Sentence examples for under proper assumptions from inspiring English sources

Next we show (gamma^_{ab}) is attained, under proper assumptions for (a,b>0).

Theoretical proof of existence and uniqueness of the solution under proper assumptions is provided.

Next, under proper assumptions, asymptotic stability is proved for constant commands, while bounded stability is proved for varying commands.

Remark 4.3 Under proper assumptions, we have obtained the threshold of the existence of positive solutions to (2.1 - 2.2 2.1 - 2.2

By using the Nehari manifold and fibering map methods, multiple positive solutions were obtained under proper assumptions.

We prove that, under proper assumptions, this significantly improves the convergence rate of the initial iterative method.

Besides, the prediction analysis guarantees that under the proper assumptions, collision between vehicles may not happen at any future time.

Moreover, we achieve some regularity result of the solution and prove the solution is positive under some proper assumptions.

The semilocal convergence of the method is proved under the proper assumptions coinciding with physical properties, which can provide the theoretical foundation for its applications.

Under the proper assumptions, we obtain the existence of nontrivial solutions to perturbed p-Laplacian system by using the variational approach.

The existence of a positive solution of a p-Laplace-like equation with critical growth is established by the generalization to the concentration-compactness principle and the Sobolev inequality under some proper assumptions.