Faintly compatible mappings and common fixed points
© Bisht and Shahzad; licensee Springer. 2013
Received: 18 January 2013
Accepted: 15 April 2013
Published: 18 June 2013
In this paper we introduce a generalization of the concept of compatible mappings, and using that condition, we obtain some new fixed point theorems under both contractive and noncontractive conditions, which may allow the existence of a common fixed point or the existence of multiple fixed or coincidence points. We also manifest that the new concept is a necessary condition for the existence of a common fixed point.
1 Introduction and preliminaries
The question of the existence of common fixed points of commuting continuous self-maps of a compact convex set had remained an open problem for a fairly long time. It was only in 1969 that Boyce  and Huneke  independently proved that there exist two continuous commuting self-maps of the unit interval without a common fixed point. Thus, the study of common fixed points of a pair of self-mappings satisfying contractive-type conditions becomes interesting in view of the fact that even commuting continuous mappings on such nicely behaved entities as compact convex sets may fail to have a common fixed point. When we extend such studies to the class of noncommuting contractive-type mapping pair, it becomes still more interesting.
In 1982, Sessa  obtained the first weaker version of commutativity by introducing the notion of weak commutativity. This concept was further generalized by Jungck when he defined the concept of compatible mappings . Extending weak commuting mappings, Pant  introduced the notion of R-weak commutativity. In 1996 Jungck again generalized the notion of compatible mappings by introducing weakly compatible mappings . In 2008, Al-Thagafi and Shahzad  weakened the notion of nontrivial weakly compatible maps by introducing a new notion of occasionally weakly compatible (in short, owc) maps. In the recent work, Pant and Pant  redefined the notion of occasionally weakly compatible mappings by conditional commutativity. Over the past few years, generalizations of weakly compatible mappings and owc have been extensively used to study common fixed points of contractive mappings. In a recent note Alghamdi et al. (, see also  and references therein) have shown that many recent results which employ several weaker noncommuting notions are not real generalizations of previously existing results on weakly compatible mappings. They have also shown that many of the generalized commutativity conditions including owc fall in the subclass of weak compatibility in the setting of a unique common fixed point (or a unique point of coincidence). Those new classes of noncommuting notions are interesting but contractive conditions do not provide an ideal setting for the application of these concepts. For proper applications of these notions, one should look to mappings satisfying nonexpansive conditions, Lipschitz-type conditions or some other general conditions .
In 2011, Haghi et al.  proved a powerful lemma and showed that some coincidence and common fixed point generalizations in fixed point theory are merely consequences of the corresponding fixed point theorems existing in the literature (for more details, see ).
Fixed point theorems are statements containing sufficient conditions that ensure the existence of a fixed point. Common fixed point theorems invariably require a commutativity condition, a condition on the ranges of the mappings, some continuity condition, and a contractive or possibly a Lipschitz-type condition and every significant fixed point theorem attempts to weaken or obtain a necessary version of one or more of these conditions (see, for instance, [12–16]).
Before proceeding further, we recall some relevant concepts.
compatible  if and only if , whenever is a sequence in X such that for some t in X;
noncompatible if there exists a sequence in X such that for some t in X, but is either non-zero or non-existent;
weakly compatible  if the pair commutes on the set of coincidence points (a point is called a coincidence point of the pair if ), i.e., whenever for some ;
occasionally weakly compatible  if there exists a coincidence point x in X such that implies ;
conditionally commuting  if the pair commutes on a nonempty subset of the set of coincidence points whenever the set of coincidences is nonempty;
subcompatible  if there exists a sequence in X such that and ;
conditionally compatible  if and only if whenever the set of sequences satisfying is nonempty, there exists a sequence such that and .
It may be observed that compatibility is independent of the notion of conditional compatibility, and in the setting of a unique common fixed point (or unique point of coincidence), conditional compatibility does not reduce to the class of compatibility. The following examples illustrate these facts.
Then it can be verified that f and g are compatible but not conditionally compatible.
In this example A and S are conditionally compatible but not compatible. To see this, we can consider the constant sequence , then , , , and . Again, if we consider the sequence , then , , , and . Thus f and g are conditionally compatible but not compatible.
It may also be observed that conditional compatibility need not imply commutativity at the coincidence points. The following example illustrates this fact.
In this example A and S are conditionally compatible, but they do not commute their only coincidence point . To see this, let us consider the sequence , then and . Thus f and g are conditionally compatible. On the other hand, we have iff and , . Then , but .
In this paper we define the notion of conditionally compatible maps in a slightly different manner as follows.
Definition 1.2 Two self-mappings A and S of a metric space will be called to be faintly compatible iff A and S are conditionally compatible and A and S commute on a nonempty subset of coincidence points whenever the set of coincidences is nonempty.
If A and S are compatible, then they are obviously faintly compatible, but the converse is not true in general.
In this example A and S are faintly compatible but not compatible. To see this, if we consider the constant sequence , then A and S are faintly compatible. On the other hand, if we choose the sequence , then , and . Thus A and S are faintly compatible, but they are not compatible.
It is also relevant to mention here that faint compatibility and noncompatibility are independent concepts. To see this, we can consider the following examples.
