The purpose of this work is to weaken the condition of completeness of the metric space in the result of Kutbi and Sintunavarat [5] by using the concept of α-completeness of the metric space.
Remark 4.2 Theorem 4.1 improves the results of Aalam et al. [[18], Theorem 3.1], Rao et al. [[12], Theorem 3.1] and Chauhan and Kumar [[20], Theorem 3.1] as Theorem 4.1 never requires conditions on completeness (or closedness) of the underlying space (or subspace), continuity of the involved mappings and containment amongst range sets of the involved mappings.
Our results never require the conditions on completeness (or closedness) of the underlying space (or subspaces) together with the conditions on continuity in respect of any one of the involved mappings.
Thus ⊥ follows semantically from Γ, but is not provable from it in GL, contradicting the condition of strong completeness.
Under these conditions, the completeness of melting of solid bed, the stability of solid bed, the melt pressure at vent zone, and the melt temperature at the end of screw were 0, 0, 0 bar, and 279.5 °C, respectively.
For instance, in 2002, Gregori and Sapena [5] have introduced a kind of contractive mappings and proved fuzzy fixed point theorems in GV-spaces and KM-spaces by using a strong condition for completeness, now called the completeness in the sense of Grabiec or G-completeness, which can be considered a fuzzy version of the Banach contraction theorem.
Recently Gregori and Sapena [6] introduced a kind of contractive mapping for proving Banach's contraction principle by using strong condition for completeness (G-completeness) in fuzzy metric spaces.
Geometric backtracking is a necessary condition for completeness, but it may lead to a dramatic computational explosion due to the large size of the space of geometric states.
