Robustness of Mann Type Algorithm with Perturbed Mapping for Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to study the robustness of Mann type algorithm in the sense that approximately perturbed mapping does not alter the convergence of Mann type algorithm. It is proven that Mann type algorithm with perturbed mapping remains convergent in a Banach space setting where , a nonexpansive mapping, , , errors and a strongly accretive and strictly pseudocontractive mapping.

Let be a nonempty closed convex subset of a real Banach space , and a nonexpansive mapping (i.e., for all ). We use to denote the set of fixed points of ; that is, . Throughout this paper it is assumed that . Construction of fixed points of nonlinear mappings is an important and active research area. In particular, iterative methods for finding fixed points of nonexpansive mappings have received vast investigation since these methods find applications in a variety of applied areas of variational inequality problems, equilibrium problems, inverse problems, partial differential equations, image recovery, and signal processing (see, e.g., [1–17]).

In 1953, Mann [18] introduced an iterative algorithm which is now referred to as Mann's algorithm. Most of the literature deals with the special case of the general Mann's algorithm; that is, for an arbitrary initial guess , the sequence is generated by the recursive manner

where is a convex subset of a Banach space is a mapping and is a sequence in the interval . It is well known that Mann's algorithm can be employed to approximate fixed points of nonexpansive mappings and zeros of (strongly) accretive mappings in Hilbert spaces and Banach spaces. Many convergence theorems have been announced and published by a large number of authors. A typical convergence result in connection with the Mann's algorithm is the following theorem proved by Ishikawa [19].

Theorem IS (see [19])

Let be a nonempty subset of a Banach space and let be a nonexpansive mapping. Let be a real sequence satisfying the following control conditions:

(a);

(b).

Let be defined by (1.1) such that for all . If is bounded then as .

The interest and importance of Theorem IS lie in the fact that strong or weak convergence of the sequence can be achieved under certain appropriate assumptions imposed on the mapping , the domain or the space . In an infinite-dimensional space , Mann's algorithm has only weak convergence, in general. In fact, it is known that if the sequence is such that , then Mann's algorithm converges weakly to a fixed point of provided that the underlying space is a Hilbert space or more general, a uniformly convex Banach space which has a Fréchet differentiable norm or satisfies Opial's property. See, for example, [20, 21].

The study of the robustness of Mann's algorithm is initiated by Combettes [22] where he considered a parallel projection method algorithm in signal synthesis (design and recovery) problems in a real Hilbert space as follows:

where for each , is the (nearest point) projection of a signal onto a closed convex subset of [23] ( is interpreted as the th constraint set of the signals), is a sequence of relaxation parameters in are strictly positive weights such that , and stands for the error made in computing the projection onto at iteration . Then he proved the following robustness result of algorithm (1.2).

Theorem 1.1 (see [22]).

Assume . Assume also

(i)

(ii).

Then the sequence generated by (1.2) converges weakly to a point in .

Define a mapping by

and put

Since is a projection, the mapping is nonexpansive. Thus and algorithm (1.2) can be rewritten as

where is given by (1.3). Note that can be written as and thus is nonexpansive. Note also that . Furthermore, conditions (i) and (ii) in Theorem 1.1 can be stated as

.

Very early, some authors had considered Mann iterations in the setting of uniformly convex Banach spaces and established strong and weak convergence results for Mann iterations; see, e.g., [24, 25]. Recently, Kim and Xu [26] studied the robustness of Mann's algorithm for nonexpansive mappings in Banach spaces and extended Combettes' robustness result (Theorem 1.1 above) for projections from Hilbert spaces to the setting of uniformly convex Banach spaces.

Theorem 1.2 (see [26, Theorem 3.3]).

Assume that is a uniformly convex Banach space. Assume, in addition, that either has the KK- property or satisfies Opial's property. Let be a nonexpansive mapping such that . Given an initial guess . Let be generated by the following perturbed Mann's algorithm:

where and satisfy the following properties:

(i),

(ii).

