Sentence examples for on a nonempty interval from inspiring English sources

Suppose that is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval, and is strictly convex on.

Let Y be a subset of an ordered metric space, ([a,b]) a nonempty interval in Y and (T:[a,b] longrightarrow[a,b]) a nondecreasing operator.

Consider the family F = ∏ β ∈ Γ A β ; A β is a nonempty interval in X β and ( A β ) is decreasing.

Let a real function f be defined on some nonempty interval I of a real line ℝ.

Let (Ysubsetmathbb{R}) be an arbitrary nonempty interval (bounded or unbounded).

For each α ∈ [ 0, 1 ], the set X α is a closed, bounded and nonempty interval of ℝ.

Let (I subset mathbb {R}) be a nonempty closed interval and ([alpha,beta]) be a nonempty functional interval in (AC(I)).

If belongs to then -level set is a nonempty compact interval for all.

Further note that Fix(g) is a nonempty closed interval if g is a nonexpansive self-map of [0,1].

But it is well known that the boundary of a nonempty (closed) interval cannot be its retract (see [19]).

It follows that at least one of f ( [ 0, a ] ) ∩ [ a, b ] and f ( [ 0, a ] ) ∩ [ b, 1 ] is a nonempty closed interval (not a singleton).