In the setting of Example 2.2, we define, besides retaining the rest of the example as it stands.
Theorem 7) remains true if we replace the condition (u0) by one of the following conditions (besides retaining rest of the hypotheses): ((mathrm{u}_{0}^{1})): ((fX,preceq)) is totally ordered, ((mathrm{u}_{0}^{2})): ((X,preceq)) is ((f,g -directed.
Remark 3 Theorem 8 remains true if condition (u0) is replaced by any one of the conditions ( u 0 1 )-( u 0 9 ) (of Corollary 14) besides retaining rest of the hypotheses.
CPF enables, in fresh concrete, drainage of excess water and air besides retaining binder particles at the concrete surface, leading to a blow-hole free surface and enhanced quality of the outer layers.
Corollary 7 Corollary 4 (also Corollary 5 or Corollary 6) remains true if the condition (e) is replaced by the following condition (besides retaining the rest of the hypotheses): (e)′ there exists α ∈ [ 0, 1 ) such that d ( f x, f y ) ≤ α d ( x, y ) ∀ x, y ∈ X with x ≺ ≻ y.
Example 4.3 In the setting of Example 4.1, replace the self-mappings A, B, S and T by the following besides retaining the rest: We obtain A ( X ) = { 1, 5 } ⊂ [ 1, 13 ] = T ( X ) and B ( X ) = { 1, 4 } ⊂ [ 1, 4 ] = S ( X ), which shows that S ( X ) and T ( X ) are closed subsets of X.
Corollary 6 Corollary 4 remains true if the conditions (c) and (d) are replaced by the conditions (c)′′ and (d)′′, respectively (besides retaining the rest of the hypotheses): (c)′′ either f is continuous or ( X, d, ⪯ ) has the MCB property, (d)′′ there exists x 0 ∈ X such that x 0 ≺ ≻ f ( x 0 ).
Theorem 1 remains true if the conditions (c) and (d) are replaced by the conditions ((mathrm{c})^{prime}) and ((mathrm{d})^{prime}), respectively (besides retaining the rest of the hypotheses): ((mathrm{c})^{prime}) : either f is continuous or ((X,d,preceq)) has the DCL property, ((mathrm{d})^{prime}) : there exists (x_{0}in X) such that (x_{0} succeq f(x_{0})).
Theorem 3 remains true if the condition (f) is replaced by the following condition (besides retaining the rest of the hypotheses): ((mathrm{f})^{prime}) : (mathrm{C} x,y,precsucc)) is nonempty for each (x,yin X). (mathrm{C} x,y,precsucc)) is nonempty for each (x,yin X).
Theorem 5 remains true if certain involved terms namely: O̅-complete, O̅-continuous and the ICC property are, respectively, replaced by (underline{mathrm{O}} -complete, (underline{mathrm{O}} -completeunderline{mathrm{O}} -continuous the assumptiO}} -continuousced by the following (besides retanding the rest): ((mathrm{DCC^{property: there exists (x_{0}in X) such that (x_{0}succeq f(x_{0})).
