Let ((mathcal X,d,preceq )) be a regular ordered metric space and let (mathcal F ) be a self-map on (mathcal X ) satisfying for all (x,yin overline{mathcal{O (x_0;mathcal F )}}) such that (x) and (y) are comparable, begin{aligned} T(d(mathcal F x,mathcal F y),d x,y),d x,mathcal F x),d y,mathcal F y),d x,mathcal F y),d y,mathcal F x))le 0 end{aligned}where (T in mathfrak T ^{prime }).
Corollary 5 Let ( X, d, ≼ ) be a regular ordered metric space and let T be a self-map on X satisfying for all x, y ∈ O ( x 0 ; T ) - such that x and y are comparable, F ( d ( T x, T y ) ) ≤ φ ( F ( M 3 [ T ] ( x, y ) ) ), where F ∈ F, φ ∈ Φ and M 3 [ T ] ( x, y ) = max { d ( x, y ), d ( T x, x ), d ( T y, y ), 1 2 [ d ( x, T y ) + d ( T x, y ) ] }.
We will denote by Φ the set of functions φ : [0 + ∞) → [0, +∞), such that φ is right continuous, φ(0) = 0 and φ(t) < t for every t > 0. The first result of this section is the following. Theorem 4 Let ( X, d, ≼ ) be a regular ordered metric space and let T, S and R be self-maps on X satisfying F ( d ( T x, S y ) ) ≤ φ ( F ( M [ T, S, R ] ( x, y ) ) ) (4.1).
Corollary 4 Let ( X, d, ≼ ) be a regular ordered metric space and let T and R be self-maps on X satisfying F ( d ( T x, T y ) ) ≤ φ ( F ( M 2 [ T, R ] ( x, y ) ) ), for all x, y ∈ O x 0 ; T, T, R - (for some x0) such that R x and R y are comparable, where F ∈ F, φ ∈ Φ and M 2 [ T, R ] ( x, y ) = max { d ( R x, R y ), d ( T x, R x ), d ( T y, R y ), 1 2 [ d ( R x, T y ) + d ( T x, R y ) ] }.
The intuition of men like Newton and Kepler that, beneath the apparent chaos of everyday life, the universe is a regular, ordered machine that can be described with a few simple equations proved amazingly to be correct.
Let ((X,leq,|cdot|)) be a regular partially ordered complete normed linear space such that the order relation ≤ and the norm (|cdot| ) in X are compatible.
Let ((E,preceq, Vert cdot Vert )) be a regular partially ordered complete normed linear space.
Unlike Verizon, AT&T offers a separate data plan for your iPad that does not require you to be a regular customer for voice service in order to take advantage of the bulk rate pricing.
I was a regular mail-order customer with my lawn-mowing earnings and allowance and ordered the record.
Materials donated by the Cremin family and Ellen Condliffe Lagemann were not in a regular order, and were physically arranged in roughly chronological order.
He said Mr Lee was a regular customer who had ordered the cake after seeing an advertising leaflet in the shop for its custom-made baking service.
