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

In this paper, using the Leggett–Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equations with impulses and parameters on time scales { ( x ( t ) + c ( t ) x ( t − r 1 ) ) Δ = a ( t ) g ( x ( t ) ) x ( t ) − ∑ i = 1 n λ i f i ( t , x ( t − τ i ( t ) ) ) , t ≠ t j , t ∈ T , j = 1 , 2 , … , q , x ( t j − ) − x ( t j + ) = I j ( x ( t j ) ) , t = t j , j = 1 , 2 , … , q , where λ i , i = 1 , 2 , … , n are parameters, T is an ω -periodic time scale, a ∈ C ( T , R + ) , c ∈ C ( T , [ 0 , 1 ) ) and both of them are ω -periodic functions, τ i ∈ C ( T , R ) , i = 1 , 2 , … , n are ω -periodic functions, f i ∈ C ( T × R + , R + ) , i = 1 , 2 , … , n are nondecreasing with respect to their second arguments and ω -periodic with respect to their first arguments, respectively; g ∈ C ( R + , R + ) and there exist two positive constants l , L such that 0 < l ≤ g ( x ) ≤ L < ∞ for all x > 0 , I j ∈ C ( R , R + ) ( j = 1 , 2 , … , q ) and is bounded, r 1 is a constant.