Theorem 1 Let ( w 1, w 2 ) be a zero-order meromorphic solution of system (1.2), where Ω k ( z, w 1, w 2 ) ( k = 1, 2 ) are homogeneous q-difference polynomials in w 1 and w 2, respectively, with meromorphic coefficients, and P k ( z, w k ) and Q ( z, w k ), k = 1, 2, are polynomials in w k ( z ) with meromorphic coefficients having no common factors.
In this paper, we study the existence and multiplicity of solutions for the following elliptic system of noncooperative type: where Ω is a bounded smooth domain in R N with N ≥ 3, λ > 0, and μ > 0, 1 < p, q < 2 and α, β > 1 satisfy α + β = 2 ⋆, where 2 ⋆ : = 2 N / N − 2 denotes the critical Sobolev exponent.
As an application of Theorem 1.1, we study the existence, nonexistence, and multiplicity of positive radial solutions for the following quasilinear system on an exterior domain: where Ω = { x ∈ R N : | x | > r 0 }, r 0 > 0, 1 < p < N, Δ p z = div ( | ∇ z | p − 2 ∇ z ), k i ∈ C ( [ r 0, ∞ ), ( 0, ∞ ) ), i = 1, 2 and f, g ∈ C ( R +, R + ) with R + = [ 0, ∞ ).
Let (x t)=0) be the equilibrium point of the fractional-order system (D^{q} x=f x,t),xinOmega), where Ω is neighborhood region of the origin.
The best performance could be obtained by initially setting ω to some relatively high value (e.g., 0.9), which corresponds to a system where particles perform extensive exploration, and gradually reducing ω to a much lower value (e.g., 0.4), where the system would be more dissipative and exploitative and would be better at homing into local optima.
where omega_{text{n}} = frac{{omega_{text{d}} }}{{sqrt {1 - xi^{2} } }}, (7) where ω n is the system natural frequency, ω d is the damped natural frequency, and ζ is the damping ratio.
In this section, we assume that all the coefficients of system (1.3) are positive sequences with common periodic ω, where ω is a fixed positive integer, stands for the prescribed common period of the parameters in system (1.3), then the system (1.3) is an ω-periodic system for this case.
System (1) can be made dimensionless and simplified by adopting a unit system, where the units for length, time and energy are given by a 0, 1 / ω 0, and ħ ω 0, respectively, with a 0 = ħ m 1 ω 0, ω 0 = min { ω j x, ω j y, ω j z } [12].
where ω represents a vector in the global coordinate system and ω l denotes its representation relative to the local coordinate system attached to bee k.
where ω 0(i) is the state vector of the entire interconnected system and X k (ω 0(i)) is a nonlinear measurement function of bus k.
This is due to the fact that the search space at each node grows as | Ω | N t as compared to |Ω| for the SISO system, where |Ω| is the alphabet size and N t is the number of transmit antennas.
