content_image/302171.jpg content_image/302172.jpg
continuous with discontinuities of the first kind at the points s$_{k}$, k ∈ Z , such that ψ − $^{1}$( s$_{k}$ +) − ψ − $^{1}$( s$_{k}$ ) = δ$_{k}$, where ψ − $^{1}$( s$_{k}$ +) = lim $^{s}$→s + k ψ − $^{1}$( s ) , the sequence { s$_{k}$ } is ( ψ ( ω ) , p ) − periodic, i.e., s$_{k}$$_{+}$$_{p}$ = s$_{k}$ + ψ ( ω ) for all k ∈ Z , and ψ ( t + ω ) = ψ ( t ) + ψ ( ω ) , t ∈ T ′ $_{0}$. Moreover, if a function h ( t ) is ω − periodic on T $_{0}$, then h ( ψ − $^{1}$( s )) is ψ ( ω ) − periodic, and vice versa. 

Let us denote by C $_{rd}$( T $_{0}$) the set of all functions which are rd-continuous on T $_{0}$, and let C 1 $_{rd}$( T $_{0}$) ⊂ C $_{rd}$( T $_{0}$) be the set of all continuously differentiable functions on T $_{0}$, assuming that the functions have a one sided derivative at θ$_{k}$, k ∈ Z . On the other hand, we say that a function defined on R is an element of the set P C 0 if it is left-continuous on R and continuous on R \ ⋃ ∞ k $^{=}$−∞ { s$_{k}$ } , and it has discontinuities of the first kind at the points s$_{k}$, k ∈ Z . Moreover, a function h : R → R n belongs to the set P C 1 0 if both h and h ′ are elements of P C $_{0}$, where h ′ ( s$_{k}$ ) = lim $^{s}$→s − k h ( s ) − h ( s$_{k}$ ) s s$_{k}$ , k ∈ Z . It was shown by Akhmet and Turan [1] that a function ϑ ( t ) belongs to C $_{rd}$( T $_{0}$) ( C 1 $_{rd}$( T $_{0}$) ) if and only if ϑ ( ψ − $^{1}$( s )) belongs to P C 0 ( P C 1 0 ) . In accordance with the equation y $^{Δ}$( θ$_{2}$$_{k}$ ) = y ( θ$_{2}$$_{k}$$_{+1}$ ) − y ( θ$_{2}$$_{k}$ ) , k Z , system (1.1) can be written as 

θ$_{2}$$_{k}$$_{+1}$ θ$_{2}$$_{k}$ ∈ 

<img src='content_image/302171.jpg'>

Applying the transformation s = ψ ( t ) to (2.5) we obtain the following impulsive system, 

<img src='content_image/302172.jpg'>

where x ( s ) = y ( ψ − $^{1}$( s )) , Δ x | s = s$_{k}$ = x ( s$_{k}$ +) − x ( s$_{k}$ ) , k ∈ Z , and x ( s$_{k}$ +) = lim $^{s}$→s + k x ( s ) . 

In what follows, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for square matrices [27]. 

The following conditions are required throughout the paper. 

(C1) det( I + δ$_{k}$A ) ̸ = 0 for all k ∈ Z , where I is the n × n identity matrix; 

(C2) All eigenvalues of the matrix e ψ ( ω ) $^{A}$Π $^{p}$− 1 j $_{=0}$( I + δ$_{j}$A ) lie inside the unit circle; 

(C3) There exist positive numbers M$_{f}$ and M$_{F}$ such that sup $^{t}$∈ T$_{0}$ $^{, y}$∈ R n ‖ f ( t, y ) ‖ ≤ M$_{f}$ and sup $^{η}$∈ $_{Λ}$‖ F ( η ) ‖ ≤ M$_{F}$ ; 

(C4) There exists a positive number L$_{f}$ such that ‖ f ( t, y$_{1}$ ) − f ( t, y$_{2}$ ) ‖ ≤ L$_{f}$ ‖ y$_{1}$ − y$_{2}$ ‖ for all t ∈ T 0 and y$_{1}$, y$_{2}$ ∈ R $^{n}$. 

