Sentence examples for weak compatibility for from inspiring English sources

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 [18] (resp., Jungck [11, 12]).

However, in [15] Abbas and Ilić obtained various common fixed-point and invariant approximation results for such mappings under the assumption of weak compatibility of maps.

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).

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

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

Next, we need to show the weak compatibility of the pair ((f_{1},f_{3})).

By (2.34) and using the weak compatibility of f and T, we obtain fw=fTv=Tfv=Tw.

The main differences are manifested in the weak compatibility of long-term trends.

Hence, weak compatibility remains the minimal commutativity condition for the existence of common fixed point for contractive type mappings.

Hence weak compatibility remains the minimal commutativity condition for the existence of a common fixed point for a contractive pair of maps.

Now, it has been shown that weak compatibility is the minimal noncommuting condition for the existence of common fixed points of contractive type mapping pairs.