id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-4	Zhang, Mengqing; Tian, Jing; Zou, Keyue	Asymptotic stability of a stochastic age-structured cooperative Lotka-Volterra system with Poisson jumps	2023	18	.pdf	application/pdf	6444	411	83	t 0 E sup s∈[0,t] |X(r)− x(r)|2ds + ((2λ1 + 1)L2 1 + 2ρ2 1 + 1)E ∫ t 0 |Ψ(s)− ψ̄(s)|2ds + E sup s∈[0,t] ∫ t 0 2(X(s)− x(s), (G1x(Ψ(s))−G1x(ψ̄(s)))dw(s)) + 2E sup s∈[0,t] ∫ t 0 (X(s)− x(s), (J1x(Ψ(s))− J1x(ψ̄(s)))dÑ(s)). (4.1) By the BDG inequality, we have E sup s∈[0,t] ∫ = − ∫ t tk ∂x(s) ∂a ds+ ∫ t tk [H1x +H2x]ds+ ∫ t tk G1xdw(s) + ∫ t tk J1xdN(s), thus |x(t)− x̄(t)|2 ≤ 4| ∫ t tk ∂x(s) ∂a ds|2 + 4| ∫ t tk [H1x +H2x]ds|2 + 4| ∫ t tk G1xdw(s)|2 + 4| ∫ t tk J1xdN(s)|2 ≤ 4∆ ∫ t tk |∂x(s) ∂a |2ds+ 4∆ ∫ t tk |H1x +H2x|2ds+ 4| ∫ t tk G1xdw(s)|2 + 8|λ1 ∫ t tk J1xds|2 ≤ 4∆ ∫ t tk |∂x(s) ∂a |2ds+ 8∆[ ∫ t tk (|H2 1x|+ |H2 2x|)ds] + 4| ∫ t tk G1xdw(s)|2 + 8| ∫ t tk J1xdN̄(s)|2 + 8|λ1 ∫ t tk J1xds|2. by Lemma 3.1, E sup t∈[0,T ] |x(t)− x̄(t)|2 ≤ 5E sup t∈[0,T ] max k=0,1,...,N−1 ∣∣ ∫ t tk G1x(ψ)dw(s) ∣∣2 + 8E sup t∈[0,T ] max k=0,1,...,N−1 ∣∣ ∫ t tk J1x(ψ)dÑ(s) ∣∣2 + 5∆ ∫ t tk |∂x(s) ∂a |2ds+ 8TC[L2 1 + 2ρ1 + 8λ2 1L 2 1]∆. 12 M. ZHANG, J. TIAN, K. ZOU EJDE-2023/02 According to the Doob inequality, we obtain E sup t∈[0,T ] |x(t)− x̄(t)|2 ≤ 5∆ ∫ t tk |∂x(s) ∂a |2ds+ 8TC[L2 1 + 2ρ1 + 8λ2 1L 2 1]∆ + 5 max k=0,1,...,N−1 ∫ (k+1)∆ k∆ E|G1x(x, y)|2ds + 8λ1 max k=0,1,...,N−1 ∫ (k+1)∆ k∆ E|J1x(x, y)|2ds ≤ 5∆ ∫ t tk |∂x(s) ∂a |2ds+ 8TC[L2 1 + 2ρ1 + 8λ2 1L 2 1]∆ + 5L2 1C∆ + 8λ1L 2 1C∆. (3.7)	cache/ejde-4.pdf	txt/ejde-4.txt