Sentence examples for using standard arguments as from inspiring English sources

The following result can be proved using standard arguments as in [26, 28].

Lemmas 1 and 2 can be proved using standard arguments as the Lemmas 1 and 2 in [13], respectively.

As an application, we have the following corollary by using standard arguments as in [15], Theorem 3.3.

When the input is (I t)=(I_{1}(t),I_{2}(t),ldots,I_{n}(t))^{T}= 0,0,ldots,0)^{T}), by using standard arguments as Theorem 3.1, we can derive the stability conditions for system (2.1).

Corollary 6 All the state components x i ( t ) of system (1) with piecewise constant feedback function (9) in case 2′, i ∈ N ˜ 3, will flow to the interval [ 3, + ∞ ) when t → + ∞. Corollaries 4-6 can be proved using standard arguments as Theorems 4-6.

Corollary 3 All the state components x i ( t ) of system (1) with piecewise constant feedback function (9) in case 1′, i ∈ N 3, will flow to the interval [ 3, + ∞ ) when t → + ∞. Corollaries 1-3 can be proved using standard arguments as Theorems 1-3. Remark 2 Theorems 1-3 and Corollaries 1-3 are obtained based on Mexican-hat-type feedback function (8) and piecewise constant feedback function (9).

Now, using standard arguments, it is easy to show that p ( x m, x n ) → 0 as m, n → ∞.

When (12) holds, the proof can be presented using standard arguments similar to the proof above.

Furthermore, using standard arguments, we obtain the functional is functional in and (3.3).

In order to prove this theorem, we use standard arguments of density.

In this paper, we introduce a new condition and prove a unique common fixed-point theorem for hybrid mappings in partial metric spaces without using any standard arguments as commutativity and continuity conditions.