Then the sequence converges weakly to a fixed point of .

Further, Kim and Xu [26] also extended the robustness to nonexpansive mappings which are defined on subsets of a Hilbert space and to accretive operators.

Theorem 1.3 (see [26, Theorem 4.1]).

Let be a nonempty closed convex subset of a Hilbert space and a nonexpansive mapping with . Let be generated from an arbitrary via one of the following algorithms (1.7) and (1.7):

where the sequences and are such that

(i),

(ii).

Then converges weakly to a fixed point of .

Theorem 1.4 (see [26, Theorem 5.1]).

Let be a uniformly convex Banach space. Assume in addition that either has the KK- property or satisfies Opial's property. Let be an -accretive operator in such that . Moreover, assume that and satisfy the following properties:

(i);

(ii);

(iii), where and are two constants;

(iv).

Then the sequence generated from an arbitrary by

converges weakly to a point of .

Let be a real reflexive Banach space. Let be a nonexpansive mapping with . Assume that is -strongly accretive and -strictly pseudocontractive with where . In this paper, inspired by Combettes' robustness result (Theorem 1.1 above) and Kim and Xu's robustness result (Theorem 1.2 above) we will consider the robustness of Mann type algorithm with perturbed mapping, which generates, from an arbitrary initial guess , a sequence by the recursive manner

where and are sequences in and in , respectively, such that

(i);

(ii);

(iii).

More precisely, we will prove under conditions (i)–(iii) the weak convergence of the algorithm (1.9) in a uniformly convex Banach space which either has the KK-property or satisfies Opial's property. This theorem extends Kim and Xu's robustness result (Theorem 1.2 above) from Mann's algorithm (1.6) with errors to Mann type algorithm (1.9) with perturbed mapping . On the other hand, we also extend Kim and Xu's robustness results (Theorems 1.3 and 1.4 above) for nonexpansive mappings which are defined on subsets of a Hilbert space and accretive operators in a uniformly convex Banach space from Mann's algorithm with errors to Mann type algorithm with perturbed mapping.

Throughout this paper, we use the following notations:

(i)stands for weak convergence and for strong convergence,

(ii) denotes the weak -limit set of .

Let be a real Banach space. Recall that the norm of is said to be Fréchet differentiable if, for each , the unit sphere of , the limit

exists and is attained uniformly in . In this case, we have

for all , where is the normalized duality map from to defined by

is the duality pairing between and , and is a function defined on such that . Examples of Banach spaces which have a Fréchet differentiable norm include and for (these spaces are actually uniformly smooth).

It is known that a Banach space is Fréchet differentiable if and only if the duality map is single-valued and norm-to-norm continuous.

We need the concept of the KK-property. A Banach space is said to have the KK-property (the Kadec-Klee property) if, for any sequence in , the conditions and imply that . It is known [27, Remark 3.2] that the dual space of a reflexive Banach space with a Fréchet differentiable norm has the KK-property.

Recall now that satisfies Opial's property [28] provided that, for each sequence in , the condition implies

It is known [28] that each enjoys this property, while does not unless . It is known [29] that any separable Banach space can be equivalently renormed so that it satisfies Opial's property.

Recall that a Banach space is said to be uniformly convex if, for each , the modulus of convexity of defined by

is positive.

We need an inequality characterization of uniform convexity.

Lemma 2.1 (see [30]).

Given a number . A real Banach space is uniformly convex if and only if there exists a continuous strictly increasing function , such that

for all and such that and .

A mapping with domain and range in is called -strongly accretive if for each ,

for some . is called -strictly pseudocontractive if for each ,

for some . It is easy to see that (2.8) can be rewritten as

The following proposition will be used frequently throughout this paper. For the sake of completeness, we include its proof.

Proposition 2.2.

Let be a real Banach space and a mapping.

