Degree of Convergence of Iterative Algorithms for Boundedly Lipschitzian Strong Pseudocontractions

Abstract

Let be a nonempty closed convex subset of a real Hilbert space and let be a boundedly Lipschitzian strong pseudo-contraction with a nonempty fixed point set. Three iterative algorithms are proposed for approximating the unique fixed point of ; one of them is for the self-mapping case, and the others are for the nonself-mapping case. Not only the strong convergence, but also the degree of convergence of the three iterative algorithms is obtained. Some numerical results corresponding to the self-mapping case are given which show advantages of our methods. As an application of our results, adopting the regularization idea, we also propose implicit and explicit algorithms for approximating a fixed point of a boundedly Lipschitzian pseudocontractive self-mapping from into itself, respectively.

1. Introduction and Preliminaries

Let be a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . Recall that a mapping is said to be pseudo-contractive if

(1.1)

for every . is said to be strongly pseudo-contractive if there exists a positive constant such that

(1.2)

for all . In this case, we also call a -strong pseudocontraction. Using (1.2), it is easy to see that every strong pseudocontraction has at most one fixed point.

is said to be Lipschitzian if there exists a positive constant such that

(1.3)

for all . In this case, is also said to be -Lipschitzian. In particular, is said to be nonexpansive if ; and it is said to be contractive if . is said to be boundedly Lipschitzian if, for each bounded subset of , there exists a positive constant depending only on such that

(1.4)

for all .

We will denote by the set of fixed points of , that is, . Let be a sequence and a point in . Then we use and to denote strong and weak convergence to of the sequence , respectively.

Among classes of nonlinear mappings, the class of pseudocontractions is one of the most important classes of mappings. This is mainly due to the fact that there is a precise corresponding relation between the class of pseudocontractions and the class of monotone mappings. A mapping is monotone (i.e., for all ) if and only if is pseudo-contractive, where and denotes the identity mapping on .

Within the past 40 years or so, mathematicians have been devoting their study to the existence and iterative construction of fixed points for pseudocontractions and of zeros for monotone mappings (see, e.g., [1–18]). However, most of these algorithms have no estimation of degree of convergence even if for strong pseudocontractions in setting of Hilbert spaces. Everyone knows that it is very important to get the degree of convergence for an algorithm in computing science.

The main purpose of this paper is to consider the iterative algorithms for approximating the unique fixed point (if the set of fixed points is not empty) of a boundedly Lipschitzian strong pseudocontraction defined on a nonempty closed convex subset of a real Hilbert space. Three iterative algorithms are proposed; one of them is for the self-mapping case, and the others are for the nonself-mapping case. Not only the strong convergence, but also the degree of convergence of the three iterative algorithms is obtained. Some numerical results corresponding to the self-mapping case are given which show advantages of our methods. As an application of our results, adopting the regularization idea, we also establish implicit and explicit algorithms for approximating a fixed point of a boundedly Lipschitzian pseudo-contractive self-mapping from into itself, respectively.

In order to give our main results, let us recall a basic existence result for fixed points for continuous strong pseudocontractions which was proved by Deimling [6] in 1974.

Theorem 1.1 (Deimling [6]).

Let be a closed subset of a real Banach space , and let be a continuous -strong pseudocontraction, and

(1.5)

for each , where denotes the distance from the point to the subset of . Then has a unique fixed point.

Corollary 1.2 (see [6]).

Let be a closed convex subset of a real Banach space , and let be a continuous -strong pseudocontraction, then has a unique fixed point.

We also need some facts which are listed as lemmas below.

Lemma 1.3 (see [7]).

Let be a real Hilbert space. Given a closed and convex subset of and points and given also a real number such that , then the set

(1.6)

is closed and convex.

Lemma 1.4 (see, e.g., [9]).

Let be a closed convex subset of a real Hilbert space , and let be the (metric or nearest point) projection from onto (i.e., for , is the only point in such that ). Given and , then if and only if there holds the relation

(1.7)

Lemma 1.5 (see [18]).

Let be a nonempty closed convex subset of a real Hilbert space and a demicontinuous pseudo-contractive self-mapping from into itself. Then is a closed convex subset of and is demiclosed at zero.

Now we are in a position to prove main results in this paper.

2. Algorithms for Strongly Pseudocontractive Self-Mappings

In this section, we propose an iterative algorithm for boundedly Lipschitzian and strongly pseudo-contractive self-mappings. Since the algorithm has nothing to do with the metric projection, it is easy to realize in practical computing.

Theorem 2.1.

Let be a nonempty closed convex subset of a real Hilbert space , and let be a boundedly Lipschitzian and strong pseudocontraction. Take arbitrarily, and let and . Define recursively by

(2.1)

where is a constant such that and is the bounded Lipschitz constant of upon . Then converges strongly to the unique fixed point of , and the estimation of degree of convergence is as follows:

(2.2)

In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.

