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Some discussion on the existence of common fixed points for a pair of maps
Fixed Point Theory and Applications volume 2013, Article number: 187 (2013)
Abstract
In this paper, the concepts of conditionally sequential absorbing and pseudo-reciprocal continuous maps are introduced in connection to giving a brief discussion on the role of various types of commutativity (e.g., weakly compatible, occasionally weakly compatible, subcompatible, pseudo-compatible, etc.) and continuity-type conditions (e.g., reciprocal, weak reciprocal, g-reciprocal, conditionally reciprocal, subsequential and sequential continuity of type and ) in the context of existence of common fixed points of a pair of maps. Here, the utility of newly introduced maps (i.e., conditionally sequential absorbing and pseudo-reciprocal continuous) in view of common fixed points for a pair of maps satisfying contractive as well as nonexpansive Lipschitz-type conditions is shown.
MSC:47H10, 54H25.
1 Introduction and preliminaries
The classical results of Banach [1] (see also [2]) and Edelstein [3] have been the inspiration for many researchers working in the area of metric fixed point theory. In 1976, Jungck [4] generalized the Banach contraction principle by introducing the idea of commuting maps and settled the historical open problem that a pair of commuting and continuous self-mappings on the unit interval need not have a common fixed point [5, 6]. This result of Jungck [4] made foundation to study and investigate common fixed points and their applications in various other branches of mathematical sciences in the last five decades. Since then many fixed point theorists have attempted to find weaker forms of commutativity and continuity that may ensure the existence of a common fixed point for a pair of self-mappings on a metric space. Systematic observations and comparison of commutativity-type mappings are available in [7].
Proving a common fixed point for mappings satisfying Banach-type contractive conditions involves the following steps: step one is to show that there exists a Cauchy sequence which converges to a point in X (where X is complete); the second step is to show the existence of a coincidence point by assuming suitable weaker forms of commutativity and continuity conditions; and step three automatically gives rise to the fact that this coincidence point is a unique common fixed point due to the contractive condition. Observing carefully step two, one finds that showing the existence of a coincidence point for involved maps is nothing but assuming the existence of a coincidence point itself by a suitable choice of weaker forms of commutativity and continuity conditions (see, for instance, [8–16]).
Keeping the above facts in mind, Jungck and Rhoades [17] utilized the notion of occasionally weakly compatible maps introduced in [18] (as a generalization of weakly compatible maps) for those pairs which do have at least one coincidence point where the maps commute (it is well known that a pair of maps without a coincidence point is always vacuously weakly compatible) and obtained fixed point theorems for such maps.
On the other hand, Singh and Mishra in [19] illustrated a technique to prove the existence of a coincidence point without assuming continuity and commutativity-type conditions. Whereas the result of Suzuki and Pathak [20] does not involve any continuity-type conditions to prove the existence of a coincidence point as well as a common fixed point for a pair of maps (but they used weaker forms of commutativity conditions). It is also worth mentioning that Suzuki and Pathak [20] did not provide any illustrative examples to discuss and highlight the above facts. It is also important to note that none of the results of Jungck [9], Singh and Mishra [19] and Suzuki and Pathak [20] can be obtained from each other due to their different characteristics. These facts are illustrated in this paper via Example 2.6 (p.10).
Motivated by the works of Jungck and Rhoades [17], Bouhadjera and Thobie [21] (respectively Hussain et al. [22] and Sintunavarat and Kumam [23]) introduced the notion of subcompatible maps (respectively the notions of occasionally weakly operator and occasionally weakly biased maps) as generalization of occasionally weakly compatible maps and obtained fixed point theorems for such maps. However, Dorić et al. in [24] (respectively Alghamdi et al. [25]) showed that in the event of a pair of single-valued maps, the notion of occasionally weakly compatible (respectively occasionally weakly operator and occasionally weakly biased maps) reduces to weak compatibility due to the unique coincidence point of the involved maps, which is always ensured by underlying contractive conditions. Hence weak compatibility remains the minimal commutativity condition for the existence of a common fixed point for a contractive pair of maps. In view of these, the various results for occasionally weakly compatible maps (occasionally weakly operator and occasionally weakly biased maps) obtained in [17, 21, 22, 26–30], which were used under contractive conditions, do not yield real generalizations (see also [31, 32]). Considering these facts, Pant and Pant [33] (see also [34]) redefined the concept of occasionally weakly compatibility by introducing the idea of conditionally commuting maps which constitute a proper setting in the context of studying non-unique common fixed points for a pair of maps.
