Sentence examples for by using the compactness from inspiring English sources

The weak formulation of the conservation law (3) is obtained by shifting the derivatives to the test functions by partial integration, and by using the compactness of the support.

Then, by using the compactness criterion and Schauder's fixed point theorem, we present an existence theorem for an asymptotically almost periodic solution to the addressed Volterra-type difference equation.

We obtain the following approximation result, because the partition of unity { β i }, subordinate to the finite covering { F i c : i = 0, 1, …, n } of Y, can be given by using the compactness of X instead of the hypothesis of metric space in the proof of Lemma 4.2.

Also in Theorem 2.1.1 [16], it is shown, by using the compactness of the unit disc of the Euclidean space, that every continuous mapping F from the Euclidean space (mathbb{R}^{n}) into (2^{mathbb{R}^{n}} ) satisfying the Chebyshev condition induces the continuity of (P_{F}).

Note that, by using the compactness of S α ( t ) and Lemma 3.1, similar to the proof of Theorem 3.3, we can prove that the mapping x ( t ) → ∫ 0 t S α ( t − s ) x ( s ) d s. from L 2 ( [ 0, b ], X ) to C ( [ 0, b ], X ) is compact, i.e., the Cauchy operator G : L 2 ( [ 0, b ], X ) → C ( [ 0, b ], X ) is also compact.

If Λ < − v ∞, then by using the concentration compactness principle of Lions, one shows that Λ is the principle eigenvalue of S with a positive eigenfunction Φ 0 : S Φ 0 = λ 0 Φ 0, Φ 0 ∈ H 2 ( R N ), Φ 0 > 0. 3. The spectrum of S in ( − ∞, − v ∞ ), namely σ ( S ) ∩ ( − ∞, − v ∞ ), is at most a countable set, which we denote by Λ = λ 0 < λ 1 < λ 2 < ⋯,  .

We have proved the existence of the uniform attractor in L 2 ( Ω ¯, d μ ) for the non-autonomous p-Laplacian evolution equations subject to dynamic nonlinear boundary conditions by using the Sobolev compactness embedding theory, and the existence of the uniform attractor in ( W 1, p ∩ L q × L q by asymptotic a priori estimate.

In this paper, by using the concentration-compactness principle and the variational method, we obtain a multiplicity result for Kirchhoff-type problems involving critical growth in bounded domains.

In this paper we consider a class of elliptic problems of p-Kirchhoff type with critical exponent in bounded domains and new results as regards the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and variational method.

One can show that the structure A has a proper elementary extension A′. (There is a proof of this using the compactness theorem and the diagram lemma — see 3.1 and 3.2 below; another proof is by ultrapowers — see 4.1 below).

end{aligned} In [15], Temam also discussed the convergence of the approximations to the incompressible Navier-Stokes equations on bounded domains by using the classical Sobolev compactness embedding theorems and the classical Lions method [16] of fractional derivatives to recover the compactness in time.