Sentence examples for data for the solutions from inspiring English sources
The case where the coefficients are Lipschitz operators is studied well, and theorems on the existence, uniqueness, and continuity with respect to initial data for the solutions in (L^{2}) are proved; see, for example, Kotelenez [3].
The inequality above has proved to be very effective in the research of boundedness, uniqueness, and continuous dependence on initial data for the solutions to certain differential equations, as it can provide explicit bounds for the unknown function u ( t ).
It is observed that except for the laminar to turbulent transition region, the turbulent friction factor data for the solutions are bounded in the area between the two universal asymptotes and the data points are linear for all concentrations.
They gave a sufficient condition on the initial data for the solution to blow up in a finite time.
end{aligned} This implies the continuous dependence on the initial data for the solution which generates a continuous process ({tilde{U}(cdot,cdot )}).
In Section 3, we apply the results established in Section 2 to research boundedness, quantitative property, and continuous dependence on the initial data for the solution to a certain fractional integral equation.
Then, in Section 3, we investigate a certain fractional differential equation, in which the boundedness, uniqueness, and continuous dependence on initial data for the solution to the fractional differential equation are investigated by use of the generalized Gronwall-type inequalities established.
The question of the largest Besov-type spaces on initial data for which the solutions of (1.1) have well-posedness and analyticity is still open.
Furthermore, two more specific conditions of how small the initial data are for the solutions to exist globally are concluded by the dilation transformation.
Moreover, two more specific conditions of how small the initial data are for the solutions to exist globally are investigated by some suitable dilation transformation.
The existence, existence and uniqueness, and data dependence for the solutions of equation (1) were analyzed by using Picard operators and weakly Picard operators methods.