Possibly the first common fixed point theorem (respectively fixed point theorem) without any continuity requirement was established by Pant [12, 35] when he introduced the idea of noncompatible and reciprocal continuous maps. (However, the origin of metric fixed point theory for a single mapping without continuity requirement can be traced back to Kannan [36].) Recently, Pant et al. [37] and Pant and Bisht [38] generalized the notion of reciprocal continuity by introducing weak reciprocal continuity and conditionally reciprocal continuity and utilized the same to obtain some common fixed point theorems. In this connection, the recent paper of Gopal et al. [39] is also readable.
Motivated by the results of Pant and Bisht [38, 40], we introduce the concept of conditionally sequential absorbing and pseudo-reciprocal continuous maps, which allows us to give a comparative study of various types of commutativity conditions (e.g., compatible, weakly compatible, occasionally weakly compatible, conditionally commuting, pseudo-compatible) and continuity-type conditions (e.g., reciprocal, weak reciprocal, g-reciprocal, conditionally reciprocal, subsequential and sequential continuity of type and ) with these newly introduced notions in the context of existence of common fixed points of a pair of maps.
The following relevant known definitions (and results) will be needed in our subsequent discussion. A pair of self-mappings defined on a metric space is said to be
-
(i)
compatible [9] iff whenever is a sequence in X such that for some t in X.
It is clear from the above definition that f and g will be noncompatible [35] if there exists a sequence in X such that for some t in X, but is either nonzero or non-existent;
-
(ii)
f-compatible [41] if whenever is a sequence in X such that for some t in X;
-
(iii)
g-compatible [41] if whenever is a sequence in X such that for some t in X;
-
(iv)
weakly compatible [42] if the mappings commute at their coincidence points, i.e., for some implies ;
-
(v)
occasionally weakly compatible [17] if there exists a point x in X that is a coincidence point of f and g at which f and g commute;
-
(vi)
subcompatible [21] iff there exists a sequence in X such that with for some ;
-
(vii)
conditionally commuting [33] if they commute on a nonempty subset of the set of coincidence points whenever the set of their coincidence point is nonempty;
-
(viii)
conditionally compatible [34] iff, whenever the set of sequences satisfying is nonempty, there exists a sequence such that (say) and ;
-
(ix)
pseudo-compatible [40] iff, whenever the set of sequences satisfying is nonempty, there exists a sequence such that (say), ; and for any associated sequence of .
We also recall that a pair of self-mappings defined on a metric space is said to be
-
(i)
reciprocally continuous [12, 43] iff and whenever is a sequence in X such that for some t in X;
-
(ii)
weakly reciprocally continuous [37] if or whenever is a sequence in X such that for some t in X;
-
(iii)
conditionally reciprocally continuous (CRC) [38] if, whenever the set of sequences satisfying is nonempty, there exists a sequence satisfying (say) such that and ;
-
(iv)
g-reciprocally continuous [40] iff and whenever is a sequence such that for some t in X;
-
(v)
sequentially continuous of type [39] iff there exists a sequence in X such that for some satisfying and ;
-
(vi)
sequentially continuous of type [39] iff there exists a sequence in X such that for some satisfying and ;
-
(vii)
subsequentially continuous [21] iff there exists a sequence in X such that and with for some .
Theorem 1.1 [40]
Let f and g be g-reciprocally continuous self-mappings of a complete metric space such that
-
(i)
;
-
(ii)
, .
If f and g are pseudo-compatible, then f and g have a unique common fixed point.
