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 / 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.