f is weakly decreasing with respect to T. The pair {T, f} is compatible.
Similarly, Lach and Moraga-González (2017) find that consumer surplus always (although weakly) decreases with increased competition.
The graphene is closed, with sheet resistivity strongly decreasing with growth temperature, weakly decreasing with the amount of deposited C, and reaching down to 2 kΩ/□.
To prove f is weakly decreasing with respect to T, let x, y ∈ X be such that y ∈ T-1(fx).
Assume that T and f satisfy the following hypotheses: (i) f is weakly decreasing with respect to T. (ii) The pair {T, f} is compatible.
In the same way, Shatanawi et al. [14] studied some new real generalizations on coincidence points for weakly decreasing mappings satisfying a weakly contractive condition in an ordered metric space.
We say that f is weakly decreasing with respect to T if the following conditions hold: (1) fX ⊆ TX. (2) For all x ∈ X, we have fy ≼ fx for all y ∈ T -1(fx). .
Now, since x1 ∈ T-1(fx0) and x2 ∈ T-1(fx1), by using the assumption that f is weakly decreasing with respect to T, we obtain f x 0 ≽ f x 1 ≽ f x 2. By induction on n, we conclude that f x 0 ≽ f x 1 ≽ ⋯ ≽ f x n ≽ f x n + 1 ≽ ⋯.
Suppose that the following hypotheses are satisfied: (i) If (x n ) is a nonincreasing sequence in X with respect to ≼ such that x n → x ∈ X as n → +∞, then x n ≽ x for all n ∈ ℕ. (ii) f is weakly decreasing with respect to T. (iii) TX is a complete subspace of X. .
If (x n ) is a nonincreasing sequence in X with respect to ≼ such that x n → x ∈ X as n → +∞, then x n ≽ x for all n ∈ ℕ. f is weakly decreasing with respect to T. TX is a complete subspace of X.
