Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales - Yongkun Li; Chao Wang (Journal of Applied Mathematics)
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, where Πi1(t,x)=t2i+1+τ2i+1(Πi2(t,x)) and Πi2(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), and fj(t,x)   (j=1,2,…,n) are ω-periodic functions of t, Bi+p=Bi,  Ji+p(x)=Ji(x) uniformly with respect to i∈Z.