In this case, we say that is a complete qpm on.
Let ((X,d)) be a complete (T_{0} -qpm space and let (f,g : Xto X) be two self-mappings such that (f(X)subset g(X)).
Corollary 2 Let ( X, d ) be a complete T 0 -qpm space, q a Q-function on ( X, d ), T X → ClCl d s ( X ) a multivalued map and r ∈ [ 0, 1 ) such that for each x, y ∈ X and u ∈ T x, there is v ∈ T y satisfying q ( u, v ) ≤ r q ( x, y ). Then T has a fixed point. Corollary 1 was proved in [[5], Theorem 3.3].
Corollary 1 Let ( X, d ) be a complete T 0 -qpm space, q a Q-function on ( X, d ), φ : [ 0, ∞ ) → [ 0, ∞ ) a Bianchini-Grandolfi gauge function and T X → ClCl d s ( X ) a multivalued map such that for each x, y ∈ X and u ∈ T x, there is v ∈ T y satisfying q ( u, v ) ≤ φ ( q ( x, y ) ). Then T has a fixed point. If we take φ ( t ) = r t where r ∈ [ 0, 1 ) we get one of the main results in [2].
He's a complete horseman".
Example 5 Let X = { 0, 1 } and let p be the partial metric on X given by p ( x, y ) = max { x, y } for all x, y ∈ X. Obviously d p is a complete T 0 -qpm on X.
Let be a weighted qpm space.
(a) Let be a weightable qpm space with weighting function.
