real sequences Open image in new window , Open image in new window , and Open image in new window have elements in Open image in new window of sum being identity, for all Open image in new window ;

(ii) Open image in new window is a representation of a left reversible semigroup with identity Open image in new window being asymptotically nonexpansive, on a compact convex subset Open image in new window of a smooth Banach space, with respect to a left-regular sequence of means defined on an appropriate invariant subspace of Open image in new window ;

(iii) Open image in new window is a contraction on Open image in new window .

It has been proven that the solution of the sequence converges strongly to a unique common fixed point of the representation Open image in new window which is the solution of a variational inequality [6]. The viscosity iteration process (3.1) generalizes that proposed in [13] for Open image in new window and Open image in new window and also that proposed in [14, 15] with Open image in new window , Open image in new window and Open image in new window ; for all Open image in new window . Halpern's iteration is obtained by replacing Open image in new window and Open image in new window in (3.1) by using the formalism of Hilbert spaces, for all Open image in new window (see, e.g., [4, 9, 10]). There has been proven the weak convergence of the sequence Open image in new window to a fixed point of Open image in new window for any given Open image in new window if Open image in new window for Open image in new window [9], also proven to converge strongly to one such a point if Open image in new window and Open image in new window as Open image in new window , and Open image in new window [10]. On the other hand, note that if Open image in new window , Open image in new window , and Open image in new window with Open image in new window , for all Open image in new window , 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 Open image in new window ; for all Open image in new window .

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 Open image in new window . Furthermore the sum of the four terms of the scalar sequences Open image in new window , Open image in new window , and Open image in new window and Open image in new window 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 Open image in new window for a sequence of given finite numbers Open image in new window with Open image in new window (if Open image in new window , then the corresponding sum is dropped off) which can be rewritten as (2.1) if Open image in new window ; for all Open image in new window (except possibly for a finite number of values of the sequence Open image in new window what implies Open image in new window ) by defining the sequence

with Open image in new window , where

(i) Open image in new window is a strongly left-regular sequence of means on Open image in new window , that is, Open image in new window . See Definition 3.5;

(ii) Open image in new window is a left reversible semigroup represented as Lipschitzian mappings on Open image in new window by Open image in new window .

The iterative scheme is subject to the following assumptions.

Assumption 1.

( Open image in new window ) Open image in new window , Open image in new window , and Open image in new window are real sequences in Open image in new window , Open image in new window is a real sequence in Open image in new window , and Open image in new window are sequences in Open image in new window , for all Open image in new window for some given Open image in new window and Open image in new window .

( Open image in new window ) Open image in new window , Open image in new window .

( Open image in new window ) Open image in new window .

( Open image in new window ) Open image in new window .

( Open image in new window ) Open image in new window ; for all Open image in new window with Open image in new window being a bounded real sequence satisfying Open image in new window and Open image in new window .

( Open image in new window ) Open image in new window is a contraction on a nonempty compact convex subset Open image in new window , of diameter Open image in new window of a Banach space Open image in new window , of topological dual Open image in new window , which is smooth, that is, its normalized duality mapping Open image in new window from Open image in new window into the family of nonempty (by the Hahn-Banach theorem [6, 11]), weak-star compact convex subsets of Open image in new window , defined by

is single valued.

( Open image in new window ) The representation Open image in new window of the left reversible semigroup Open image in new window with identity is asymptotically nonexpansive on Open image in new window (see Definition 3.3) with respect to Open image in new window , with Open image in new window which is strongly left regular so that it fulfils Open image in new window .

( Open image in new window ) Open image in new window .

( Open image in new window ) Open image in new window is a complete metric space and Open image in new window is a self-mapping satisfying the inequality

where Open image in new window , for all Open image in new window are continuous monotone nondecreasing functions satisfying Open image in new window if and only if Open image in new window ; for all Open image in new window .

( Open image in new window ) Open image in new window is a sequence in Open image in new window generated as Open image in new window , Open image in new window with Open image in new window and Open image in new window is a finite given number.

Note that Assumption 1( Open image in new window ) is stronger than the conditions imposed on the sequence Open image in new window 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 Open image in new window is added to the set of usual coefficient sequences and that the exact constraint for the sum of coefficients Open image in new window being unity for all Open image in new window is replaced by a limit-type constraint Open image in new window as Open image in new window while during the transient such a constraint can exceed unity or be below unity at each sample (see Assumption 1( Open image in new window ). Another generalization is the inclusion of a nonnegative term with generalized contractive mapping Open image in new window involving another iterative scheme evolving on another, and in general distinct, complete metric space Open image in new window (see Assumptions 1( Open image in new window ) and 1( Open image in new window ). Some boundedness and convergence properties of the iterative process (4.1) are formulated and proven in the subsequent result.

