Sentence examples for has compact values from inspiring English sources

Assume that there exists a function φ : [ 0, ∞ ) → [ 0, 1 ) such that lim r → t + sup φ ( r ) < 1 for each  t ∈ ( 0, ∞ ). and H ( T x, T y ) ≤ φ ( d ( x, y ) ) d ( x, y ) for all  x, y ∈ X. Then there exists z ∈ X such that z ∈ T z. The multi-valued mapping T considered by Reich [7] in Theorem 1.2 has compact values, that is, Tx is a nonempty compact subset of X for all x ∈ X.

Therefore Γ has compact values.

due to Lemma 2.1, and therefore has compact values.

By a similar argument, we find that Γ has compact values.

It is clear that F has approximative values if it has compact values.

Then, has compact values on, that is, is a compact set for each.

Assume that there exists a function φ : ( 0, ∞ ) → [ 0, 1 ) such that lim r → t + sup φ ( r ) < 1 for each  t ∈ [ 0, ∞ ). and H ( T x, T y ) ≤ φ ( d ( x, y ) ) ( d ( x, y ) ) for all  x, y ∈ X. Then there exists z ∈ X such that z ∈ T z. The multi-valued mapping T considered by Reich in Theorem 2 has compact value, that is, Tx is a nonempty compact subset of X for all x ∈ X.

Assume that there exists a function φ : [ 0, ∞ ) → [ 0, 1 ) such that lim sup r → t + φ ( r ) < 1. for each t ∈ ( 0, ∞ ) and H ( T x, T y ) ≤ φ ( d ( x, y ) ) d ( x, y ). for all x, y ∈ X. Then there exists z ∈ X such that z ∈ T z. The multi-valued mapping T studied by Reich [14] in Theorem 1.2 has compact value, that is, Tx is a nonempty compact subset of X for all x in the spaces X.

It follows from the compactness of X and Y that both (mathcal{F}_{2}) and (mathcal{F}_{3}^) have compact values.

Remark 3.2 Theorem 3.1 is an extension of Theorem 1.13 in the case of a mapping T : X → C B ( X ) having compact values.

The two processes ( C V, U ) and ( S V 2 q, U ) are semi-continuous and that U ( t, τ ) : S V 2 q → P ( S V 2 q ) and U ( t, τ ) : C V → P ( C V ) have compact values in their respective topologies.