Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces

Let K be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and be a continuous strictly hemicontractive mapping. Under some conditions, we obtain that the Mann iteration method with error term converges strongly to a unique fixed point of T and is almost T-stable on K. As an application of our results, we establish strong convergence of a multi-step iteration process.

Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudo-contractive mapping from a bounded closed convex subset of (or ) into itself. Schu [2] generalized the result in [1] to both uniformly continuous strongly pseudo-contractive mappings and real smooth Banach spaces. Park [3] extended the result in [1] to both strongly pseudocontractive mappings and certain smooth Banach spaces. Rhoades [4] proved that the Mann and Ishikawa iteration methods may exhibit different behavior for different classes of nonlinear mappings. Harder and Hicks [5,6] revealed the importance of investigating the stability of various iteration procedures for various classes of nonlinear mappings. Harder [7] established applications of stability results to first-order differential equations. Afterwords, several generalizations have been made in various directions (see, for example, [2,4,8-21].

Let K be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and be a continuous strictly hemicontractive mapping. Under some conditions, we obtain that the Mann iteration method with error term converges strongly to a unique fixed point of T and is almost T-stable on K. As an application, we shall also establish strong convergence of a multi-step iteration process. The results presented here generalize the corresponding results in [2-4,10,11,22].

Let K be a nonempty subset of an arbitrary Banach space X and be its dual space. The symbols , and stand for the domain, the range and the set of fixed points of respectively (x is called a fixed point of T iff ). We denote by J the normalized duality mapping from X to defined by

Let T be a self-mapping of K.

Definition 1 The mapping T is called Lipshitzian if there exists such that

for all . If , then T is called non-expansive and if , T is called contraction.

Definition 2[10,22]

1. The mapping T is said to be pseudocontractive if the inequality

holds for each and for all .

2. T is said to be strongly pseudocontractive if there exists such that

for all and .

3. T is said to be local strongly pseudocontractive if for each , there exists such that

for all and .

4. T is said to be strictly hemicontractive if and if there exists such that

for all , and .

Clearly, each strongly pseudocontractive operator is local strongly pseudocontractive.

Definition 3[5-7]

Let K be a nonempty convex subset of X and be an operator. Assume that and defines an iteration scheme which produces a sequence . Suppose, furthermore, that converges strongly to . Let be any bounded sequence in K and put .

(1) The iteration scheme defined by is said to be T-stable on K if implies that .

(2) The iteration scheme defined by is said to be almost T-stable on K if implies that .

It is easy to verify that an iteration scheme which is T-stable on K is almost T-stable on K.

Lemma 4[3]

LetXbe a smooth Banach space. Suppose one of the following holds:

(1) Jis uniformly continuous on any bounded subsets ofX,

(2) for allx, yinX,

(3) for any bounded subsetDofX, there is asuch that

for all, wherecsatisfies

Then for anyand any bounded subsetK, there existssuch that

for alland.

Lemma 5[10]

Letbe an operator with. ThenTis strictly hemicontractive if and only if there existssuch that for alland, there existssatisfying

Lemma 6[4]

LetXbe an arbitrary normed linear space andbe an operator.

(1) IfTis a local strongly pseudocontractive operator and, thenis a singleton andTis strictly hemicontractive.

(2) IfTis strictly hemicontractive, thenis a singleton.

We now prove our main results.

Lemma 7Let, andbe nonnegative real sequences, and letbe a constant satisfying

where, for alland. Then, .

Proof By a straightforward argument, for ,

where we put . Note that . It follows from (3.1) that

For a given , there exists a positive integer k such that . Thus (3.2) ensures that

Letting yields . □

Remark 8

(i) If for each , then Lemma 7 reduces to Lemma 1 of Park [3].

(ii) If , then Lemma 7 reduces to Lemma 2.1 of Liu et al.[4].

Theorem 9LetXbe a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. LetKbe a nonempty closed bounded convex subset ofXandbe a continuous strictly hemicontractive mapping. Suppose thatis an arbitrary sequence inKand, andare any sequences insatisfying conditions (i) , (ii) , (iii) and (iv) .

For a sequenceinK, suppose thatis the sequence generated from an arbitraryby

and satisfying.

Letbe any sequence inKand defineby

where, such that.

Then

(a) the sequenceconverges strongly to a unique fixed pointqofT,

(b) implies that, so thatis almostT-stable onK,

(c) implies that.

Proof From (ii), we have , where as .

It follows from Lemma 6 that is a singleton. That is, for some .

Set . For all , it is easy to verify that

For given any and the bounded subset K, there exists a satisfying (2.6). Note that (ii), (iii), and the continuity of T ensure that there exists an N such that

where and t satisfies (2.7). Using (3.3) and Lemma 4, we infer that

for all .

