Sentence examples for bounded subset of the from inspiring English sources

Suppose that for each bounded subset of, the ordered pair satisfies either condition AKTT or condition.

If is a bounded subset of, the Chebyshev radius of relative to is defined by (2.4).

Then Ω is an open and bounded subset of the Banach space C 1 [ 0, T ].

Clearly, S is a closed, convex, and bounded subset of the Banach space X.

Suppose that for each bounded subset of, the ordered pair satisfies condition AKTT.

It is easy to check that S̃ is a nonempty, closed, convex, and bounded subset of the Banach space X̃.

We will denote by the family of all subsets of the family of nonempty closed and bounded subsets of the family of nonempty compact subsets of.

We will use the following notations, throughout this paper, where is a symmetric space, and, and is the class of all nonempty bounded subsets of The diameter of is denoted and defined by.

For any 0 < α < β < 1, 0 ≤ p ≤ q, the bounded subsets of the space C q + β ( Q T ) are precompact subsets of C p + α ( Q T ).

The function (chi mathcal{M}_{X}to[0,infty)) is called the Hausdorff measure of noncompactness. We now recall some basic properties of the Hausdorff measure of noncompactness. Let F, (F_{1}), and (F_{2}) be bounded subsets of the metric space ((X,d)).

Using the notion of a partial metric on a set X, Aydi et al. [18] defined a partial Hausdorff metric on the set of closed and bounded subsets of the set X.