Some Common Fixed Point Theorems for Weakly Compatible Mappings in Metric Spaces
© M. A. Ahmed. 2009
Received: 23 October 2008
Accepted: 18 January 2009
Published: 3 February 2009
We establish a common fixed point theorem for weakly compatible mappings generalizing a result of Khan and Kubiaczyk (1988). Also, an example is given to support our generalization. We also prove common fixed point theorems for weakly compatible mappings in metric and compact metric spaces.
In the last years, fixed point theorems have been applied to show the existence and uniqueness of the solutions of differential equations, integral equations and many other branches mathematics (see, e.g., [1–3]). Some common fixed point theorems for weakly commuting, compatible, -compatible and weakly compatible mappings under different contractive conditions in metric spaces have appeared in [4–15]. Throughout this paper, is a metric space.
where , for some and , for some .
If for some , we denote , and for , and , respectively. Also, if , then one can deduce that .
We need the following definitions and lemmas.
Definition 1.1 (see ).
A sequence of nonempty subsets of is said to be convergent to if:
(i)each point in is the limit of a convergent sequence , where is in for ( := the set of all positive integers),
(ii)for arbitrary , there exists an integer such that for , where denotes the set of all points in for which there exists a point in , depending on such that .
is then said to be the limit of the sequence .
Definition 1.2 (see ).
A set-valued function is said to be continuous if for any sequence in with , it yields .
Lemma 1.3 (see ).
If and are sequences in converging to and in , respectively, then the sequence converges to .
Lemma 1.4 (see ).
Let be a sequence in and let be a point in such that . Then the sequence converges to the set in .
Lemma 1.5 (see ).
For any , it yields that .
Lemma 1.6 (see ).
Let be a right continuous function such that for every . Then, for every , where denotes the -times repeated composition of with itself.
Definition 1.7 (see ).
The mappings and are weakly commuting on if and for all .
Definition 1.8 (see ).
The mappings and are said to be -compatible if whenever is a sequence in such that , and for some .
Definition 1.9 (see ).
The mappings and are weakly compatible if they commute at coincidence points, that is, for each point such that , then (note that the equation implies that is a singleton).
If is a single-valued mapping, then Definition 1.7 (resp., Definitions 1.8 and 1.9) reduces to the concept of weak commutativity (resp., compatibility and weak compatibility) for single-valued mappings due to Sessa  (resp., Jungck [11, 12]).
Throughtout this paper, we assume that is the set of all functions satisfying the following conditions:
(i) is upper semi-continuous continuous at a point from the right, and non-decreasing in each coodinate variable,
(ii)For each , .
Theorem 1.10 (see ).
where satisfies (i) and for each , and , with . Then and have a unique common fixed point such that .
In the present paper, we are concerned with the following:
(1)replacing the commutativity of the mappings in Theorem 1.10 by the weak compatibility of a pair of mappings to obtain a common fixed point theorem metric spaces without the continuity assumption of the mappings,
(2)giving an example to support our generalization of Theorem 1.10,
(3)establishing another common fixed point theorem for two families of set-valued mappings and two single-valued mappings,
(4)proving a common fixed point theorem for weakly compatible mappings under a strict contractive condition on compact metric spaces.
2. Main Results
In this section, we establish a common fixed point theorem in metric spaces generalizing Theorems 1.10. Also, an example is introduced to support our generalization. We prove a common fixed point theorem for two families of set-valued mappings and two single-valued mappings. Finally, we establish a common fixed point theorem under a strict contractive condition on compact metric spaces.
First we state and prove the following.
Let be two sefmaps of a metric space and let be two set-valued mappings with
then there is a point such that .
then . Therefore, is a Cauchy sequence.
as . So, is a Cauchy sequence. Hence, for some . But by (2.3), so that . Consequently, . Moreover, we have, for , that . Therefore, . So, we have by Lemma 1.4 that . In like manner it follows that and .
We must conclude that .
This contradiction demands that . Similarly, if the pair is weakly compatible, one can deduce that . Therefore, we get that .
The proof, assuming the completeness of , is similar to the above.
which is inadmissible. So, .
Now, we give an example to show the greater generality of Theorem 2.1 over Theorem 1.10.
for all . It is clear that is a complete metric space. Since is a closed subset of , so is complete. We note that is a -compatible pair and therefore a weakly compatible pair. Also, and , that is, and are weakly compatible. On the other hand, if , so that even though , that is, is not a -compatible pair. We know that is the unique common fixed point of and . Hence the hypotheses of Theorem 2.1 are satisfied. Theorem 1.10 is not applicable because for all , and the maps I, J and G are not continuous at .
In Theorem 2.1, if the mappings and are replaced by and , where is an index set, we obtain the following.
then there is a point such that for each .
This yields that .
Inspired by the work of Chang , we state the following theorem on compact metric spaces.
holds whenever the right-hand side of (2.20) is positive, then there is a unique point in such that .
The author wishes to thank the refrees for their comments which improved the original manuscript.
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