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

Yongkun Li and Chao Wang

Journal of Applied Mathematics, 2012, vol. 2012, issue 1

Abstract: 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)=t21i++τ21i+(Πi2(t,x)) and Πi2(t,x)=Bix+Ji(x)+x, i=1,21,2,…. λj (j=,…,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.

Date: 2012
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https://doi.org/10.1155/2012/516476

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