Periodic boundary value problems for first-order impulsive dynamic equations on time scales

Nonlinear Analysis 69 (2008) 4074–4087 www.elsevier.com/locate/na

Periodic boundary value problems for first-order impulsive dynamic equations on time scalesI Fengjie Geng a,b,∗ , Yancong Xu b , Deming Zhu b a School of Information Engineering, China University of Geosciences (Beijing), Beijing, 100083, China b Department of Mathematics, East China Normal University, Shanghai, 200062, China

Received 16 June 2006; accepted 18 October 2007

Abstract This paper deals with the periodic boundary value problem for a class of first-order impulsive dynamic equations on time scales. We introduce a new notion of lower and upper solutions, which extends the classical lower and upper solutions. The existence of extremal solutions is presented by virtue of the method of lower and upper solutions coupled with monotone iterative technique. An example is also included to illustrate the main results. c 2007 Elsevier Ltd. All rights reserved.

MSC: 34B37 Keywords: Time scales; Jump operators; Impulsive dynamic equations; Lower and upper solutions; Periodic boundary value problem

1. Introduction Recently, the theory of dynamic equations on time scales has gained much attention (see, for example, [2,7,8]). This is not only because it can unify continuous and discrete calculus, but also because the time-scales’ calculus has a tremendous potential for applications. For instance, it can model insect populations, neural networks, epidemic models and so on [7]. Especially, the impulsive dynamic equations on time scales have attracted current interest from a few authors, the readers are referred to [4–6,13,16] and references therein. These works focus mainly on initial boundary value problems of first-order or two-point boundary value problems of second-order impulsive dynamic equations, and most of them are concerned with the existence of either one or two positive solutions via some fixed-point theorems, such as the Schaefer’s fixed-point theorem and nonlinear alternative of Leray–Schauder type [5], and the double fixed-point theorem due to Avery and Henderson [16]. While, in 2005, Belarbi et al. [4] considered the following initial value problem: \  1 T, t 6= tk , k = 1, 2, . . . , m,  y (t) + p(t)y σ (t) = f (t, y(t)), t ∈ J := [0, b] (1.1) y(tk+ ) − y(tk− ) = Ik (y(tk− )), k = 1, 2, . . . , m,  y(0) = η, I Supported by National Natural Science Foundation of China (#10671069). ∗ Corresponding author at: School of Information Engineering, China University of Geosciences (Beijing), Beijing, 100083, China.

E-mail addresses: gengfengjie [email protected] (F. Geng), [email protected] (Y. Xu), [email protected] (D. Zhu). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.10.038

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where T is a time scale, f : J × R → R is a given function, Ik ∈ C(R, R), tk ∈ (0, b) ∩ T and 0 = t0 < t1 < · · · < tm < tm+1 = b, y σ (t) = y(σ (t)). They gave an existence criterion for the extremal solution of (1.1) by way of the Heikkila and Lakshmikantham fixed-point theorem [15]. Since there are few works about periodic boundary value problems of impulsive dynamic equations on time scales, motivated by [4,13,14,16], in this paper, we investigate the following periodic boundary value problem: \  1 T, t 6= tk , k = 1, 2, . . . , m,  y (t) = f (t, y(t)), t ∈ J := [0, T ] (1.2) Imp(y)(tk ) := Ik (y(tk− )), k = 1, 2, . . . , m,  y(0) = y(σ (T )), where T is a time scale, σ is the jump operator, which will be defined later, f : J × R → R is continuous in the second variable, Ik ∈ C(R, R), tk ∈ (0, T ) ∩ T and 0 < t1 < · · · < tm < T , Imp(y)(tk ) = y(tk+ ) − y(tk− ), y(tk+ ) = limh→0+ y(tk + h) and y(tk− ) = limh→0+ y(tk − h) represent the right and left limits of y(t) at t = tk in the sense of time scales, that is, tk + h ∈ (0, T ) ∩ T for each h in a neighbourhood of 0 and in addition, if tk is right scattered, then y(tk+ ) = y(tk ), whereas, if tk is left scattered, then y(tk− ) = y(tk ). Periodic boundary value problems (PBVPs) for impulsive differential (resp. difference) equations have drawn much attention, see [9–12,14,19–28] and references therein. In 2000, Nieto and Rodriguez-Lopez [21,22] introduced a new concept of lower and upper solutions for the boundary value problem  0 y (t) = g(t, y(t), y(w(t))), 0 ≤ w(t) ≤ t, t ∈ J = [0, T ], (1.3) y(0) = y(T ). Following them, Ding et al. [10] present a similar definition of lower and upper solutions to a class of first-order impulsive functional differential equations. Inspired by these two works, in this paper, we extend the new concept of lower and upper solutions for the first-order impulsive dynamic equation (1.2). By utilizing the method of lower and upper solutions coupled with the monotone iterative technique, we establish the existence of extremal solutions for PBVP (1.2). To the best of our knowledge, the problems concerning the existence of extremal solutions to periodic boundary value problems for impulsive dynamic equations on time scales have not been considered yet. The remainder of the paper is organized as follows. Section 2 contains the necessary definitions and some fundamental properties of the exponential function. In Section 3, first we present some new comparison principles and then establish the existence of extremal solutions. Finally, an example is provided in Section 4 to illustrate the sharpness of our main results. 2. Preliminaries For convenience, we first introduce some basic knowledge and several definitions on time scales, which also can be found in [1–3,7,8,17,18]. By a time scale we mean an arbitrary nonempty closed subset of R and we denote the time scale by T throughout the paper. Examples of time scales are N, Z, R, Cantor set and so on. The handicap that time scales are not necessarily connected is eliminated by utilizing the notion of jump operators which we shall next define. Definition 2.1. For t < sup T and r > inf T, the mappings σ, ρ : T → T, σ (t) = inf{s ∈ T : s > t},