Proof.

Firstly, it concludes by using Corollary 1.2 that has a unique fixed point, denoted by , in . We also assert that holds. Indeed, since is a -strong pseudocontraction, we get that

(2.3)

holds for all . Taking and in (2.3), we have

(2.4)

That is, .

Now we prove by mathematical induction that

(2.5)

and hold for all . For , observing that , is -Lipschitzian restricted to , and is -strongly pseudo-contractive, it is easy to get that

(2.6)

hence

(2.7)

Noting that the condition implies

(2.8)

we have from (2.4), (2.7), and (2.8) that

(2.9)

That is, .

Suppose that and

(2.10)

Similar to (2.7), we have from that

(2.11)

Thus (2.11) together with (2.10) leads to (2.5). On the other hand, we have from (2.4), (2.5), and (2.8) that , that is, .

By (2.7), we have

(2.12)

Consequently

(2.13)

Thus (2.2) is obtained by using (2.5) and (2.13).

Finally, we show that (2.2) is the optimal estimation of degree of convergence in the sense of ignoring constant factors. For this purpose, it suffices to find an example such that

(2.14)

Indeed, taking with the usual inner product and norm and taking , let be a rotation operator defined by

(2.15)

where such that . Obviously, has a unique fixed point . Moreover, is nonexpansive, that is, Lipschitz constant . Since is a linear operator, using (2.15), we have

(2.16)

hence is a -strong pseudocontraction.

Taking an initial value and a control parameter such that , it follows from direct calculating that

(2.17)

This shows that the estimation (2.2) cannot be improved.

Remark 2.2.

If , then reaches the minimum when , so is said to be optimal control parameter of process (2.1). If , then it is not difficult to verify that the optimal control parameter is zero. The same result also applies to all of the following algorithms.

If is Lipschitzian on the whole , that is, there exists a positive constant such that for all , then we obtain the following result as a special case of Theorem 2.1.

Theorem 2.3.

Let be a nonempty closed convex subset of a real Hilbert space , and let be an -Lipschitzian and -strong pseudocontraction. Take an initial guess arbitrarily, and define a sequence as follows:

(2.18)

where is a constant such that . Then converges strongly to the unique fixed point , and the estimation of degree of convergence is obtained as follow:

(2.19)

In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.

In order to test the computing effect of the algorithm (2.1), some numerical results for the function

(2.20)

are given as follows. Using the mean value theorem, it is easy to verify that is a -strongly pseudo-contractive and boundedly Lipcshitzian function. For each constant , the bounded Lipschitz constant of upon the interval is . Choosing the initial guess in (2.1), it follows by using Theorem 2.1 that , , , and the control parameter such that . Since we do not know the exact fixed point of , we propose the relative rate of convergence to test the computing effect of algorithm (2.1) for . All the numerical results are in Tables 1–3.

Table 1 .

Table 2 .

Table 3 .

3. Algorithms for Strongly Pseudo-Contractive Nonself-Mappings

In this section, we turn to designing two iterative algorithms for boundedly Lipschitzian and strongly pseudo-contractive nonself-mappings. In this case, a boundedly Lipschitzian strong pseudocontraction may not have a fixed point, so we assume that the mapping has a unique fixed point (noting that each strong pseudocontraction has at most one fixed point). In addition, we will have to use the metric projection in the algorithms.

In fact, the first algorithm is a modification of process (2.1) as follows. We omit its proof, which is very similar to the proof of Theorem 2.1.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space , and let be a boundedly Lipschitzian and strong pseudocontraction with a unique fixed point. Take arbitrarily, and let and . Define a sequence via the recursive formula

(3.1)

where is a constant such that and is the bounded Lipschitz constant of upon . Then converges strongly to the unique fixed point of , and the estimation of degree of convergence is as follows:

(3.2)

In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.

If is Lipschitzian on the whole , we have the following result as a special case of Theorem 3.1.

Theorem 3.2.

Let be a nonempty closed convex subset of a real Hilbert space , and let be an -Lipschitzian and -strong pseudocontraction. Let has a unique fixed point . Take an initial guess arbitrarily, and define recursively by

(3.3)

where is a constant such that . Then converges strongly to the unique fixed point , and the estimation of degree of convergence is as follows:

(3.4)

In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.

Now we give the second algorithm for the strongly pseudo-contractive nonself-mapping case.

Theorem 3.3.

Let be a nonempty closed convex subset of a real Hilbert space , and let be a boundedly Lipschitzian and strong pseudocontraction . Let have a unique fixed point . Take arbitrarily, and let . Define a sequence of as follows:

(3.5)

where is the bounded Lipschitz constant of upon and is a constant such that . Then converges strongly to . One also has the estimation of degree of convergence

(3.6)

In addition, this estimation of degree of convergence is optimal in the sense of ignoring constant factors.

