Sentence examples for proximal property from inspiring English sources

T satisfies the proximal property.

(b) satisfies the proximal property.

So T satisfies the proximal property.

Since satisfies the proximal property, we have.

(b) T satisfies the proximal property.  .

T is weakly continuous; T satisfies the proximal property.

A seminal result of Aronszajn [34] establishes that such solution sets satisfy a more general proximal contractibility property tantamount to being contractible in each of their open neighborhoods (sets with trivial shape and (R_{delta}) sets14).

F is continuous having the proximal mixed monotone property and proximally coupled weak contraction on A. There exist ( x 0, y 0 ) and ( x 1, y 1 ) in A × A such that x 1 = F ( x 0, y 0 ) with x 0 ≤ x 1 and y 1 = F ( y 0, x 0 ) with y 0 ≥ y 1.

(i) F is continuous having the proximal mixed monotone property and proximally coupled weak contraction on A.   (ii) There exist ( x 0, y 0 ) and ( x 1, y 1 ) in A × A such that x 1 = F ( x 0, y 0 ) with x 0 ≤ x 1 and y 1 = F ( y 0, x 0 ) with y 0 ≥ y 1.  .

Then, it can be seen that F is continuous such that F ( A 0 × A 0 ) ⊆ B 0. The only comparable pairs of points in A are x ⪯ x for x ∈ A, hence the proximal mixed monotone property and the proximally coupled weak contraction on A are satisfied trivially.

One can see that, if (A = B) in the above definition, the notion of the proximal mixed monotone property reduces to that of the mixed monotone property.