ρ(r ) = sup{s ∈ T : s < r }

are called the forward jump operator and backward jump operator, respectively. In the case T = R, σ (t) = ρ(t) = t, if T = Z, then σ (t) = t + 1, ρ(t) = t − 1. Definition 2.2. For t ∈ T, t is said to be right scattered if σ (t) > t, right dense if σ (t) = t, left scattered if ρ(t) < t, left dense if ρ(t) = t. Definition 2.3. If T has a right-scattered minimum m, define Tk = T \ {m}; otherwise set Tk = T. If T has a leftscattered maximum M, define Tk = T \ {M}; otherwise let Tk = T. The notations [0, T ], [0, T ), and so on, will denote time scale intervals such as [0, T ] = {t ∈ T : 0 ≤ t ≤ T } etc., where 0, T ∈ T with 0 < ρ(T ).

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Definition 2.4. For x : T → R and t ∈ Tk , we define the delta derivative of x(t), x 1 (t), to be the number (when it exists) with the property that, for any  > 0, there is a neighbourhood U of t such that |[x(σ (t)) − x(s)] − x 1 (t)[σ (t) − s]| < |σ (t) − s|,

for all s ∈ U.

Note that if T = R, x 1 (t) = x 0 (t), and if T = Z, x 1 (t) = x(t + 1) − x(t). Definition 2.5. If F 1 (t) = f (t), then the delta integral is defined by Z t f (s)1s = F(t) − F(a). a

Definition 2.6. A function f : T → R is called rd-continuous provided it is continuous at right-dense points in T and its left-side limits exist (finite) at left-dense points in T, write f ∈ Crd (T) = Crd (T, R). Definition 2.7. A function p : T → R is called regressive if 1 + µ(t) p(t) 6= 0 for all t ∈ Tk , where µ(t) = σ (t) − t, which is said to be the graininess function. The set of all regressive and rd-continuous functions f : T → R will be denoted in this paper by R = R(T) = R(T, R). Definition 2.8. If p ∈ R, we define the exponential function by ( log(1 + hz) Z t  e p (t, s) = exp ξµ(τ ) ( p(τ ))1τ for s, t ∈ T, with ξh (z) = h s z

if h 6= 0, if h = 0.

It is known that y(t) = e p (t, t0 ) is the unique solution of the initial value problem y 1 (t) = p(t)y, y(t0 ) = 1. Let p, q ∈ R, define p ⊕ q = p + q + µpq,

p := −

p , 1 + µp

p q := p ⊕ ( q).

We now give some fundamental properties of the exponential function. Theorem 2.1. Assume that p, q ∈ R, then the following hold: (i) e0 (t, s) ≡ 1 and e p (t, t) ≡ 1; (ii) e p (σ (t), s) = (1 + µ(t) p(t))e p (t, s); 1 = e p (t, s); (iii) e p (t,s) 1 (iv) e p (t, s) = e p (s,t) = e p (s, t); (v) e p (t, r )e p (r, s) = e p (t, s); (vi) e p (t, s)eq (t, s) = e p⊕q (t, s);

(vii)

e p (t,s) eq (t,s)

= e p q (t, s).

