Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces
© Razani and Bagherboum; licensee Springer. 2013
Received: 19 February 2013
Accepted: 13 November 2013
Published: 3 December 2013
The purpose of this paper is to investigate some strong convergence as well as stability results of some iterative procedures for a special class of mappings. First, this class of mappings called weak Jungck -contractive mappings, which is a generalization of some known classes of Jungck-type contractive mappings, is introduced. Then, using an iterative procedure, we prove the existence of coincidence points for such mappings. Finally, we investigate the strong convergence of some iterative Jungck-type procedures and study stability and almost stability of these procedures. Our results improve and extend many known results in other spaces.
MSC:47H06, 47H10, 54H25, 65D15.
Czerwik  initiated the study of multivalued contractions in b-metric spaces.
if and only if ;
Then the pair is called a b-metric space.
It is clear that normed linear spaces, (or ) spaces (), (or ) spaces, Hilbert spaces, Banach spaces, hyperbolic spaces, ℝ-trees and spaces are examples of b-metric spaces.
Throughout this paper, is the set of nonnegative real numbers and Y is a nonempty arbitrary subset of a b-metric space . Moreover, will be denoted as the set of fixed points of . Approximately, all the concepts and results in metric spaces are extended to the setting of b-metric spaces (for more details, see ).
The first result on stability of T-stable mappings was introduced by Ostrowski  for the Banach contraction principle. Harder and Hicks  proved that the sequence generated by the Picard iterative process in a complete metric space converges strongly to the fixed point of T and is stable with respect to T, provided that T is a Zamfirescu mapping. Rhoades  extended the stability results of  to more general classes of contractive mappings. Ding  constructed the Ishikawa-type iterative process in a convex metric space. He showed that this process converges to the fixed point of T, provided that T belongs in the class which is defined by Rhoades.
for all and .
Osilike  considered a mapping T from a metric space X into itself satisfying the condition for some and for all . Furthermore, he extended some of the stability results in . Indeed, he proved T-stability for such a mapping with respect to Picard, Kirk, Mann, and Ishikawa iterations. Thereafter, Olatinwo  improved this concept to the context of multivalued weak contraction for the Jungck iteration in a complete b-metric space. In  this contractive condition was generalized by replacing this condition with , where and is monotone increasing with , and some stability results were proved. Recently, Olatinwo  extended this condition to , where is a subadditive comparison function and is monotone increasing with . He studied this contractive condition as a particular case of the class of φ-quasinonexpansive mappings (see ). Also, he proved some stability results as well as strong convergence results for the pair of nonself mappings in a complete metric space.
In 1968, Goebel  generalized the well-known Banach contraction principle by taking a continuous mapping S in place of the identity mapping, where S commuted with T and . In fact, he used two mappings for introducing the contractive condition as follows.
for all . In addition, Jungck , using a constructive method, proved the existence of a unique common fixed point of S and T, where .
A mapping T is said to be a Jungck-Zamfirescu contraction (JZ) if there exist real numbers α, β, and γ satisfying , such that for each , one has at least one of the following:
It is worth mentioning that a Jungck-Zamfirescu mapping is a (JR) mapping. In [, Proposition 3.3], a comparison of the above contractive conditions is established as follows.
(JC) ⇒ (JS) ⇒ (JQC);
(JC) ⇒ (JR) ⇒ (JQC);
(JS) and (JR) are independent;
(JR) ⇒ (WJC);
(JS) and (WJC) are independent;
(JQC) and (WJC) are independent;
reverse implications of (i), (ii), and (iv) are not true.
In this paper, a special class of mappings called a weak Jungck -contraction is introduced, and it is shown that it contains other known classes of Jungck-type contractive mappings. Then, using a Jungck-Picard iterative procedure, we investigate the existence of coincidence points and the uniqueness of the coincidence value of weak Jungck -contractive mappings. Also, some strong convergence as well as stability results of some Jungck-type iterative procedures (such as Jungck-Ishikawa etc.) are studied. These results play a crucial role in numerical computations for approximation of coincidence values of two nonlinear mappings.
In , Berinde introduced the concepts of comparison function and -comparison function with respect to the function . A function φ is called a comparison function if it satisfies the following:
(i φ ) φ is monotone increasing, i.e., ;
(ii φ ) The sequence for all , where stands for the n th iterate of φ.
If φ satisfies (i φ ) and
(iii φ ) converges for all ,
then φ is said to be a -comparison function.
