Sentence examples for as those in the proof of from inspiring English sources

The rest of the proof can be obtained from the same arguments as those in the proof of Theorem 2.1.

To prove the remaining statements, we follow the same steps as those in the proof of Theorem 3.1.

Following the same lines as those in the proof of Theorem 2.3, we can construct a sequence ({x_{n}}) in (A_{0}) satisfying d_{S}(x_{n+1},Tx_{n}) = d_{S}(A,B), quad forall ninmathbb {N}cup {0}.

Similar analysis to that in Zhang et al. ([3], Lemma 2.3) leads to the conclusion that, for all (tgeq t_{1}), there exist two possible cases (1) and (2) (as those in the proof of Theorem 3.1), where (t_{1}geq t_{0}) is sufficiently large.

By the arguments as those in the proof of Theorem 2.1, we get u t − ∫ R N J ( x − y ) ( u ( y, t ) − u ( x, t ) ) d y − m p 2 ∫ Ω ψ p ( x ) d x ∫ Ω u p d x + u p ≤ 0. According to the above results, the solution u ( x, t ) of equation (1.1) vanishes in finite time.

The same arguments as those in the proof of Theorem 7 show that (15) holds, i.e. limsup_{trightarrow+infty} x t)leq k. (31) Hence, for (varepsilon>0) sufficiently small, satisfying (alpha _{2}e^{-d_{1 1} k+varepsilonrepsilon)< d_{1 2}), there is a (T_{1}>0) such that (x t)leq k+varepsilon) for (t>T_{1}).

Proof We proceed similarly as in the proof of Lemma 58.

Proof As observed in the proof of Corollary 3, (20) changes into (22).

Proof As described in the proof of Theorem 3.1, T : P → P is completely continuous.

as in the proof of.

Proof We argue as in the proof of Corollary 1.