(i)If is a -strictly pseudocontractive, then is Lipschitz continuous with constant

(ii)If is -strongly accretive and -strictly pseudocontractive with , then for each fixed , the mapping has the following property:

Proof.

(i) From (2.9), we derive

which implies that

Thus

and so is Lipschitz continuous with constant .

(ii) From (2.8) and (2.9), we obtain

Since , we have

Consequently, for each fixed , we have

This shows that inequality (2.10) holds.

Proposition 2.3.

Let be a uniformly convex Banach space and a nonempty closed convex subset of .

(i)Reference [31] (demiclosedness principle). If is a nonexpansive mapping and if is a sequence in such that and , then .

(ii)Reference [32]. If is also bounded, then there exists a continuous, strictly increasing, and convex function (depending only on the diameter of ) with and such that

for all , and nonexpansive mappings .

We also use the following elementary lemma.

Lemma 2.4 (see [33]).

Let and be sequences of nonnegative real numbers such that and for all . Then exists.

Let be a real reflexive Banach space. Let be a nonexpansive mapping with . Assume that is -strongly accretive and -strictly pseudocontractive with . We now discuss the robustness of Mann type algorithm with perturbed mapping, which generates, from an initial guess , a sequence as follows:

where and are sequences in and in , respectively, such that

(i);

(ii);

(iii).

We remark that Mann type algorithm with perturbed mapping is based on Mann iteration method and steepest-descent method. Indeed, in algorithm (3.1), one iteration step "" is taken from Mann iteration method, and another iteration step "" is taken from steepest-descent method.

We first discuss some properties of algorithm (3.1).

Lemma 3.1.

Let be generated by algorithm (3.1) and let Then exists.

Proof.

We have

The conclusion of the lemma is a consequence of Lemma 2.4.

Proposition 3.2.

Let be a uniformly convex Banach space.

(i)For all and , exists.

(ii)If, in addition, the dual space of has the -property, then the weak -limit set of , , is a singleton.

Proof.

(i) For integers , define the mappings and as follows:

and . It is easy to see that . First, let us show that and are nonexpansive. Indeed, for all , using Proposition 2.2 no. (ii) we have

Thus is nonexpansive (due to ) and so is .

Second, let us show that for each ,

Indeed, whenever , we have

This implies that inequality (3.5) holds for . Assume that inequality (3.5) holds for some . Consider the case of . Observe that

This shows that inequality (3.5) holds for the case of . Thus, by induction, we know that inequality (3.5) holds for all .

Now set

By Proposition 2.3 no. (ii) and noticing inequality (3.5) we deduce that

Therefore,

Since exists and and are convergent, we conclude from (3.10) that

Also, since, for all ,

it follows from (3.11) and (3.12) that exists.

(ii) This is Lemma 3.2 of [27].

Now we can state and prove the main result of this section.

Theorem 3.3.

Assume that is a uniformly convex Banach space. Assume, in addition, that either has the -property or satisfies Opial's property. Let be a nonexpansive mapping such that and -strongly accretive and -strictly pseudocontractive with . Given an initial guess . Let be generated by the following Mann type algorithm with perturbed mapping

where and satisfy the following properties:

(i);

(ii);

(iii).

Then the sequence converges weakly to a fixed point of .

Proof.

Fix and select a number large enough so that for all . Let satisfy for all . By Lemma 2.1, we have

It follows that

This implies that

In particular, . Due to condition (i), we must have that . Hence

However, since

we have

and, by Lemma 2.4, exists and hence, by (3.17),

Notice that, by the demiclosedness principle of , we obtain

Hence to prove that converges weakly to a fixed point of , it suffices to show that is a singleton. We distinguish two cases. First assume that has the KK-property. Then that is a singleton is guaranteed by Proposition 3.2 no. (ii).

Next assume that satisfies Opial's property. Take and let and be subsequences of such that and , respectively. If , we reach the following contradiction:

This shows that is a singleton. The proof is therefore complete.

