Sentence examples for which satisfies for from inspiring English sources

Assume that is a surjective self-mapping which is continuous everywhere in X which satisfies for some real constant, some,.

where and is a delay function, has a solution which satisfies for all for some fixed, then the coefficient satisfies, where satisfies for all.

Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence of which satisfies for all.

Let us define the interpolation operator Π h : V → V h, which satisfies: for any q ∈ V, ∫ T ( q − Π h q ) ⋅ v h d x d y = 0, ∀ v h ∈ V h, T ∈ T h.

We assume that is a quotient of odd positive integers, and are positive, real-valued rd-continuous functions defined on, is strictly increasing, and is a time scale, and as, which satisfies for some positive constant, for all nonzero.

Next, we define the standard L 2 -orthogonal projection Q h : K → K h, which satisfies: for any u ˜ ∈ K, ( u ˜ − Q h u ˜, u ˜ h ) = 0, ∀ u ˜ h ∈ K h, (2.24).

Denote by the class of analytic functions given by which satisfy for all.

Corollary Assume that the following conditions are satisfied: (i) The holomorphic semigroup e − z L, 0 ≤ | arg ( z ) | < π / 2 − θ is represented by the kernels a z ( x, y ) which satisfy, for all ν > θ, an upper bound | a z ( x, y ) | ≤ c ν h | z | ( x, y )  .

Theorem 4 Assume the following conditions are satisfied: (i) The holomorphic semigroup e − z L, 0 ≤ | arg ( z ) | < π / 2 − θ is represented by the kernels a z ( x, y ) which satisfy, for all ν > θ, an upper bound | a z ( x, y ) | ≤ c ν h | z | ( x, y )  .

Corollary 2 Assume the following conditions are satisfied: (i) The holomorphic semigroup e − z L, 0 ≤ | arg ( z ) | < π / 2 − θ is represented by the kernels a z ( x, y ) which satisfy, for all ν > θ, an upper bound | a z ( x, y ) | ≤ c ν h | z | ( x, y )  .

(i) ; (ii)for which satisfies, ; (iii)for, and,.