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

[摘要] 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.

[学科分类] 应用数学