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Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 331 (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.
Definition 1.1 Let X be a set and let be a given real number. A function is said to be a b-metric if and only if for all the following conditions are satisfied:
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.
A mapping T is said to be a φ-quasinonexpansive if and there exists a function such that
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.
A mapping T is called a Jungck contraction if there exists a real number such that
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:
A mapping T is said to be a contractive mapping satisfying (JS), (JR) or (JQC) if there exists a constant such that for any ,
A mapping T is said to be a weak Jungck contraction if there exist two constants and such that for all ,
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:
There exist , and a convergent series of nonnegative terms such that
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
(iv φ ) There exist , , and a convergent series of nonnegative terms such that
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 ()
Let be a comparison function, and let be a sequence of positive numbers such that . Then
Lemma 2.3 ()
Let , , and be sequences of nonnegative real numbers satisfying the inequality
If , and , then exists.
Lemma 2.4 Suppose that and are two sequences of nonnegative numbers such that
where φ is a subadditive comparison function. If , then .
Proof The monotone increasing and the subadditivity of φ together with inequality (2.1) imply that
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 . □
Lemma 2.5 Let be a real sequence in , let be a sequence of positive numbers such that converges, and let be a sequence of nonnegative numbers such that
where φ is a convex subadditive comparison function. If , then .
Proof Since for all , using a straightforward induction and (2.3), one can obtain
for all . Now, yields that . Then
which implies that
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.
Definition 3.1 Let Y be an arbitrary subset of a b-metric space , and let be such that z is a coincidence point of S and T, i.e., . We say that T is a Jungck φ-quasinonexpansive mapping with respect to S if there exists a function such that
for all .
The above definition was used in  when S is the identity mapping on .
Definition 3.2 Let Y be an arbitrary subset of a b-metric space and . A mapping T is said to be a weak Jungck -contractive mapping with respect to S if there exist an s-comparison function and a monotone increasing function with upper semicontinuity from the right at such that for all ,
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.
Example 3.1 Let be given by and
where is endowed with the usual metric. It is easy to see that T satisfies the following property:
for all , , and . But T is not a weak Jungck -contractive mapping. Indeed, if there exist a 1-comparison function φ and a monotone increasing function ψ with upper semicontinuity from the right at such that for all ,
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 .
Proof If , then for all ,
which implies that
Similarly, if , then 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 .
Proof If , then according to the inequality
for all . Moreover,
On the other hand, if , then
for all . Also
Now, we take
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.
Definition 4.1 Let be a b-metric space. A mapping is said to be a convex structure on X if for each and ,
A b-metric space X equipped with the convex structure W is called a convex b-metric space, which is denoted by .
Example 4.1 The space () consisting of all the sequences of real numbers for which converges, with the function given 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.
Let be the sequence generated by an iterative procedure involving the mapping T and S, that is,
where is the initial approximation and f is a function.
In the sequel, we discuss several special cases of (4.2):
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 ;
The Jungck-Krasnoselskij iteration is defined by (4.2) with(4.3)
The Jungck-Mann iteration is stated by (4.2) with(4.4)
where is a sequence of real numbers such that ;
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.
Proof First, we prove that S and T have at least one coincidence point in Y. To do this, let be the Jungck-Picard iterative process defined by and . Taking and in (3.1), we obtain
which implies that
Since for all , is a Cauchy sequence. Also, is complete, so has a limit in , that is, there exists such that . Hence, and
Taking the upper limit in the above inequality, we obtain . Hence, , i.e., z is a coincidence point.
Now, we show that S and T have a unique coincidence value. Assume that S and T have two coincidence values such that . Then there exist such that and . Thus, we conclude that
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.
Proof Theorem 4.3 states the existence of coincidence points in Y and one can obtain the uniqueness of coincidence value in a similar way. We now show that the Jungck-Ishikawa iteration given by , where for each , converges to , where z is a coincidence point of S and T. Using (3.1), we have
Substituting (4.7) in (4.6), it follows that
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.
Example 4.2 Let be given by and
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.
Definition 5.1 Let be a convex b-metric space, let Y be a subset of X, and let be such that . For any , let the sequence , generated by iterative procedure (4.2), converges to p. Also, let be an arbitrary sequence and let , . Then
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 .
Example 5.1 Let be given by and
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.
Proof Note that, by Theorem 4.2, there exists a coincidence point such that converges to . Suppose that and define , where . Assume that . Then we have
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.
Example 5.2 Let be given by and , where ℝ is again endowed with the usual metric. Then T is a weak Jungck -contraction. Let be a sequence generated by the Ishikawa iterative process with and . Then
Suppose that . As and , Lemma 2 of  implies that (the unique coincidence value of S and T).
To prove the fact that the Ishikawa iteration is not -stable, we use the sequence given by . Then
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.
Proof In view of Theorem 4.3, there exists a coincidence point such that converges to . Suppose that , , , where . Assume that . Then
From (5.1) and (5.2), we conclude that
Since φ is an s-comparison function, sφ is a comparison function. Thus, inequality (5.3) implies that
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.
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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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Razani, A., Bagherboum, M. Convergence and stability of Jungck-type iterative procedures in convex b-metric spaces. Fixed Point Theory Appl 2013, 331 (2013). https://doi.org/10.1186/1687-1812-2013-331
- weak Jungck -contractive mapping
- iterative procedure
- coincidence point
- convex b-metric space