Sentence examples for there exists sufficiently from inspiring English sources

Combining with (2.45), we need only to show that for any there exists sufficiently large such that (2.48).

Let (varepsilon >0) be fixed, since ( {limnolimits _{nrightarrow infty }} omega _0(U_{n}) = 0), there exists sufficiently large (m_1in mathbb {N}) such that (omega _0(U_{m_1})

Thus, if (k_{2}>K_{2}) and (d>D), then there exists sufficiently large N such that Re(gamma_{n}<0) for (n>N).

If x is an eventually positive solution of (1.1), then there exists sufficiently large T ≥ t 0 such that: (1) S n Δ ( t, x ( t ) ) < 0 for t ≥ T.   (2) Either lim t → ∞ x ( t ) = 0 or S i ( t, x ( t ) ) > 0 for t ≥ T and 0 ≤ i ≤ n.  .

This, together with (9) and (10), implies that { A u ( t 1 ) 1 + t 1 α − 1 : u ∈ U } are equicontinuous on any finite subinterval of J. Now, we are going to prove that for any given ε > 0, there exists sufficiently large τ > 0, which satisfies ∥ A u ( t 1 ) 1 + t 1 α − 1 − A u ( t 2 ) 1 + t 2 α − 1 ∥ ≤ ε. for all u ∈ U and t 1, t 2 ≥ τ.

Assume that there exists sufficiently large, such that (2.3).

From the conditions, there exist sufficiently big positive constants such that (4.13).

Then there exist sufficiently large initial data u 0, v 0, u 1, v 1 such that any solutions of problem (4.1) blew up at finite time T ∗.

Then (3.30) implies that, for any given (lambda>0), there exist sufficiently small (s_{0lambda}in 0,1)) and (varepsilon>0) such that mathcal{J}(s_{0lambda}u)< -varepsilon,quad forall uin K_{m}.

In addition, the author employs the large system analysis and variational analysis to obtain the optimal power and bandwidth profiles for the case where there exist sufficiently many users in each cell.

It only requires that, for any potential target point q, there exist sufficiently rich sets that are close such that the cost-to-go function ({mathcal{V}}) still has some upper continuity property along sequences converging to q in those sets.