Due to the special structure of the nonlinear Schrodinger equation, the Jacobian operator exhibits one eigenvalue that moves to zero when the Newton iterate converges to a nontrivial solution
and is exactly zero at a solution.
Let the functions r(t) > 0 and u(t) exist for every nontrivial solution
x(t), t [member of] [J.
A nontrivial solution
of system (8) exists if the characteristic equation ([[alpha].
2] [less than or equal to] 0, then every nonnegative, nontrivial solution
While the Q that minimizes either [Mathematical Expression Omitted] or the log-price variance will not also minimize [Mathematical Expression Omitted], which is what is relevant here, since [Mathematical Expression Omitted] the nontrivial solution
will be influenced in the direction of that Q.
And it is easy to see that u is a nontrivial solution
i] : Z [right arrow] R, i = 1, 2, is said to satisfy property (U) on Z, provided there is no nontrivial solution
, r(m), of (3.
We know that there exists a minimizer on S which is a critical point of J(u) and so a nontrivial solution
1) is disconjugate on the interval [c, d], if there is no nontrivial solution
has a nontrivial solution
having two distinct zeros on [a, b], then q+(t) = [[q(t) + [absolute value of q(t)]] / 2] must satisfy the Lyapunov inequality
does not admit a nontrivial solution
, or letting z = Gw, that the equation
Then, for any [lambda] [member of] [0,[infinity][, there exists a nonnegative nontrivial solution
(u, v) to the problem (1.