Theorem 4.1.

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) Open image in new window . Also, Open image in new window and Open image in new window for any norm defined on the smooth Banach space Open image in new window and there exists a nonempty bounded compact convex set Open image in new window such that the solution of (4.2) is permanent in Open image in new window , for all Open image in new window and some sufficiently large finite Open image in new window with Open image in new window .

(iv)Assume that Open image in new window such that each sequence element Open image in new window (the first closed orthant of Open image in new window ); for all Open image in new window , for some Open image in new window so that (4.1) is a positive viscosity iteration scheme. Then,

(iv.1) Open image in new window is a nonnegative sequence (i.e., all its components are nonnegative for all Open image in new window , for all Open image in new window ), denoted as Open image in new window ; for all Open image in new window .

(iv.2)Property (i) holds for Open image in new window and Property (ii) also holds for a limiting point Open image in new window .

(iv.3)Property (iii) becomes
what implies that either

Proof.

From (4.2) and substituting the real sequence Open image in new window from the constraint Assumption 1( Open image in new window ), we have the following:
where Open image in new window is an arbitrary finite sufficiently large integer, and
with Open image in new window as Open image in new window . One gets from (4.12) into (4.10),
what implies that
(see [8]) since Open image in new window as Open image in new window since Open image in new window is in Open image in new window and Open image in new window is a strongly left-regular sequence of means on Open image in new window such that Open image in new window ; furthermore, Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window as Open image in new window and Open image in new window as Open image in new window . Thus, from (4.14) and using the above technical result in [8] for difference equations of the class (2.1) (see also [2]), it follows that
Using Assumption 1 and using (4.15) into (4.17) yield
since Open image in new window , Open image in new window , Open image in new window , Open image in new window as Open image in new window . Also, it follows that Open image in new window as Open image in new window from (4.15) and (4.18). Note that it has not been proven yet that the sequences Open image in new window and Open image in new window converge to a finite limit as Open image in new window since it has not been proven that they are bounded. Thus, the four sequences Open image in new window and Open image in new window converge asymptotically to the same finite or infinite real limit. Proceed recursively with the solution of (4.1). Thus, for a given sufficiently large finite Open image in new window and for all Open image in new window , one gets
for sufficiently large Open image in new window and a sufficiently small Open image in new window which exists from Assumptions 1( Open image in new window ) and 1( Open image in new window ). Note that the sequences Open image in new window and Open image in new window may be chosen to satisfy Open image in new window and Open image in new window ; for all Open image in new window . Now, proceed by complete induction by assuming that Open image in new window for given sufficiently large Open image in new window and finite Open image in new window . Then, one gets from (4.20) that Open image in new window for any prescribed Open image in new window if

with Open image in new window and Open image in new window which always holds for sufficiently large finite Open image in new window since Open image in new window as Open image in new window . It has been proven by complete induction that the first part of Property (i) holds with the set Open image in new window being built such that Open image in new window for the given initial condition Open image in new window . For a set of initial conditions Open image in new window with any set Open image in new window convex and bounded, a common set Open image in new window might be defined for any initial condition of (4.1) in Open image in new window with a redefinition of the constant Open image in new window as Open image in new window . The second part of Property (i) follows for any norm on Open image in new window from the property of equivalence of norms. Furthermore, the real sequences Open image in new window and Open image in new window converge strongly to a finite limit in Open image in new window 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. Open image in new window ) follows since Open image in new window is a nonnegative Open image in new window -vector sequence provided that Open image in new window if Open image in new window what follows from simple inspection of (4.1). Properties (iv. Open image in new window )-(iv. Open image in new window ) 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 Open image in new window which is the representation as Lipschitzian mappings on Open image in new window of a left reversible semigroup Open image in new window with identity.

Theorem 4.2.

The following properties hold.
  1. (i)

    Let Open image in new window be the set of fixed points of the asymptotically nonexpansive semigroup Open image in new window on Open image in new window . Then, the common strong limit Open image in new window of the sequences Open image in new window and Open image in new window in Theorem 4.1(ii) is a fixed point of Open image in new window located in Open image in new window and, thus, a stable equilibrium point of the iterative scheme (4.1) provided that Open image in new window , and then Open image in new window , is sufficiently large.

     

(ii) Open image in new window .

