The line shows the estimates with linear restrictions imposed on the coefficients.
For the SDDE with Markovian switching again, the following hypothesis is imposed on the coefficients f and g.
Further, the method used here permits us to relax the restrictions usually imposed on the coefficients of equation (1.1).
As it is shown in [34] for the sublinear terms case, we can also remove the sign condition imposed on the coefficients of the sub-half-linear terms to obtain interval criterion which is applicable for the case when some or all of the functions,, are nonpositive.
In general, if the polynomial in Eq. (3) is of degree m, then it has to be implied that the side conditions (orthogonality) imposed on the coefficients λ i are mathop sum limits_{i = 1}^{N} lambda_{i} q(x_{i} ) = 0, (4)for all polynomials q with m being its highest degree.
Our main results do not require common factors of the coefficients of the system, relax the conditions imposed on the diffusion coefficient matrix, and generalize some early results.
To guarantee the positive solutions without explosion (i.e., the global positive solutions), some conditions are in general needed to impose on the coefficients of system (1.2).
Next, we impose on the coefficients of (1) a pointwise (inxandt) weak law of large numbers and a weak invariance principleh tε−10 k 0k |k|⩽2p⇒{Θk}|k|⩽2p ((3))inC([0, T], H1), H1being a separable Hilbert space of functions andh∈ 0, 1) denoting a constant.
On the basis of less restrictive assumptions imposed on the neutral coefficient, some new criteria are presented.
Our results do not require differentiability of the delay functions and have relaxed the conditions imposed on the diffusion coefficient matrix.