3. Main results We will assume for the remainder of the paper that, the points of impulse tk , k = 1, 2, . . . , m are right dense (where we only consider pulses in right-dense points, other cases may be considered similarly). Consider the space PC = {y : [0, σ (T )] → R | yk ∈ C(Jk , R), k = 1, 2, . . . , m and there exist y(tk+ ) and y(tk− ) with y(tk− ) = y(tk ), k = 1, 2, . . . , m}, which is a Banach space with norm kyk PC = max{kyk k Jk , k = 0, 1, . . . , m}, where yk is the restriction of y to Jk := (tk , tk+1 ] ⊂ (0, σ (T )], k = 1, 2, . . . , m and J0 := [0, t1 ], tm+1 = σ (T ).

F. Geng et al. / Nonlinear Analysis 69 (2008) 4074–4087

Definition 3.1. A function y ∈ PC y satisfies the dynamic equation

T

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C 1 (J \ {t1 , t2 . . . , tm }, R) is said to be a solution of PBVP (1.2) if and only if

y 1 (t) = f (t, y(t)) everywhere on J \ {tk }, k = 1, 2, . . . , m, the impulsive conditions Imp(y)(tk ) = y(tk+ ) − y(tk− ) = Ik (y(tk− )) for each k = 1, 2, . . . , m, and the periodic boundary condition y(0) = y(σ (T )). We introduce the following concept of lower and upper solutions for PBVP (1.2). T Definition 3.2. A function α ∈ PC C 1 (J \ {t1 , t2 . . . , tm }, R) is called a lower solution of PBVP (1.2) if  1 α (t) ≤ f (t, α(t)) − rα(t) , t ∈ J, t 6= tk , Imp(α)(tk ) ≤ Ik (α(tk )) − dαk , k = 1, 2, . . . , m, where rα(t)

 0 α(0) ≤ α(σ (T )); = Mt + 1 [α(0) − α(σ (T ))] α(0) > α(σ (T )),  σ (T )

and dαk

 0 α(0) ≤ α(σ (T )); = L k tk [α(0) − α(σ (T ))] α(0) > α(σ (T )),  σ (T )

for 1 − µ(t)M > 0 and L k < 1. T Definition 3.3. A function β ∈ PC C 1 (J \ {t1 , t2 . . . , tm }, R) is called an upper solution of PBVP (1.2) if  1 β (t) ≥ f (t, β(t)) + r¯β(t) , t ∈ J, t 6= tk , Imp(β)(tk ) ≥ Ik (β(tk )) + r¯βk , k = 1, 2, . . . , m, where r¯β(t)

 0 β(0) ≥ β(σ (T )); = Mt + 1 [β(σ (T )) − β(0)] β(0) < β(σ (T )),  σ (T )

and r¯βk

 0 β(0) ≥ β(σ (T )); = L k tk [β(σ (T )) − β(0)] β(0) < β(σ (T )),  σ (T )

for 1 − µ(t)M > 0 and L k < 1. Remark. The above definitions reduce to the classical lower and upper solutions if α(0) ≤ α(σ (T )) and β(0) ≥ β(σ (T )) hold simultaneously. In fact, functions α and β need not satisfy α(0) ≤ α(σ (T )) and β(0) ≥ β(σ (T )) in the above two definitions. Lemma 3.1. Suppose h : T → R is rd-continuous and µ < M1 , then (i) y ∈ PC is a solution of the following periodic boundary value problem  1 t ∈ J \ {tk }, k = 1, 2, . . . , m,  y (t) + M y(t) = h(t), Imp(y)(tk ) = −L k y(tk ) + Ik (η(tk )) + L k η(tk ), k = 1, 2, . . . , m,  y(0) = y(σ (T )),

(3.1)

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if and only if Z y(t) =

σ (T )

G(t, s)h(s)1s + 0

m X

G(t, tk )[−L k y(tk ) + Ik (η(tk )) + L k η(tk )],

t ∈ [0, σ (T )],

k=1

where η ∈ PC and  e (t, σ (s)) −M  ,  1 − −M (σ (T ), 0) G(t, s) = e e(σ (T ), 0)e−M (t, σ (s))   −M , 1 − e−M (σ (T ), 0)

0 ≤ σ (s) < t ≤ σ (T ),

(ii) furthermore, for 0 ≤ L k < 1, if

Pm

1 1−e−M (σ (T ),0)

0 ≤ t ≤ σ (s) ≤ σ (T ); k=1

L k < 1, then PBVP (3.1) has a unique solution.

