Sentence examples for a unique solution of the Cauchy from inspiring English sources

If we differentiate this identity, we obtain that x* is a unique solution of the Cauchy problem (16).

Moreover, x* is continuously differentiable and it is a unique solution of the Cauchy problem for the additively modified differential equation d x ( t ) d t = f ( t, x ( t ) ) + 1 T [ C - 1 d - ( C - 1 A + I 2 ) z ] - 1 T ∫ 0 T f ( s, x ( s ) ) d s, t ∈ ( 0, T ), x ( 0 ) = z.

The function y ∞ is the unique solution of the Cauchy problem (6.15).

The function u ∞ ( ⋅, z ) is the unique solution of the Cauchy problem (3.8)  .

The function x ∞ is the unique solution of the Cauchy problem (6.8).

Denote by x the unique solution of the Cauchy problem left { textstylebegin{array}{l} ^{mathrm{c}}D_{t}^{alpha}x t)=lambda x t)+f t,x t)),quad tin J, x 0)=y(0), end{array}displaystyle right.

Therefore, there exists a unique solution to the Cauchy problem (1.1).

Then there exists a unique solution u of the Cauchy problem (1.1) in (C [0, T]; H)) for some (Tin 0,infty]) (maximal existence time).

For such a problem, when f ( u ) = 1 2 u 2, by introducing the smooth approximation w ̃ ( t, x ) of the rarefaction wave solution w R (t, x) as the unique solution of the following Cauchy problem w ̃ t + w ̃ w ̃ x = w ̃ x x, t > - t 0, x ∈ R, w ̃ ( - t 0, x ) = w 0 R ( x ), x ∈ R (1.6).

That means that there is a unique solution which solves the Cauchy problem (1.4) for T < δ.

There exists an unique solution (u t,x)) of the Cauchy problem (1.1 - 1.2 1.1 - 1.2aximal time ([0,T)) such that (u(t,x)in C([0,T);H^{1}times L^{2})).