- Research Article
- Open Access
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.
We need the following definitions and lemmas.
Definition 1.1 (see ).
Definition 1.2 (see ).
Lemma 1.3 (see ).
Lemma 1.4 (see ).
Lemma 1.5 (see ).
Lemma 1.6 (see ).
Definition 1.7 (see ).
Definition 1.8 (see ).
Definition 1.9 (see ).
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]).
Theorem 1.10 (see ).
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.
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 .
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 .
Inspired by the work of Chang , we state the following theorem on compact metric spaces.
The author wishes to thank the refrees for their comments which improved the original manuscript.
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