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concerning Duffing equations on a time scale is presented in Section 5 to support the theoretical results. Finally, some concluding remarks are given in Section 6. 

## 2 Preliminaries 

The basic concepts that are needed in the present paper about differential equations on time scales are as follows [16, 31, 32, 33]. A time scale is a nonempty closed subset of R . On a time scale T , the forward and backward jump operators are defined as σ ( t ) = inf { s ∈ T : s > t } and ρ ( t ) = sup { s ∈ T : s < t } , respectively. We say that a point t ∈ T is right-scattered if σ ( t ) > t and right-dense if σ ( t ) = $_{t}$. In a similar way, if ρ ( t ) < t, then t ∈ T is called left-scattered, and otherwise it is called left-dense. Besides, a function h : T × R n → R n is called rd-continuous if it is continuous at each ( t, u ) ∈ T × R n with right- dense t, and the limits lim ( r,ν $^{)}$→( t − ,u $_{)}$h ( r, ν ) = h ( t − , u ) and lim $^{ν}$→$_{u}$h ( t, ν ) = h ( t, u ) exist at each ( t, u ) with left-dense t. At a right-scattered point t ∈ T , the $_{Δ}$-derivative of a continuous function ϑ is defined as ϑ $^{Δ}$( t ) = ϑ ( σ ( t )) − ϑ ( t ) σ ( t ) t . On the other hand, at a right-dense point t, we have ϑ $^{Δ}$( t ) = lim $^{r}$→t ϑ ( t ) − ϑ ( r ) t r provided that the limit exists. 

It is worth noting that on the time scale T 0 used in system (1.1) the points $^{θ$_{2}$$_{k}$}$− $_{1}$, k ∈ Z , are left-scattered and right-dense, and the points $_{θ$_{2}$$_{k}$}$, k ∈ Z , are right-scattered and left-dense. Moreover, σ ( θ$_{2}$$_{k}$ ) = $_{θ$_{2}$$_{k}$$_{+1}$}$, ρ ( θ$_{2}$$_{k}$$_{+1}$ ) = $_{θ$_{2}$$_{k}$}$, k ∈ Z , and σ ( t ) = ρ ( t ) = t for any t ∈ T 0 except at the points θ$_{k}$, k ∈ Z . 

Suppose that the time scale T 0 used in the description of equation (1.1) satisfies the ω − property. That is, there exists a number ω > 0 such that t + ω ∈ T 0 whenever t ∈ T $_{0}$. In this case, there exists a natural number p such that δ$_{k}$$_{+}$$_{p}$ = δ$_{k}$ for all k ∈ Z , where δ$_{k}$ = θ$_{2}$$_{k}$$_{+1}$ − θ$_{2}$$_{k}$ [1]. Suppose that p is the minimal among those numbers. 

We assume without loss of generality that $^{θ}$− 1 < 0 < θ$_{0}$. Define on the set T ′ 0 = T 0 \ ⋃ ∞ k $^{=}$−∞ { $^{θ$_{2}$$_{k}$}$− 1 } the ψ − substitution [1] as 

<img src='content_image/319864.jpg'>

The function ψ ( t ) is one-to-one, ψ (0) = 0 , ψ ( T ′ $_{0}$) = R and lim $^{t}$→∞$^{, t}$∈ T ′ 0 ψ ( t ) = ∞ . According to the results of the paper [1], dψ ( t ) /dt = 1 , t ∈ T ′ $_{0}$, and dψ − $^{1}$( s ) /ds = 1 provided that s ̸ = s$_{k}$, k ∈ Z , where 

<img src='content_image/319865.jpg'>

and the sequence { s$_{k}$ } , k ∈ Z , is defined through the equation s$_{k}$ = ψ ( θ$_{2}$$_{k}$ ) . The function ψ − 1 is piecewise 

