"This is possibly a unique couple of years with a certain kind of actor and a certain kind of film nominated," counters Berry.
Hence, ( g x, g x ) is a unique coupled point of coincidence of mappings F and g.
Thus, ( g x, g x ) is a unique coupled point of coincidence of mappings F and g.
If for every there exists that is comparable to and with respect to ordering in, then there exists a unique coupled point of coincidence of and.
Hence, (fx, fx) is a unique coupled point of coincidence of mappings f, S, and T. Now, we show that f, S, and T have common coupled fixed point.
Since g and F are w-compatible, we have g u = g ( g x ˆ ) = g ( F ( x ˆ, y ˆ ) ) = F ( g x ˆ, g y ˆ ) = F ( u, u ), which implies that ( u, u ) is a coupled coincidence point of g and F. Since g and F have a unique coupled point of coincidence, we can conclude that g u = g x ˆ, i.e., g u = u.
Then F and g have a unique coupled point of coincidence.
Theorem 4 Let ( X, p ) be a 0-complete partial metric space, let F : X × X → C B p ( X ) and g : X → X be mappings such that all the conditions of Theorem 3 are satisfied and, for any coupled coincidence point ( w, z ) of F and g, we have F ( w, z ) = { g w } and F ( z, w ) = { g z }, then F and g have a unique coupled point of coincidence.
Therefore, ( s, s ) is a unique coupled coincidence point of F and g. □.
Then F has a unique coupled fixed point of the form ( u, u ).
