So far it remains to investigate whether Theorem 3.20 can be extended to a complete PM-space?
Let A and B be non-void closed subsets of a complete PM-space ((X,F,*)) with ∗ of Hadžić-type such that (A_{0}) and (B_{0}) are non-void.
In fact, given nonempty closed subsets A and B of a complete PM-space ((X, F,*)), a contraction non-self-mapping (T : A to B) does not necessarily has a fixed point.
Let ((M,F,tau_{T})) be a complete pms under a continuous t-norm T of H-type such that (operatorname{Ran}F subset D^).
In all cases, the genetic and physical positions were expressed as percentages of the total genetic distance of the LG or the complete PM length.
Finally, functionally important residues for MLO susceptibility proteins have been inferred by the association of naturally occurring and induced mutations with partial or complete PM resistance [ 11, 12, 21– 25].
In 1972, Sehgal and Bharucha-Reid [2] obtained a generalization of the Banach contraction principle on a complete Menger PM-space, which is a milestone in developing fixed point theory in a Menger PM-space.
Let ((X,F,Delta)) be a complete Menger PM-space, where Δ is of Hadžić-type.
Then ((X,F,Delta_{m})) is a complete Menger PM-space.
Theorem 3.9 Let ( S, F, Δ ) be a complete Menger PM-space under a t-norm Δ of ℋ-type.
Then ((X,F,Delta_{m})) is a complete Menger PM-space, (Delta_{m}) is a continuous t-norm.
