Sentence examples for of weak compatibility from inspiring English sources

Since f z = g z and f and g are weakly compatible, we obtain, by the definition of weak compatibility, that f g z = g f z.

To study common fixed points, Jungck [19] initiated the concept of weak compatibility of maps thus: (f_{1}) and (f_{2}) on Y are weakly compatible maps if (f_{1}f_{2}w=f_{2}f_{1}wheneverver (f_{1}w=f_{2}w), for some (win Y).

In 1996, Jungck [4] introduced the concept of weak compatibility.

Finally, using the notion of weak compatibility of the pair together with (1.6), we get.

In 1998, Jungck and Rhoades [16] introduced the following concept of weak compatibility.

Hence, the notion of weak compatibility is more general than compatibility.

Moreover, it was shown by Dorić et al. in [92] that in the presence of a unique point of coincidence, the OWC condition reduces to weak compatibility.

(It is also well known that pointwise R-weak commutativity is equivalent to commutativity at coincidence points and in the setting of metric spaces this notion is equivalent to weak compatibility).

Now, using the notion of the weak compatibility of the pair and (3.5), we get.

In the event of single valued pair of maps, the notion of occasional weak compatibility reduces to weak compatibility due to unique coincidence point of the underlying maps (see [10]) which is always ensured by underlying contraction condition.

In fact, it was claimed by some authors (e.g., [11, 12]) that the notion of occasional weak compatibility relaxes the requirement of completeness as well as closedness condition on underlying space or subspaces in proving common fixed point theorems for contractive type mappings.