We observe that if the domain is a proper closed convex subset of , then the vectors and may not belong to . In this case the next iterate may not be well defined by (3.13). In order to consider this situation, we will use the nearest projection and for the projection to be nonexpansive, we have to restrict our spaces to be Hilbert spaces.

Let be a real Hilbert space with inner product and norm . Given a closed convex subset of . Recall that the (nearest point) projection from onto assigns each point with its (unique) nearest point in which is denoted by . Namely, is the unique point in with the property

Note that is nonexpansive.

Let be a nonexpansive mapping with and -strongly monotone and -strictly pseudocontractive with . Starting with and after in is defined, we have two ways to define the next iterate : either applying the projection to the vectors and and defining as the convex combination of and , or projecting a convex combination of and onto to define . More precisely, we define as follows:

or

Theorem 4.1.

Let be a nonempty closed convex subset of a Hilbert space . Let be a nonexpansive mapping with and -strongly monotone and -strictly pseudocontractive with . Let be generated by either (4.2) or (4.3) where the sequences and are such that

(i);

(ii);

(iii).

Then converges weakly to a fixed point of .

Proof.

Given . Assume that is generated by (4.2). Then

Hence exists; in particular, is bounded. Let be a constant such that for all .

We compute

That is,

This implies that

In particular (noticing assumption (i)),

We also have

Moreover, noticing

we have

Similarly, if is generated by algorithm (4.3), then relations (4.4)–(4.11) still hold.

It is now readily seen that (4.11) together with Lemma 2.4 implies that exists, which together with (4.8) further implies that

Equation (4.12) implies that , due to the demiclosedness principle. Finally, repeating the last part of the proof of Theorem 3.3 in the case of Opial's property, we see that converges weakly to a fixed point of . The proof is therefore complete.

Finally in this section, we consider the case of accretive operators. Recall that a multivalued operator with domain and range in a Banach space is said to be accretive if, for each and , there is such that

where is the duality map from to the dual space . An accretive operator is -accretive if for all .

Denote by the zero set of ; that is,

Throughout the rest of this paper it is always assumed that is -accretive and is nonempty.

Denote by the resolvent of for :

It is known that is a nonexpansive mapping from to which will be assumed convex (this is so if is uniformly convex). It is also known that for .

Now consider the problem of finding a zero of an -accretive operator in a Banach space ,

We will study the convergence of the following algorithm:

where is a perturbed mapping, the initial guess is arbitrary, and are two sequences in is a sequence of positive numbers, and is an error sequence in .

Theorem 4.2.

Let be a uniformly convex Banach space. Assume in addition that either has the -property or satisfies Opial's property. Let be an -accretive operator in such that and let be -strongly accretive and -strictly pseudocontractive with . Moreover, assume that and satisfy the following properties:

(i);

(ii);

(iii);

(iv), where and are two constants;

(v).

Then the sequence generated by algorithm (4.17) converges weakly to a point of .

Proof.

The proof is a refinement of that of Theorem 3.3 given in Section 3 and [34, Theorem 3.3] together with Proposition 3.2. So we only sketch it.

Let . By (4.17), we have

By Lemma 2.4, we see that exists.

With slight modifications of the proof of Theorem 3.3 (replacing by ), we can obtain that

Now noticing

and letting for all , we deduce that

By mimicking the proof of Theorem 3.3 in [34], we can show that, in the case of ,

and in the case of ,

where is such that for all . In either case we conclude from (4.22) and (4.23) that satisfies

where fulfills . By Lemma 2.4, (4.24) implies that () exists. This together with the assumption (iv) and (4.19) implies that . So, by Lemma 3.3 in [34], we have

By the demiclosedness principle, (4.25) ensures that . Repeating the last part of the proof of Theorem 3.3, we conclude that converges weakly to a point of .

This research was partially supported by Grant no. NSC 98-2923-E-110-003-MY3 and was also partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405).

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