Any -comparison function is a comparison function;
Any comparison function satisfies and for all ;
Any subadditive comparison function is continuous;
Condition (iii φ ) is equivalent to the following one:
holds for all and any .
Berinde  expanded the concept of -comparison functions in b-metric spaces to s-comparison functions as follows.
Definition 2.1 Let be a real number. A mapping is called an s-comparison function if it satisfies (i φ ) and
holds for all and any .
Applying results 4 and 1 regarding comparison functions, it is easy to conclude that every s-comparison function is a comparison function.
In the sequel, some lemmas which are useful to obtain our main results are stated.
Lemma 2.2 ()
Lemma 2.3 ()
If , and , then exists.
where φ is a subadditive comparison function. If , then .
where (identity mapping). Moreover, since any comparison function satisfies (ii φ ), hence . Also, we have from Lemma 2.2. Thus, inequality (2.2) implies that . □
where φ is a convex subadditive comparison function. If , then .
Therefore, there exists such that . Assume that . Since φ is continuous and converges, letting in (2.4), we get that , which is a contradiction. Hence and the desired conclusion follows. □
3 Weak Jungck -contractive mappings
In this section, the class of weak Jungck -contractive mappings which contains the class of Jungck φ-quasinonexpansive mappings is studied. Furthermore, it is showed that this class includes the various classes of contractive mappings which is introduced in Section 1.
for all .
The above definition was used in  when S is the identity mapping on .
It is obvious that any weak Jungck -contraction is also Jungck φ-quasinonexpansive, but the reverse is not true. The next example illustrates this matter.
then, taking , , we have . This shows that the class of φ-quasinonexpansive mappings properly includes the class of weak Jungck -contractive mappings.
In what follows, we prove that all the mappings introduced in Section 1 are in the class of weak Jungck -contractive mappings. It is clear that every Jungck contractive mapping is a weak Jungck -contractive mapping with and , where .
Proposition 3.3 Let be a b-metric space with parameter s, let Y be an arbitrary subset of X, and let . If T is a Jungck-Zamfirescu contraction (JZ), then T is a weak Jungck -contractive mapping if and . Moreover, it is a weak Jungck -contraction with and for all .
for all . It is clear that φ is an s-comparison function, where and and ψ is a monotone increasing function which is continuous from the right at . □
The following result shows that this fact is still true for a more general class of mappings.
Proposition 3.4 Let X, Y and be as in the above proposition. If T satisfies (JS), then T is a weak Jungck -contractive mapping, provided that . Furthermore, it is a weak Jungck -contraction with and for all .
for all . It shows that φ is an s-comparison function provided that and ψ is a monotone increasing function which is continuous at . □
Similar arguments illustrate that every (JR) mapping is a weak Jungck -contractive mapping, provided that . In fact, it is a weak Jungck -contraction with for all . Also, every (JQC) mapping is a weak Jungck -contractive mapping with for all , provided that .
4 Convergence results
In 1970, Takahashi  defined a convex structure on metric spaces. In this section a version of the convexity notion in b-metric spaces is stated. Then, using some Jungck-type iterative procedures, we prove the existence of coincidence points as well as the strong convergence theorems for the weak Jungck -contractive mappings.
A b-metric space X equipped with the convex structure W is called a convex b-metric space, which is denoted by .
for all , is a b-metric space with . Also, regarding the convexity of , we obtain that for all , that is, () is a convex b-metric space with . (In a similar way, the space () is a convex b-metric space.)
Now, the iterative procedures in a convex b-metric space are ready to be illustrated. From now on, it is assumed that is a b-metric space (resp. is a convex b-metric space) with parameter s and that are two nonself mappings on a subset Y of X such that , where is a complete subspace of X.
where is the initial approximation and f is a function.
The Jungck iteration (or Jungck-Picard iteration) is given from (4.2) for . This process was essentially introduced by Jungck  and it reduces to the Picard iterative process, when S is the identity mapping on ;
- 2.The Jungck-Krasnoselskij iteration is defined by (4.2) with(4.3)
- 3.The Jungck-Mann iteration is stated by (4.2) with(4.4)
where is a sequence of real numbers such that ;
- 4.The Jungck-Ishikawa iteration is introduced by (4.2) with(4.5)
where and are two sequences of real numbers such that .
It is worth noting that Olatinwo and Postolache  used the above iterative procedures in the setting of convex metric spaces.
