Sentence examples for valued solutions from inspiring English sources

has real valued solutions,, where (28).

For example, Bogachev and Röckner [14] considered the existence of measure valued solutions for the equation involving second order partial differential operators in infinite dimensional spaces.

The author [19] also studied the finitely valued and infinitely valued solutions to the parabolic Monge-Ampère equation (-u_{t}det(D^{2}u)=f).

We establish a new connection between L2ρ(Rd Rd)⊗L2ρ(Rd;Rd) valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs.

As an application, we show that a result concerning removable singularities for -harmonic functions satisfying a Lipschitz condition or of bounded mean oscillation extends to Clifford valued solutions to corresponding Dirac equations.

Then by the technique of removable singularities, we can find that solutions to an A-harmonic system satisfying a Lipschitz condition or in the case of a bounded mean oscillation can be extended to Clifford valued solutions to the corresponding A-Dirac system.

3, the main results on the well-posedness of solutions of the nonlocal balance equations presented in the introduction are stated, first for (L^p -valued soL^p -valued then for Radon measolutionsed solutions.

In order to obtain the desired result, we use the method of removability theorems, which proved that under suitable condition, a result concerning removable singularities for equations defined by the A-harmonic operator satisfying the Lipschitz condition or of bounded mean oscillation extends to Clifford-valued solutions to the corresponding Dirac equation.

In order to obtain the desired results, we prove that under natural growth condition, a result concerning removable singularities for equations defined by an A-harmonic operator satisfying the Lipschitz condition or of bounded mean oscillation extends to Clifford-valued solutions to corresponding Dirac equations.

Then, since the determinant of the matrix { b i j ′ } i, j = 0 m − 1 does not vanish, the system ∑ j = 0 m − 1 b i j ′ ( ξ ′, q ) v j ( ξ ′, q ) = h i, i = 0, …, m − 1, has unique E-valued solutions { v j ( ξ ′, q ) } j = 0 m − 1 which depend on ξ ′ and q.

In Table 1, we give the real-valued solutions to (7) and the corresponding value of κ 4 ( y ).