Sentence examples for proof of this remark from inspiring English sources
For the proof of this remark and further information about Gronwall's type inequality for fractional order differential equations, one can see [23].
We think the contrary is true; and we might cite in proof of this remark not only the present play, but the conclusion of Lear, of Romeo and Juliet, of Macbeth, of Othello, even of Hamlet, and of other plays of less moment, in which the last act is crowded with decisive events brought about by natural means".
In 2008, Dhompongsa and Panyanak [11] gave the following result, and the proof is similar to the proof of the remark in [[12], p. 374].
Remark 6 Our proof of this theorem differs from that in ([7], Theorem 2.26).
By Remark 1.4, the proof of this lemma is standard, and we omit it here.
Then for all y ∈ X satisfying 4 η ∥ y ∥ X < 1, the equation x = y + B ( x, x ) has a solution x ∈ X satisfying and uniquely defined by the condition ∥ x ∥ X ≤ ∥ y ∥ X. Remark 1 The proof of this lemma also shows that x = lim k → ∞ x k, where the approximate solutions x k are defined by x 0 = y and x k = y + B ( x k − 1, x k − 1 ).
Moreover, if the pair ( f, g ) is weakly compatible, then f and g have a unique common fixed point in C. Remark 2.1 From the proof of this theorem, it is easy to see that condition (2.2) can be weakened to α + β + 2 γ < 1.
Proof of Remark A.1 Rewrite equation (2.1) in form (6.6).
Alternative proof of Remark 15 Let x 3 = d 3 = 1.
The proof of Remark 3.3 reproduces that of Theorem 3.1 and Theorem 3.2.