Theorem 4.2 Suppose that is a b-metric space, and let be such that T is a weak Jungck -contractive mapping. Then S and T have a coincidence point. Moreover, for any , the sequence generated by the Jungck-Picard iterative process converges strongly to the coincidence value.
Taking the upper limit in the above inequality, we obtain . Hence, , i.e., z is a coincidence point.
From our assumptions on φ, it is impossible unless , that is, , which is a contradiction. □
Theorem 4.3 Let be a convex b-metric, and let be such that T is a weak Jungck -contractive mapping such that φ is a convex subadditive function. Let be a real sequence in such that . Then, for any , the sequence defined by the Jungck-Ishikawa iterative process converges strongly to the coincidence value of S and T.
Since φ is a convex subadditive comparison function, we have the desired result from Lemma 2.5. □
Based on Theorem 4.3, it is clear that the Jungck-Mann iterative process as well as the Jungck-Krasnoselskij iterative process converge;
In Hilbert spaces, assuming that in (JQC), Theorem 4.3 is an extension of the results in .
The following example shows that condition (3.1) in Theorem 4.3 is necessary.
where is endowed with the usual metric. Let and for . Then , which implies that if and if . Therefore, the Krasnoselskij iteration associated to T does not converge strongly to the coincidence value.
5 Stability results
This section is devoted entirely to the stability of some various iterative procedures in b-metric spaces. This concept was first proposed by Ostrowski  in metric spaces. Then, Czerwik et al. [26, 27] extended Ostrowski’s classical theorem in the setting of b-metric spaces. In addition, Singh et al.  introduced the stability and almost stability of Jungck-type iterative procedures in metric spaces. Below, we state these concepts in convex b-metric spaces.
Iterative procedure (4.2) will be called -stable if implies that .
Iterative procedure (4.2) will be called almost -stable if implies that .
The above definition reduces to the concept of the stability of iterative procedure due to Harder and Hicks  when S is the identity mapping on .
where is endowed with the usual metric. Let and for . If , then , and if , we have and for all . Thus ; i.e., the Picard iteration converges strongly to the coincidence value. But the Picard iteration is not -stable. Indeed, take the sequence given by , . One can see easily that the sequence does not converge to the coincidence value, while as .
Our next theorem is presented for a pair of mappings on a nonempty subset with values in b-metric spaces under a condition more general than the condition stated by Singh and Prasad [, Theorem 4.2]. Further, this theorem reduces the condition to the condition .
Theorem 5.2 Let be a b-metric space and T be a weak Jungck -contractive mapping such that φ is subadditive. For , let be the Picard iterative process defined by . Then the Jungck-Picard iteration is -stable.
Since φ is a subadditive s-comparison function, we get that sφ is a subadditive comparison function. Therefore, Lemma 2.4 yields that , that is, . □
Remark 5.1 Theorem 5.2 is a generalization of Theorem 3.2 of Singh and Alam , Theorem 3.4 of Singh et al. , Theorems 4.1 and 4.2 of Singh and Prasad , Theorem 1 of Osilike , Theorem 2 of Berinde , Theorem 2.1 of Bosede and Rhoades  as well as Corollary 2 of Qing and Rhoades .
The following example shows that the Ishikawa iterative process is not -stable.
Suppose that . As and , Lemma 2 of  implies that (the unique coincidence value of S and T).
It is clear that and , while . Therefore, the Ishikawa iterative procedure is not -stable, but it is almost -stable. (The almost -stability is shown in the following.)
The following theorem states that Jungck-Mann iterative and Jungck-Ishikawa iterative process are almost -stable provided that .
Theorem 5.3 Let be a convex b-metric space and let T be a weak Jungck -contractive mapping such that φ is a convex subadditive function. Let be a real sequence in such that . For , let be the Ishikawa iterative process given by (4.5). Then the Jungck-Ishikawa iteration is almost -stable.
Now, according to Lemma 2.3, exists. Therefore, there exists such that . Assume that . Since sφ is a subadditive comparison function, φ is continuous and for all . Then, letting in (5.3), we get , which is a contradiction. Hence, and this completes the proof. □
In a similar way, using Lemma 1 of  in place of Lemma 2.3 in the previous proof, by omitting the condition , one can prove that Theorem 5.3 holds in convex metric spaces. This indicates that the Ishikawa iterative process given Example 5.2 is almost -stable.
The authors express their gratitude to the referees for reading this paper carefully, providing valuable suggestions and comments, which improved the contents of this paper.
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