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Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales

机译:具可变时标的非线性中立型泛函微分方程的三个正周期解

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Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale(d/dt)(x(t)+c(t)x(t-α))=a(t)g(x(t))x(t)-∑j=1nλjfj(t,x(t-vj(t))),(t,x)∈𝕋0(x),Δt|(t,x)∈𝒮2i=Πi1(t,x)-t,Δx|(t,x)∈𝒮2i=Πi2(t,x)-x, whereΠi1(t,x)=t2i+1+τ2i+1(Πi2(t,x))andΠi2(t,x)=Bix+Ji(x)+x,  i=1,2,….  λj  (j=1,2,…,n)are parameters,𝕋0(x)is a variable time scale with(ω,p)-property,c(t),  a(t),vj(t),andfj(t,x)  (j=1,2,…,n)areω-periodic functions oft,Bi+p=Bi,  Ji+p(x)=Ji(x)uniformly with respect toi∈ℤ.
机译:使用两个连续的归约法:将可变时间尺度上的系统的B等价度与时间尺度上的系统的等价度以及一个脉冲微分方程的归约关系,并通过Leggett-Williams不动点定理,我们研究了三个正周期解的存在性时标上两个连续部分之间具有过渡条件的可变时标上的非线性中立泛函微分方程(d / dt)(x(t)+ c(t)x(t-α))= a(t)g( x(t))x(t)-∑j =1nλjfj(t,x(t-vj(t))),(t,x)∈𝕋 0(x),Δt|(t,x) ∈𝒮 2i =Πi1(t,x)-t,Δx|(t,x)∈𝒮 2i =Πi2(t,x)-x,其中Πi1(t,x)= t2i + 1 + τ2i+ 1(Πi2(t,x))和Πi2(t,x)= Bix + Ji(x)+ x,i = 1,2,...。 λj(j = 1,2,…,n)是参数,𝕋 0(x)是具有(ω,p)-属性,c(t),a(t),vj(t ),并且fj(t,x)(j = 1,2,…,n)是t的ω周期函数,Bi + p = Bi,Ji + p(x)= Ji(x)关于i∈ℤ是一致的。

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