Theorem 1.2 [40]
Let f and g be g-reciprocally continuous noncompatible self-mappings of a metric space such that
-
(i)
;
-
(ii)
, where ;
-
(iii)
,
whenever the right-hand side is nonzero. If f and g are pseudo-compatible, then f and g have a unique common fixed point.
Theorem 1.3 [34]
Let f and g be conditionally compatible self-mappings of a metric space satisfying
whenever the right-hand side is nonzero. If f and g are noncompatible and reciprocally continuous, then f and g have a common fixed point.
Theorem 1.4 [38]
Let f and g be conditionally reciprocal continuous self-mappings of a complete metric space such that
-
(i)
;
-
(ii)
, .
If f and g are either compatible or g-compatible or f-compatible, then f and g have a unique common fixed point.
Theorem 1.5 [44]
Let be a complete metric space, let f and g be two noncompatible self-mappings on X satisfying
where is a continuous from right and nondecreasing function such that for all . Assume that
-
(i)
,
-
(ii)
for all and
-
(iii)
, whenever .
Then f and g have a unique common fixed point. Also, f and g are discontinuous at the common fixed point.
2 Main results
We begin with the following example.
Example 2.1 Let and d be the usual metric on X. Define self-mappings f and g on X as follows:
Then we can see that and the pair is g-reciprocally continuous. It can be verified that for all with . Thus, f and g satisfy all the conditions of Theorem 1.1 except pseudo-compatibility. For the pseudo-compatibility, consider the only existent sequence , then we have , but , , and so . Also note that the pair is not compatible. Here, has no coincidence point therefore it is also not an occasionally weakly compatible but vacuously weakly compatible pair.
This suggests that pseudo-compatible is stronger than weakly compatible (and occasionally weakly compatible) in the context of Theorem 1.1 (such an observation is missing in [40]).
The above example motivated us to define the following.
Definition 2.1 Two self-mappings f and g of a metric space are called conditionally sequential absorbing if, whenever the set of sequences satisfying is nonempty, there exists a sequence satisfying (say) such that and .
Example 2.2 Let and let d be the usual metric on X. Define as follows:
Then the maps are conditionally sequential absorbing. To view this, consider the constant sequence . However, the pair is not weakly compatible as they do not commute at their coincidence point . It may be noted that and are two coincidence points of f and g. But in respect of the unique coincidence point (common fixed point), conditionally sequential absorbing always implies weakly compatible and hence occasionally weakly compatible and pseudo-compatible, because the maps naturally commute at their unique coincidence point (common fixed point).
Example 2.3 Let and let d be the usual metric on X. Define as follows:
Then f and g are weakly compatible but not conditionally sequential absorbing. Here, and are two coincidence points.
Remark 2.1 In Example 2.1, the pair is vacuously weakly compatible but not conditionally sequential absorbing and not pseudo-compatible. Note that f and g do not have any coincidence point. In Example 2.2, the pair is conditionally sequential absorbing but not weakly compatible. In Example 2.3, the pair is weakly compatible but not conditionally sequential absorbing.
Thus, as definitions, weakly compatible, pseudo-compatible and conditionally sequential absorbing are very different. However, in the context of a unique coincidence point, conditionally sequential absorbing is stronger than weakly compatible, which will be shown in our Example 2.6.
Example 2.4 Let and let d be the usual metric on X. Define as follows:
Let us consider the sequence for . Then
Thus f and g are conditionally reciprocal continuous and subsequentially continuous. We can see that f and g are neither weak reciprocal continuous nor g-reciprocal continuous. To see this, consider the sequence for , then
Note that f and g do not have a coincidence point.
Example 2.5 Let and let d be the usual metric on X. Define as follows:
Then it is easy to see that the pair is reciprocal continuous, weak reciprocally continuous and conditionally reciprocally continuous but neither subsequentially continuous nor sequentially continuous of type and . Note that the pair has no coincidence point.
In view of the above examples, we observe that in the event of no coincidence point, subsequential continuity as well as sequential continuity of type and are different from reciprocal continuity (respectively g-reciprocal and conditionally reciprocal continuity). However, in the context of a unique coincidence point (common fixed point), subsequential continuity as well as sequential continuity of type and are equivalent to conditionally reciprocal continuity.
