We assume that the functions g,f,q i (i = 1,2) satisfy the conditions so that the solution of this equation exists and is unique.
The master curve corresponding to that equation exists only for θ ⩾ θLC limit.
Moreover, under the trilinear condition, the Wiener measure twisted by the diffusion matrix becomes stationary for the limit and we show that the solution of the limit equation exists globally in time when the initial value is sampled from the stationary measure.
It is shown that the solution of the Helmholtz equation exists in form of a uniformly convergent sequence of iterations of the equivalent Volterra integral equation.
If the exact solution of the partial differential equation exists, the approximated solution via the method converges to the exact solution [ 26].
end{aligned} (3.3) The solution (x^{n}(t)) of the above equation exists according to Lemma 3.4.
We then show that if a solution to the HJB equation exists, then it must have a strictly positive uniform lower bound.
From Equation 9, we conclude that a solution of Zoomeron equation exists if the following conditions on the parameters are satisfied: w > 0, w ≠ 1, μ ≠ 0, R > 0, A 4 2 − 4 2 A 5 w R > 0. Open image in new window (10).
Theorem 3.1 (Regularity) Let θ 0 ∈ Ḣ m , m ≥ 1, α = 1, then the solution for the regularized problem of quasi-geostrophic equation exists a solution θ ( t ) ∈ C ( ( 0, + ∞ ), Ḣ m ).
The solutions of the above equation exist, hence 4-factor operator is a surjection on the element H. Summing up, we conclude from the generalized Lyusternic theorem: for (x^{ast}= 0,0,0)), the tangent cone (T_{xast}M) to the solutions set M coincides with the kernel of the fourth derivative of the mapping (Psi_{4}), i.e., with the set (H_{4}(x^{ast})= operatorname{Ker} ^{4}Psi_{4}(H)).
