非线性二阶中立型时滞微分方程正解的存在性

考虑非线性二阶中立型微分方程,a(t)x(t)-∑ from i=1 to m (p_i(t)x(τi(t)))]″-∫from n=a to b (f(t,ξ,xg(t,ξ)])dσ(ξ))=0,t≥t_0,和相应不等式a(t)x(t)-∑ from i=1 to m (p_i(t)x(τi(t)))]″-∫from n=a to b (f(t,ξ,xg(t,ξ)])dσ(ξ))≥0,t≥t_0.存在正解是相互等价的.其中a(t),pi(t)∈C(t0,∞),R+),a(t)>0,τi(t)∈C(R~+,R~+),τi(t)t,limt→∞τi(t)=∞(i=1,2,…,m).g(t,ξ)∈C(t_0,∞)×a,b],R+).g(t,ξ)是分别关于t和ξ的增函数.g(t,ξ)t,ξ∈a,b],limt→∞,ξ∈a,b]g(t,ξ)=∞.f(t,ξ,x)∈C(t_0,∞)×a,b]×R,R+).当x>0时,xf(t,ξ,x)>0.σ(ξ)∈C(a,b],R),且σ(ξ)非减.

Eventually Positive Solutions of Second-order Nonlinear Neutral Delay Differential Equations

Consider the Second-order Nonlinear Neutral Delay Differential Equationsa a(t)x(t)-∑ from i=1 to m(p_i(t)x(τi(t)))]″-∫from n=a to b(f(t,ξ,xg(t,ξ)])dσ(ξ))=0,t≤t_0,and the associated inequalitya a(t)x(t)-∑ from i=1 to m(p_i(t)x(τi(t)))]″-∫from n=a to b(f(t,ξ,xg(t,ξ)])dσ(ξ))≥0,t≤t_0,Where a(t),pi(t)∈C(t0,∞),R+),a(t)>0,τi(t)∈C(R~+,R~+),τi(t)t,limt→∞ τi(t)=∞(i=1,2,…,m).g(t,ξ)∈C(t0,∞)× a,b ],R+).g(t,ξ)about t and ξ are nondecreasing real functions,respectiyily.g(t,ξ)≤t,ξ∈ a,b ],limt→∞,ξ∈ a,b ]g(t,ξ)=∞.f(t,ξ,x)∈C(t0,∞)× a,b ]×R,R+).As x>0,then xf(t,ξ,x)>0.σ(ξ)∈C(a,b ],R),and σ(ξ)about ξ are not decreasing.A necessary and sufficient condition for the existence of eventually bounded positive solutions are obtained.