Sentence examples for using a standard argument from inspiring English sources

Using a standard argument as in the classical case in [18, 22], we have the following.

(3.2) Using a standard argument, it is sufficient to show that ({u_{n}}_{nin mathbb{N}}) is bounded in (H_{T}^{1}), and this will be enough to derive the (PS -condition.

In fact, we can show the stability of the closed-loop control system by using a standard argument in the stability analysis of model predictive control with a terminal constraint (e.g., ([33], Chapter 6), ([34], Chapter 2), or ([35], Chapter 5)).

By Lemma 2, we get ψ ( d ( f n x, f n + 1 x ) ) ≤ c n r ( x, f x ). for all n ∈ N. Hence ∑ n = 0 ∞ ψ ( d ( f n x, f n + 1 x ) ) < ∞. and if we use a standard argument, then ( f n x ) n ∈ N is obtained as a Cauchy sequence.

It can be obtained using a standard monotonicity argument following ideas from [16].

Recently, Zhang and Yuan [15] obtained the existence of a nontrivial homoclinic solution for (HS) by using a standard minimizing argument.

Moreover, we announce that by using a standard dilation argument in (5 - 6) we see that such kernels must be homogeneous of degree −1.

By using a standard dilation argument it is seen that the inequalities considered in Theorem 5 can hold if and only if (lambda=-1).

Proof By a contradiction, we may assume u ≥ 0. By using a standard regularity argument and [[17], Lemmas 3.2 and 3.3], we have u ∈ C 1, α for some α ∈ ( 0, 1 ).

If F is bounded from below, then c = inf E F. is a critical point of F. In order to obtain the existence of non-trivial homoclinic solutions of (1.1) by using a standard minimizing argument, we will establish the corresponding variational functional of (1.1).

Using a standard truncation argument, we get begin{aligned} P Biggl(sum_{k=1}^{n} { xi_{k}}geq x Biggr) &=P Biggl(sum_{k=1}^{n} {xi_{k}}geq x, max_{1leq kleq n} xi_{k} > vx Biggr)+P Biggl(sum_{k=1}^{n} { xi_{k}}geq x, max_{1leq kleq n} xi_{k} leq vx Biggr) &leqsum_{k=1}^{n} overline{V_{k}} vx)+P Biggl(sum_{k=1}^{n} {widetilde{ xi_{k}}}geq x Biggr).