Three positive periodic solutions to nonlinear neutral functional differential equations with parameters on variable time scales

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 / d t) (x (t) + c (t) x (t - α)) = a (t) g (x (t)) x (t) - j = 1 n j f j (t,x (t - v j (t))),(t,x) t 0 (x),t | (t,x) s 2 i = i 1 (t,x) - t,x | (t,x) s 2 i = i 2 (t,x) - x,where i 1 (t,x) = t 2 i + 1 + 2 i + 1 ( i 2 (t,x)) and i 2 (t,x) = b i x + j i (x) + x,i = 1,2,. j (j = 1,2,n) are parameters,t 0 (x) is a variable time scale with (,p) -property,c (t),a (t),v j (t),and f j (t,x) (j = 1,2,n) are -periodic functions of t,b i + p = b i,j i + p (x) = j i (x) uniformly with respect to i z. copyright © 2012 yongkun li and chao wang.