Proof.
  1. (i)
    Proceed by contradiction by assuming that Open image in new window so that there exists Open image in new window such that
     

since Open image in new window , where the above two limits exist and are zero from Theorem 4.1(ii). Then, Open image in new window , with Open image in new window being nonempty since, at least one such finite fixed point exists in Open image in new window .

Property (ii) follows directly from Theorem 4.1(iii)-(iv).

Remark 4.3.

Note that the boundedness property of Theorem 4.1(i) does not require explicitly the condition of Assumption 1( Open image in new window ) that Open image in new window is asymptotically nonexpansive. On the other hand, neither Theorem 4.1 nor Theorem 4.2 requires Assumption 1( Open image in new window ).

Definition 4.4 (see [8]).

Let the sequence of means Open image in new window be in Open image in new window , and let Open image in new window be a representation of a left reversible semigroup Open image in new window . Then Open image in new window is Open image in new window -stable if the functions Open image in new window and Open image in new window on Open image in new window are also in Open image in new window ; for all Open image in new window , for all Open image in new window .

Definition 4.5 (see [8, 11]).

Let Open image in new window and Open image in new window be convex subsets of the Banach space Open image in new window , with Open image in new window under proper inclusion, and let Open image in new window be a retraction of Open image in new window onto Open image in new window . Then Open image in new window is said to be sunny if Open image in new window ; for all Open image in new window , for all Open image in new window provided that Open image in new window .

Definition 4.6.

Open image in new window is said t be a sunny nonexpansive retract of Open image in new window if there exists a sunny nonexpansive retraction Open image in new window of Open image in new window onto Open image in new window .

It is known that if Open image in new window is weakly compact, Open image in new window is a mean on Open image in new window (see Definition 3.5), and Open image in new window is in Open image in new window for each Open image in new window , then there is a unique Open image in new window such that Open image in new window for each Open image in new window . Also, if Open image in new window is smooth, that is, the duality mapping Open image in new window of Open image in new window is single valued then a retraction Open image in new window of Open image in new window onto Open image in new window is sunny and nonexpansive if and only if Open image in new window , for all Open image in new window [6, 11].

Remark 4.7.

Note that Theorem 4.2 proves the convergence to a fixed point in Open image in new window , with Open image in new window being constructively proven to be nonempty by first building a sufficiently large convex compact Open image in new window so that the solution of the iterative scheme (4.1) is always bounded on Open image in new window . Note also that Theorems 4.1 and 4.2 need not the assumption of Open image in new window being a left-invariant Open image in new window -stable subspace of containing " Open image in new window " and to be a left-invariant mean on Open image in new window , although it is assumed to be strongly left regular so that it fulfils Open image in new window ; for all Open image in new window (Assumption 1( Open image in new window ), see Definition 3.6. However, the convergence to a unique fixed point in the set Open image in new window is not proven under those less stringent assumptions. Note also that Assumption 1( Open image in new window ) required by Theorem 4.1 and also by Theorem 4.2 as a result is one of the two properties associated with the Open image in new window -stability of Open image in new window .

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 Open image in new window , Open image in new window as follows.

Theorem 4.8.

If Assumption 1 hold and, furthermore, Open image in new window is a left-invariant Open image in new window -stable subspace of Open image in new window then the sequence Open image in new window , generated by (4.1), converges strongly to a unique Open image in new window ; for all Open image in new window , for all Open image in new window , for all Open image in new window which is the unique solution of the variational inequality Open image in new window . Equivalently, Open image in new window where Open image in new window is the unique sunny nonexpansive retraction of Open image in new window onto Open image in new window .

The proof follows under similar tools as those used in [6] since Open image in new window is a nonempy sunny nonexpansive retract of Open image in new window which is unique since Open image in new window is nonexpansive for all Open image in new window .

Proof.

Let Open image in new window be the sequence solution generated by the particular iterative scheme resulting from (4.1) for any initial conditions Open image in new window when all the functions Open image in new window and Open image in new window are zeroed. It is obvious by the calculation of the recursive solution of (4.1) from (4.19) that the error from both solutions satisfies
Since the convergence of the solution to fixed points of Theorems 4.1, 4.2, and 4.8 follows also for the sequence Open image in new window it follows that a unique fixed point exists satisfying
where Open image in new window is unique since Open image in new window is also unique from Theorem 4.8. Assume that Open image in new window with Open image in new window . Then,
If Open image in new window and Open image in new window ; for all Open image in new window and the Open image in new window -functions are zero then both fixed points are related by the constraint Open image in new window . Thus, consider a representation Open image in new window of a left reversible semigroup Open image in new window as Lipschitzian mappings on Open image in new window (see Definitions 3.2 and 3.3), a nonempty compact subset of the smooth Banach space Open image in new window with Lipschitz constants Open image in new window which is asymptotically nonexpansive. Consider the iteration scheme:

with Open image in new window , where

(i) Open image in new window is a strongly left-regular sequence of means on Open image in new window , that is, Open image in new window (the dual of Open image in new window ). See Definitions 3.5 and 3.6;

(ii) Open image in new window is a left reversible semigroup represented as Lipschitzian mappings on Open image in new window by Open image in new window .

