Sentence examples for mappings are compatible from inspiring English sources

Moreover, R-weakly mappings are compatible, and compatible mappings are weakly compatible.

Clearly, weakly commuting mappings are compatible, but neither implication is reversible.

Clearly, weakly commuting mappings are compatible, but in view of Example 2.9 the converse does not hold.

In our result, we do not require that the t-norm is of Hadžić-type [19], the mappings are compatible [16], commutable, continuous or monotonic increasing.

In general, commuting mappings are weakly commuting and weakly commuting mappings are compatible, but the converses are not necessarily true and some examples can be found in [25, 27 29].

Jungck [15] defined the notion of compatible mappings to generalize the concept of weak commutativity and showed that weakly commuting mappings are compatible but the converse is not true [15].

It is easy to see that two compatible mappings are weakly compatible, but the converse is not true.

Commutative mappings must be weak commutative mappings, weak commutative mappings must be compatible, compatible mappings must be weakly compatible, but the converse is not true.

where ψ 2 ∈ F 4. Hence, Theorem 1 can be considered an extension of [23], Theorem 2.1] to the frame of partial metric spaces (since semi-compatible mappings are weakly compatible).

Let X = [ 0, ∞ ) and d be the usual metric on X. Define f, g : X → X by f x = x 3 for all  x and g x = 2 x 3 for all  x. Then d ( f g x, g f x ) > d ( f x, g x ). Therefore f and g are not weakly commuting mappings. However, f and g are compatible mappings.

In [16], Choudhury and Kundu established a similar result under the condition that F and g are compatible mappings and the function g is monotone increasing.