Put

we have from (3.7)

Observe that , for all . It follows from Lemma 7 that

Letting , we obtain that , which implies that as .

On the same lines, we obtain

for all .

Suppose that . In view of (3.4) and (3.8), we infer that

for all .

Now, put

and we have from (3.9)

Observe that , and for all . It follows from Lemma 7 that

Letting , we obtain that , which implies that as .

Conversely, suppose that , then (iii) and (3.8) imply that

as , that is, as . □

Using the methods of the proof of Theorem 9, we can easily prove the following.

Theorem 10LetX, K, Tand, be as in Theorem 9. Suppose that, andare sequences insatisfying conditions (i), (iii)-(iv) and

If, , , andare as in Theorem 9, then the conclusions of Theorem 9 hold.

Corollary 11LetXbe a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. LetKbe a nonempty closed bounded convex subset ofXandbe a Lipschitz strictly hemicontractive mapping. Suppose thatis an arbitrary sequence inKand, andare any sequences insatisfying conditions (i) , (ii) , (iii) and (iv) .

For a sequenceinK, suppose thatis the sequence generated from an arbitraryby

and satisfying.

Letbe any sequence inKand defineby

where, such that.

Then

(a) the sequenceconverges strongly to a unique fixed pointqofT,

(b) implies that, so thatis almostT-stable onK,

(c) implies that.

Corollary 12LetX, K, Tandbe as in Corollary 11. Suppose that, andare sequences insatisfying conditions (i), (iii)-(iv) and

If, , , andare as in Corollary 11, then the conclusions of Corollary 11 hold.

Corollary 13LetXbe a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. LetKbe a nonempty closed bounded convex subset ofXandbe a continuous strictly hemicontractive mapping. Suppose thatis a sequence insatisfying conditions (i) and (ii) .

For a sequenceinK, suppose thatis the sequence generated from an arbitraryby

and satisfying.

Letbe any sequence inKand defineby

where, such that.

Then

(a) the sequenceconverges strongly to a unique fixed pointqofT,

(b) implies that, so thatis almostT-stable onK,

(c) implies that.

Corollary 14LetXbe a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. LetKbe a nonempty closed bounded convex subset ofXandbe a Lipschitz strictly hemicontractive mapping. Suppose thatis a sequence insatisfying conditions (i) and (ii) .

For a sequenceinK, suppose thatis the sequence generated from an arbitraryby

and satisfying.

Letbe any sequence inKand defineby

where, such that.

Then

(a) the sequenceconverges strongly to a unique fixed pointqofT,

(b) implies that, so thatis almostT-stable onK,

(c) implies that.

Khan et al.[23] have introduced and studied a multi-step iteration process for a finite family of selfmappings. We now introduce a modified multi-step process as follows:

Let K be a nonempty closed convex subset of a real normed space E and () be a family of selfmappings.

Algorithm 1 For a given , compute the sequence by the iteration process of arbitrary fixed order ,

which is called the modified multi-step iteration process, where , .

For , we obtain the following three-step iteration process:

Algorithm 2 For a given , compute the sequence by the iteration process:

where , and are three real sequences in .

For , we obtain the Ishikawa [24] iteration process:

Algorithm 3 For a given , compute the sequence by the iteration process

where and are two real sequences in .

If , , in (4.3), we obtain the Mann iteration process [14]:

Algorithm 4 For any given , compute the sequence by the iteration process

where is a real sequence in .

Theorem 15LetKbe a nonempty closed bounded convex subset of a smooth Banach space Xand () be selfmappings ofK. Letbe a continuous strictly hemicontractive mapping. Let, be real sequences insatisfying, and. For arbitrary, define the sequenceby (4.1). Thenconverges strongly to a point in.

Proof By applying Corollary 13 under assumption that is continuous strictly hemicontractive mapping, we obtain Theorem 15 which proves strong convergence of the iteration process defined by (4.1). We will check only the condition by taking and ,

Now, from the condition , it can be easily seen that . □

Corollary 16LetKbe a nonempty closed bounded convex subset of a smooth Banach space Xand () be selfmappings ofK. Letbe a Lipschitz strictly hemicontractive mapping. Let, be real sequences insatisfying, and. For arbitrary, define the sequenceby (4.1). Thenconverges strongly to a point in.

Remark 17 Similar results can be found for the iteration processes with error terms, we omit the details.

The authors declare that they have no competing interests.

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The third author gratefully acknowledges the support from the Ministry of Education and Science of Republic Serbia. The fourth author gratefully acknowledges the financial support provided by the University of Tabuk through the project of international cooperation with the University of Texas at El Paso. We are also thankful to the editor and the referees for their suggestions for the improvement of the manuscript.

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