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 λ j f j ( t , x ( t - v j ( t ) ) ) , ( t , x ) ∈ T 0 ( x ) , Δ t | ( t , x ) ∈ S 2 i = Πi 1 ( t , x ) - t , Δ x | ( t , x ) ∈ S 2 i = Πi 2 ( t , x ) - x , where Πi 1 ( t , x ) = t 2 i + 1 + τ 2 i + 1 ( Πi 2 ( t , x ) ) and Πi 2 ( t , x ) = B i x + J i ( x ) + x ,    i = 1,2 , … .    λ j    ( j = 1,2 , … , n ) are parameters, T 0 ( x ) is a variable time scale with ( ω , p ) -property, c ( t ) ,    a ( t ) , v j ( t ), and f j ( t , x )    ( j = 1,2 , … , n ) are ω -periodic functions of t , B i + p = B i ,    J i + p ( x ) = J i ( x ) uniformly with respect to i ∈ Z .