Sentence examples for proved based on some from inspiring English sources

By defining a continuous operator on a Banach space and taking advantage of the cone theory and some fixed point theorems, the existence of multiple positive solutions for the BVPs is proved based on some properties of Green's functions and under the circumstance that the continuous functions f satisfy certain hypothesis.

Some related mathematical theorems are proved, based on which the basic ideas of the generalized wavelet-Galerkin method are described in details.

Besides, some of our results, as those in [3], were rigorously proved based on the assumption of no depletion of susceptibles during the epidemic process.

It has been proved based on Besov space norm theory.

Existence and equivalence of the solution are analyzed and proved based on Brouwer fixed point theorem.

The convergence of the switching control is proved based on state sample trajectories.

The significance of velocity uniformity is proved based on the analysis and experiment.

The stability of the proposed scheme is proved based on the Lyapunov analysis method.

Then, the asymptotic stability of the system is proved based on the Lyapunov method.

Their robustness is proved based on the proposed Lyapunov equation approach.

The stability of the overall closed-loop system is proved based on Lyapunov theory.