In this example A and S are noncompatible, but not faintly compatible. To see this, let us consider the sequence , then , , but . Thus A and S are noncompatible, but not faintly compatible.
In this example A and S are faintly compatible, but not noncompatible.
Examples 1.5 and 1.6 clearly show that faint compatibility and noncompatibility are independent of each other.
If A and S are weakly compatible, then they are also faintly compatible, but the converse is not true in general (see examples on the following pages).
It is worth mentioning here that if f and g are owc, then they are also faintly compatible, but the converse is not true in general.
Then it can be verified that A and S are trivially faintly compatible but not owc.
It may be pointed out that the notion of owc implies commutativity at some coincidence points, but it does not help in establishing the existence of coincidence points, whereas the new notion is useful in establishing the existence of coincidence points.
2 Main results
If either A or S is continuous, then A and S have a unique common fixed point.
Proof Noncompatibility of A and S implies that there exists some sequence in X such that and for some , but is either non-zero or non-existent. Since A and S are faintly compatible and , there exists a sequence in X satisfying (say) such that . Further, since A is continuous, then and . The last three limits together imply . Since implies that for some and , . Also, using (ii), we get . On letting , we get . Thus v is a coincidence point of A and S. Further, faint compatibility implies , and hence . If , then using (ii) we get , a contradiction. Hence is a common fixed point of A and S. The same conclusion is obtained when S is assumed to be continuous since the continuity of S implies the continuity of A. The uniqueness of the common fixed point theorem is an easy consequence of the condition (ii). □
We now give an example to illustrate Theorem 2.1.
Example 2.1 Let and let d be the usual metric on X. Define as follows:
if , if ,
, if .
Then A and S satisfy all the conditions of Theorem 2.1 and have a unique common fixed point at . It can be verified in this example that A and S satisfy the condition (ii) with . Furthermore, A and S are faintly compatible. Also, A and S are noncompatible. To see that, let us consider an increasing sequence in such that . Then , , and as . Therefore, A and S are noncompatible.
It is well known that the strict contractive condition does not ensure the existence of common fixed points unless the space taken to be compact or some sequence of iterates is assumed to be a Cauchy sequence. The next theorem illustrates the applicability of faintly compatible mappings satisfying the strict contractive condition.
If either A or S is continuous, then A and S have a unique common fixed point.
which gives a contradiction implying thereby . Hence, Aw is a common fixed point of A and S. The same conclusion can be drawn when S is assumed to be continuous since the continuity of S implies the continuity of A. The uniqueness of the common fixed point follows from (i). □
Remark 2.1 We have proved Theorem 2.2 under the simplest contractive condition . Analogous results for commuting or compatible mappings require a Banach-type or or Meir-Keeler-type contractive condition .
We now show that the notion of faint compatibility is also useful in studying fixed points of mappings satisfying Lipschitz-type conditions.
whenever the right-hand side is non-zero.
Suppose either A or S is continuous, then A and S have a common fixed point.
which gives a contradiction. Hence and Aw is a common fixed point of A and S. The same conclusion is obtained when S is assumed to be continuous since the continuity of S implies the continuity of A. This completes the proof. □
We now give an example  to illustrate Theorem 2.3.
Then A and S satisfy all the conditions of Theorem 2.3 and have a common fixed point and a coincidence point at which A and S do not commute. It may be verified in this example that A and S satisfy the condition (i) for together with the condition (ii). The mappings A and S are faintly compatible (take a constant sequence ) and they commute at the coincidence point at . Moreover, A and S are noncompatible (consider a sequence ).
In Example 2.2, A and S do not commute at the coincidence point , hence they do not satisfy the condition of weakly compatible mappings.
The next example also illustrates Theorem 2.3.
Then A and S satisfy all the conditions of Theorem 2.3 and have two common fixed points and .
As an application of faint compatible mappings, we now prove a common fixed point theorem under a more general condition that may hold for mappings satisfying contractive as well as nonexpansive and Lipschitz-type conditions.
whenever the right-hand side is non-zero.
Suppose A and S are continuous, then A and S have a common fixed point.
Proof Noncompatibility of A and S implies that there exists some sequence in X such that and for some , but or nonexistent. The continuity of A and S implies that , and . In view of faint compatibility and continuity of A and S, we can easily obtain a common fixed point as has been proved in the corresponding part of Theorem 2.3. □
Examples 2.2 and 2.3 also illustrate Theorem 2.4.
Remark 2.2 It may be in order to point out here that Theorems 2.1, 2.2, 2.3 and 2.4 have been proved under a noncomplete metric space. Further, it would be interesting to apply the technique of Haghi et al.  in the above results.
whenever the right-hand side is non-zero.
Remark 2.4 Faint compatibility is a necessary condition for the existence of common fixed points of given mappings A and S satisfying contractive or more general Lipschitz-type mapping pairs. Let A and S be a Lipschitz-type pair of self-mappings of a metric space and let A and S have a common fixed point x. Then and . If we choose the constant sequence , then and , that is, A and S are faintly compatible.
The authors are thankful to the referees for their valuable suggestions for improving the presentation of the paper. The research of N. Shahzad was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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