The motivation of the following definition can be predicted from the proof of the last step in our Theorem 2.1.
Definition 2.2 Two self-mappings f and g of a metric space are called pseudo-reciprocal continuous (PRC) (with respect to conditionally sequential absorbing) if, whenever the set of sequences satisfying is nonempty, there exists a sequence (satisfying (say), and ) such that and .
Common fixed point theorems
Assume that are two functions such that
-
(a)
ϕ is nondecreasing, continuous and for every ;
-
(b)
ψ is nondecreasing, right-continuous, and for every .
To prove our first result, we use the following lemma.
Lemma 2.1 [45]
For every function , let be the iterate of ψ. Then the following hold:
-
(i)
if ψ is nondecreasing, then for each , implies ;
-
(ii)
if ψ is right-continuous with for , then .
Theorem 2.1 Let f and g be two pseudo-reciprocal continuous (w.r.t. conditionally sequential absorbing) self-mappings of a complete metric space such that , and let be two functions satisfying (a) and (b). If for all ,
where
then f and g have a unique common fixed point provided is conditionally sequential absorbing.
Proof Let and since , so we have a sequence defined by
Now we show that is a Cauchy sequence. We have
If we suppose , then
which is a contradiction. Therefore
Similarly,
If for some n we have either or , then by condition (2.1) we obtain that the sequence is definitely constant and thus it is a Cauchy sequence. Suppose for each n, then from condition (2.1) we have
and for all ,
Now, we have
and then, by Lemma 2.1(ii),
Now we prove that is Cauchy.
Suppose not, then such that for infinite value of m and n with . This assumes that there exist two sequences , of natural numbers with such that
It is not restrictive to suppose that is the least positive integer exceeding and satisfying (2.5). We have
Then
We note
and therefore
Now, we have
where as and for all k. Then from
as , ϕ being continuous and ψ right-continuous, we get
This is a contradiction. Therefore is a Cauchy sequence. Since is a complete metric space, therefore such that
Since the pair is conditionally sequential absorbing, therefore there exists a sequence in X such that , (say) satisfying
and by pseudo-reciprocal continuity (w.r.t. conditionally sequential absorbing) of , we have
In view of (2.11) and (2.12), we get , i.e., u is a common fixed point of f and g. The uniqueness of a common fixed point follows easily from contractive condition (2.1). □
Example 2.6 Let with the usual metric d. Define self-maps f and g as follows:
Then f and g satisfy all the conditions of Theorem 2.1 with . Here, f and g are conditionally sequential absorbing and pseudo-reciprocal continuous (w.r.t. conditionally sequential absorbing) in respect of the constant sequence . Let us consider the sequence , then
Thus, is not a reciprocal as well as not a g-reciprocal continuous pair. Also the pair is neither compatible, f-compatible nor g-compatible.
If we take and , , then it can be verified that f and g satisfy contraction condition (2.1) with . Here, is the unique common fixed point of f and g, which is also a point of discontinuity.
On the other hand, notice that at , f and g do not satisfy the condition
used in Theorem 1.5. Here, it is worth noting that none of the earlier relevant theorems, for example, Theorem 1.1, Theorem 1.4 and Theorem 1.5, can be used in the context of this example. One more interesting part of this example is that neither nor is closed. Thus the result of Singh and Mishra [19] cannot be applicable in the context of this example.
Theorem 2.2 Let f and g be pseudo-reciprocal continuous (w.r.t. conditionally sequential absorbing) and noncompatible self-mappings of a metric space satisfying
where . If f and g are conditionally sequential absorbing, then f and g have a unique common fixed point.
Proof Since f and g are noncompatible maps, there exists a sequence in X such that and for some t in X but either or the limit does not exist. Also, the pair is conditionally sequential absorbing; therefore, there exists a sequence in X such that (say) with and . Now, by the pseudo-reciprocal continuity (w.r.t. conditionally sequential absorbing) of the pair , we have and . In view of these limits, we get u is a common fixed point of f and g.