Assumption 2.

The iterative scheme (4.27) keeps the applicable parts of Assumptions 1( Open image in new window )–1( Open image in new window ), 1( Open image in new window ) for the nonidentically zero parameterizing sequences Open image in new window , Open image in new window and Open image in new window . Assumptions 1( Open image in new window ) and 1( Open image in new window ) are modified with the replacements Open image in new window , Open image in new window , and Open image in new window .

Theorems 4.1 and 4.8 result in the following result for the iterative scheme (4.27) for Open image in new window , Open image in new window :

Theorem 4.9.

The following properties hold under Assumption 2.

(i) Open image in new window ; for all Open image in new window . Also, Open image in new window and Open image in new window for any norm defined on the smooth Banach space Open image in new window and there exists a nonempty bounded compact convex set Open image in new window such that the solution of (4.2) is permanent in Open image in new window , for all Open image in new window and some sufficiently large finite Open image in new window with Open image in new window .

(ii) Open image in new window and Open image in new window as Open image in new window .

(iii) Open image in new window

(iv)Assume that the nonempty convex subset Open image in new window of the smooth Banach space Open image in new window , which contains the sequence Open image in new window of means on Open image in new window , is such that each element Open image in new window ; for all Open image in new window , for some Open image in new window so that (4.1) is a positive viscosity iteration scheme (4.27). Then,

(iv.1) Open image in new window is a nonnegative sequence (i.e., all its components are nonnegative for all Open image in new window , for all Open image in new window ), denoted as Open image in new window ; for all Open image in new window .

(iv.2)Property (i) holds for Open image in new window and Property (ii) also holds for a limiting point Open image in new window .

(iv.3)Property (iii) becomes
what implies that either
(v)If, furthermore, Open image in new window is a left-invariant Open image in new window -stable subspace of Open image in new window , then the sequence Open image in new window , generated by (4.27), converges strongly to a unique Open image in new window ; for all Open image in new window , for all Open image in new window which is the unique solution of the variational inequality Open image in new window . Equivalently, Open image in new window where Open image in new window is the unique sunny nonexpansive retraction of Open image in new window onto Open image in new window . Furthermore, the unique fixed points of the iterative schemes (4.1) and (4.27) are related by

If, in addition, Open image in new window and Open image in new window and the Open image in new window -functions are identically zero in the iterative scheme (4.1), then Open image in new window .

Remark 4.10.

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.

Corollary 4.11.

Assume that

( Open image in new window ) Open image in new window are both contractive mappings with Open image in new window being compact and convex, Open image in new window , such that Open image in new window is a left-invariant Open image in new window -stable subspace of Open image in new window with Open image in new window being a left reversible semigroup;

( Open image in new window ) Open image in new window , with Open image in new window being compact and convex, Open image in new window , Open image in new window , Open image in new window and Open image in new window ; for all Open image in new window for some real constants Open image in new window , and Open image in new window if Open image in new window ;

( Open image in new window ) Open image in new window and Open image in new window .

Then, the sets of fixed points of the positive iteration schemes (4.1) and (4.27) contain a common stable equilibrium point Open image in new window which is a unique solution to the variational equations of Theorems 4.8 and 4.9; that is, Open image in new window and that Open image in new window .

Outline of Proof

The fact that the mappings Open image in new window are both contractive, Open image in new window and Open image in new window imply that the generated sequences Open image in new window , Open image in new window are both nonnegative and bounded for any Open image in new window and they have unique zero limits from Theorem 2.1(v).

The following result is obvious since if the representation Open image in new window is nonexpansive, contractive or asymptotically contractive (Definitions 3.3 and 3.4), then it is also asymptotically nonexpansive as a result.

Corollary 4.12.

If the representation Open image in new window is nonexpansive, contractive or asymptotically contractive, then Theorems 4.1, 4.2, and 4.8 still hold under Assumption 1, and Theorem 4.9 still holds under Assumption 2.

Notes

Acknowledgments

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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Copyright information

© M. De la Sen. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.IIDP. Faculty of Science and TechnologyUniversity of the Basque Country, Campus of Leioa (Bizkaia)BilbaoSpain

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