Sentence examples for continuous dependence on from inspiring English sources

Proof The continuity of P follows immediately from the continuous dependence on data stated in Corollary 2.2 and by the continuity of the map ψ ↦ ψ ˜ of Lemma 3.5 and of the map that associates to any φ ∈ M ˜ its restriction to the interval [ − T, 0 ].

In this section, we present one example for the results established above, in which the boundedness, quantitative property, and continuous dependence on the initial value for the solutions to one certain fractional integral equation are researched.

As for applications, we have presented one example, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solution to a certain fractional integral equation are investigated.

Next, we will discuss the continuous dependence on the Forchheimer coefficient λ.

The uniqueness of solution is often a direct consequence of the continuous dependence on parameters.

Then we can obtain the continuous dependence on the initial data for mild solutions.

Finally, the same conditions of Lipschitz continuity guarantee the continuous dependence of solutions on given data, i.e., initial states and coefficients, which then follows, essentially, from Gronwall's inequality.

An application of the Schauder fixed point theorem requires the continuity of our mapping A. This means that we need continuous dependence of solutions on the data.

When regarding these points as initial conditions in (mathbb {T}) for the one-dimensional system submitted to stimulus f, the continuous dependence of solutions on initial conditions implies lim_{nto+infty}bigl| R^{ntau}theta_{1}^{(f)}( vartheta_{1},cdot )|_{[0,tau]}bigr| =bigl| R^{alpha}f|_{[0,tau]}bigr |

The property of continuous dependence of solutions on parameters forces the deformation to pass continuously through this explosion of limit cycle canards in order to transition from equilibrium to the full-blown relaxation oscillations in these planar systems.

We will prove the following main theorem on the continuous dependence of the solution on the given data.