Now, suppose that there exists another common fixed point w of f and g such that . Then, on using (2.13), we have
Thus, we have , a contradiction and hence . □
Example 2.7 Again consider Example 2.6 wherein the pair satisfies all the conditions of Theorem 2.2 for all . Note that at , f and g do not satisfy the condition
whenever the right-hand side is nonzero. Thus Theorem 2.2 is a genuine extension and improvement of Theorem 1.2 due to Pant and Bisht [40].
Observation The proof of Theorem 2.1, Theorem 2.2 and examples above immediately suggest us defining another type of continuity as follows.
Definition 2.3 Two self-mappings f and g of a metric space are called pseudo-reciprocal continuous (PRC) (with respect to pseudo-compatible) if whenever the set of sequences satisfying is nonempty, there exists a sequence (satisfying (say); ; and for any associated sequence of ) such that and .
However, the notions of pseudo-compatibility and pseudo-reciprocal continuity (w.r.t. pseudo-compatibility) are no more applicable in the context of the existence of non-unique common fixed points for a pair of maps. This fact is illustrated in Example 2.11 below. At the same time, conditionally sequential absorbing and pseudo-reciprocal continuity (w.r.t. conditionally sequential absorbing) are easily applicable.
Theorem 2.3 Let f and g be reciprocal (or g-reciprocal) continuous and noncompatible self-mappings of a metric space satisfying (2.13). Then the pair has a unique common fixed point provided it is conditionally sequential absorbing. Moreover, f and g are discontinuous at the common fixed point.
Proof Since f and g are noncompatible, there exists a sequence in X such that for some , but is either nonzero or not existent. Also, since f and g are conditionally sequential absorbing and , there exists a sequence in X, satisfying (say), such that and . The reciprocal continuity of the pair implies that and . Thus, in view of these limits, we obtain . If we consider the pair g-reciprocal continuous, then we have and . Since , so we have . Now, suppose . On using (2.13), we get , a contradiction and hence . Thus u is a common fixed point of f and g. Applying (2.13), we can show the uniqueness of the common fixed point.
We now show that f and g are discontinuous at the common fixed point u. If possible, suppose f is continuous at u. Then, considering the sequence of the present theorem and on using (2.13), we get and hence by the continuity of f, we have and . Now, reciprocal (or g-reciprocal) continuity of the pair implies that . This further yields that , which contradicts the fact that is either nonzero or non-existent. Hence f is discontinuous at the fixed point.
Next, suppose that g is continuous at u. Then, for the sequence , we get and . If is reciprocal continuous, then we have , and if it is g-reciprocal continuous, then on using (2.13), we get . Thus, we obtain , a contradiction. Therefore f and g are discontinuous at their common fixed point. □
Example 2.8 Let with the usual metric d. Define as follows:
Then f and g satisfy all the conditions of Theorem 2.3. It can be verified in this example that f and g satisfy contractive condition (2.13) for all . To see that f and g are noncompatible, consider the sequence in X such that , then , , , , and so . Also here the pair is g-reciprocal continuous. To see this, let be a sequence in X such that for some t in X. Then , or , and . The pair is conditionally sequential absorbing in respect of the constant sequence given by . Here, is the unique common fixed point where f and g are discontinuous.
Note that at , f and g do not satisfy the condition
whenever the right-hand side is nonzero. Also notice that at , f and g do not satisfy
Thus, Theorem 2.3 is a genuine extension and improvement of Theorem 1.2 due to Pant and Bisht [40] and Theorem 1.5 due to Rezapour and Shahzad [44].
In the absence of contractive condition (2.13), the following corollaries are straightforward from Theorems 2.2 and 2.3.
Corollary 2.1 Let f and g be pseudo-reciprocal continuous (w.r.t. conditionally sequential absorbing) and noncompatible self-mappings of a metric space . Then f and g have a common fixed point provided it is conditionally sequential absorbing.
Corollary 2.2 Let f and g be reciprocal (or g-reciprocal) continuous and noncompatible self-mappings of a metric space . Then the pair has a common fixed point provided it is conditionally sequential absorbing.