Proof.

By the proof of Theorem 2.1, we have

(3.7)

that is, .

Now we verify by mathematical induction that and the sequence generated by (3.5) is well defined for each . For , observing that , and is a -strong pseudocontraction, we have

(3.8)

Hence

(3.9)

and this means . Noting that the condition implies that , and by using Lemma 1.3, is nonempty, closed, and convex. Using Lemma 1.4, there exists a unique element such that . Suppose that has been obtained and for some . Likewise, observing that , and is a -strong pseudocontraction, we also have

(3.10)

The definition of and (3.10) imply that . Using Lemma 1.3 again, the condition guarantees that is nonempty, closed, and convex. So there exists a unique element such that .

Finally, we prove that (3.6) holds and converges strongly to . Observing process (3.5), , and for all , we have from an argument similar to getting (3.10) that

(3.11)

By induction step, we have

(3.12)

By (3.11) and triangular inequality, we have

(3.13)

hence

(3.14)

Thus the combination of (3.12) and (3.14) leads to (3.6), and converges strongly to due to the fact that .

By the same argument in the proof of Theorem 2.1, we assert that (3.6) is the optimal estimation of degree of convergence.

Remark 3.4.

The formulation of process (3.1) is simpler than that of process (3.5). But process (3.5) is believed to have faster rate of convergence than that of process (3.1) due to the fact that , in general.

4. Algorithms for Pseudo-Contractive Self-Mappings

Let be a nonempty closed convex subset of a real Hilbert space , and let be a boundedly Lipschitzian pseudocontraction with a nonempty fixed point set , that is, . It follows from Lemma 1.5 that is closed and convex, so the metric projection operator is well defined.

In this section, adopting the regularization idea, we propose implicit and explicit algorithms for approximating a fixed point of , respectively. More precisely, given an arbitrary element , for each , it is easy to show that defined by

(4.1)

is a boundedly Lipschitzian -strong pseudocontraction. Then we have from Corollary 1.2 that has a unique fixed point. Denote by the unique fixed point of . Namely, is the only solution of the fixed point equation

(4.2)

Firstly, we prove that converges strongly to a fixed point of , as . Next, we give our explicit method based on this implicit method and Theorem 2.1.

Theorem 4.1.

Let be a nonempty closed convex subset of a real Hilbert space , and let be a boundedly Lipschitzian pseudocontraction with a nonempty fixed point set . Let , and is determined by (4.2). Then is bounded, and . Moreover, .

Proof.

First we show that is bounded. Take arbitrarily; noting that is a pseudocontraction, we have from (4.2) and the fact that

(4.3)

Hence

(4.4)

Clearly, . This says that is bounded. Since is boundedly Lipschitzian, so it is not difficult to show that is also bounded. Thus we can assert that the set of weak cluster points , where .

Next, we prove ; namely, if is a null sequence in such that as , then . To see this, using (4.2), we get

(4.5)

Clearly, this together with the boundedness of implies that as . Using Lemma 1.5, we have , that is, . Taking and in (4.4), we have

(4.6)

Taking the limit as , we see that .

Finally, we turn to proving that . Since is monotone, for any , we have

(4.7)

Observing , it follows from (4.2) that

(4.8)

Thus, we have

(4.9)

Since is continuous and , we obtain by taking the limit that

(4.10)

By Lemma 1.4, we get . This means that . Thus we have proved that .

Our following explicit method is motivated by Theorem 4.1, Theorem 2.1, and Zhou's iterative method in [17]. Given a sequence such that as . Denote by the unique fixed point of the mapping . Namely,

(4.11)

Theorem 4.1 says that . Observe that is boundedly Lipschitzian and -strongly pseudo-contractive.

Theorem 4.2.

Let be a nonempty closed convex subset of a real Hilbert space , and let be a boundedly Lipschitzian pseudocontraction with a nonempty fixed point set . Let , such that as . For arbitrary initial datum . Define iteratively a sequence in an explicit manner as follows:

(4.12)

where is the least positive integer satisfying

(4.13)

is the bounded Lipschitz constant of upon ,

(4.14)

and the control parameter sequence such that . Then converges strongly to .

Proof.

Recalling that, for each , is a boundedly Lipschitzian -strong pseudocontraction, we have by using Theorem 2.1 that

(4.15)

(4.16)

Since the condition implies , so there exists a least positive integer satisfying condition (4.13). Using Theorem 4.1, we have

(4.17)

In order to complete the proof, it suffices to show that as . To this end, we estimate . By the proof of Theorem 2.1, we assert that and for all . Thus we have from (4.11)–(4.16) that

(4.18)

Hence as , since as . Consequently, .