The following examples illustrate the above corollaries.
Example 2.9 Consider and let d be the usual metric on X. Define as
Here f and g satisfy all the conditions of Corollary 2.1. In view of the constant sequence or , the pair is conditionally sequential absorbing and pseudo-reciprocal continuous (w.r.t. conditionally sequential absorbing). For noncompatibility as well as non-reciprocal continuity, let us consider the sequence , then we have
and so . Here, 2 and 11 are two common fixed points of f and g. Also the pair is not weakly compatible as f and g do not commute at their coincidence point .
Note that at and the present example does not satisfy condition (2.13) for any and also Lipschitz-type condition used in [33] for any . Also notice that at , the involved maps do not satisfy any of the conditions:
-
(i)
,
-
(ii)
,
-
(iii)
, and
-
(iv)
,
whenever the right-hand side is nonzero. Here, it is worth noting that none of the Theorem 1.3 due to Pant and Bisht [34] and the main results contained in Pant and Pant [33] and Gopal et al. [46] can be used in the context of Corollary 2.1.
Example 2.10 Consider and let d be the usual metric on X. Define as
In this example the pair is noncompatible as well as reciprocal continuous and satisfies all the conditions of Corollary 2.2. Let us consider the sequence , then , and
therefore , and so is noncompatible. Here, 2 and 11 are two common fixed points of f and g.
Finally, we present an example which shows that the requirement of conditionally sequential absorbing property is necessary for producing common fixed points of mappings satisfying non-expansive or Lipschitz-type conditions besides exhibiting the limitations of commuting properties of the pairs utilized in earlier related results of Pant and Bisht [34], Pant and Pant [33] and Jungck and Rhoades [17].
Example 2.11 Let endowed with the usual metric d and by
Then by a routine calculation, it can be verified that and for all , where . Also, f and g are a noncompatible and weakly commuting (and hence occasionally weakly compatible and conditionally commuting) pair. In order to show that is noncompatible, the sequence ; , satisfies the requirements. Also, it is straightforward to verify that the pair is pseudo-compatible as well as pseudo-reciprocal continuous (w.r.t. pseudo-compatible), but the pair is not conditionally sequential absorbing in respect of or . On the other hand, at , it can be verified that the mappings f and g do not satisfy any one of the conditions described by (i), (ii), (iii) or (iv) mentioned earlier. Notice that the estimated pair has no common fixed point.
Observations The following definitions can be considered as variants of conditionally sequential absorbing. Two self-mappings f and g of a metric space are called conditionally sequential absorbing
-
(i)
of type (A) if, whenever the set of sequences satisfying is nonempty, there exists a sequence satisfying (say) such that and ;
-
(ii)
of type (B) if, whenever the set of sequences satisfying is nonempty, there exists a sequence satisfying (say) such that and ;
-
(iii)
of type (C) if, whenever the set of sequences satisfying is nonempty, there exists a sequence satisfying (say) such that and .
We can have some more variants by interchanging the place of f and g. In respect of these variants, we can also define the corresponding pseudo-reciprocal continuity, for example, two self-mappings f and g of a metric space are called pseudo-reciprocal continuous of type (A) if, whenever the set of sequences satisfying is nonempty, there exists a sequence (satisfying (say), and ) such that and .
Remark 2.2 The conclusion of our previous results will remain true if we replace the conditionally sequential absorbing and pseudo-reciprocal continuity by any one of the above variants of conditionally sequential absorbing and corresponding pseudo-reciprocal continuity. However, in the context of a unique coincidence or common fixed point, all these variants coincide with each others.
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Acknowledgements
The second author thanks for the support of the King Mongkut’s University of Technology Thonburi (KMUTT) and the third author is supported by CSIR, Govt. of India, grant number 25(0215)/13/EMR-II.
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Patel, D.K., Kumam, P. & Gopal, D. Some discussion on the existence of common fixed points for a pair of maps. Fixed Point Theory Appl 2013, 187 (2013). https://doi.org/10.1186/1687-1812-2013-187
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DOI: https://doi.org/10.1186/1687-1812-2013-187
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