Proof. For notational convenience, denote ω(tk ) = −L k y(tk ) + Ik (η(tk )) + L k η(tk ),

k = 1, 2, . . . , m.

Notice that y 1 (t) + M y(t) = h(t) is equivalent to y 1 (t) + M(y σ (t) − µ(t)y 1 (t)) = h(t), i.e. y 1 (t) + (−M)y σ (t) =

h(t) . 1 − µ(t)M

(3.2)

So if y is a solution of PBVP (3.1), then it satisfies (3.2). Multiplying (3.2) by e (−M) (t, 0) and integrating both sides from 0 to t (here t ∈ (0, t1 ]) yields Z t Z t e (−M) (s, 0) h(s)1s, [y(s)e (−M) (s, 0)]1 1s = 0 0 1 − µ(s)M Z t e (−M) (s, 0) y(t)e (−M) (t, 0) − y(0) = h(s)1s. 0 1 − µ(s)M It then follows from Theorem 2.1 that Z t e (−M) (s, t) y(t) = y(0)e (−M) (0, t) + h(s)1s 0 1 − µ(s)M Z t e−M (t, s) = y(0)e−M (t, 0) + h(s)1s 0 1 − µ(s)M Z t = y(0)e−M (t, 0) + e−M (t, σ (s))h(s)1s, t ∈ [0, t1 ].

(3.3)

0 e−M (t,s) Here we have used the equations 1−µ(s)M = Now assume that y|(t1 ,t2 ] is a solution to

y 1 (t) + M y(t) = h(t),

e−M (t,s) e−M (σ (s),s)

= e−M (t, σ (s)).

t ∈ (t1 , t2 ],

(3.4)

and y(t1+ ) − y(t1− ) = ω(t1 ).

(3.5)

Multiply (3.4) by e (−M) (t, 0) and integrate from t1 to t ∈ (t1 , t2 ], we arrive at Z t y(t) = y(t1+ )e−M (t, t1 ) + e−M (t, σ (s))h(s)1s. t1

(3.6)

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Combining (3.6) with (3.5), one may easily acquire Z t y(t) = [y(t1− ) + ω(t1 )]e−M (t, t1 ) + e−M (t, σ (s))h(s)1s.

(3.7)

According to (3.3), one knows that Z − y(t1 ) = y(0)e−M (t1 , 0) +

(3.8)

t1

t1

e−M (t1 , σ (s))h(s)1s.

0

Based on (3.7) and (3.8) and Theorem 2.1, we achieve Z t y(t) = y(0)e−M (t, 0) + e−M (t, σ (s))h(s)1s + e−M (t, t1 )ω(t1 ),

t ∈ (t1 , t2 ].

0

If we continue with the above procedure, we obtain for t ∈ (tk , tk+1 ] that Z t − y(t) = y(tk )e−M (t, tk ) + e−M (t, σ (s))h(s)1s + e−M (t, tk )ω(tk ), tk

with y(tk− ) = y(0)e−M (tk , 0) +

Z

tk

X

e−M (tk , σ (s))h(s)1s +

0

e−M (t j , tk )ω(t j ).

0
As a result, we have for t ∈ J y(t) = y(0)e−M (t, 0) +

Z

t

e−M (t, σ (s))h(s)1s +

0

X

e−M (t, tk )ω(tk ).

(3.9)

0
Since y(0) = y(σ (T )), (3.9) then produces that Z σ (T ) y(0) = y(0)e−M (σ (T ), 0) + e−M (σ (T ), σ (s))h(s)1s + 0

X

e−M (σ (T ), tk )ω(tk ),

0
which implies 1 y(0) = (1 − e−M (σ (T ), 0))

σ (T )

Z

! e−M (σ (T ), σ (s))h(s)1s +

0

X

e−M (σ (T ), tk )ω(tk ) .

(3.10)

0
Substituting (3.10) into (3.9), one reaches ! Z σ (T ) X e−M (t, 0) y(t) = e−M (σ (T ), σ (s))h(s)1s + e−M (σ (T ), tk )ω(tk ) (1 − e−M (σ (T ), 0)) 0 0
e−M (t, 0) = (1 − e−M (σ (T ), 0))

0
Z

e−M (σ (T ), σ (s))h(s)1s +

Z

0

σ (T )

! e−M (σ (T ), σ (s))h(s)1s

t

!

e−M (t, 0) + (1 − e−M (σ (T ), 0))

X

e−M (σ (T ), tk )ω(tk ) +

0
1 − e−M (σ (T ), 0) + 1 − e−M (σ (T ), 0)

Z

1 = (1 − e−M (σ (T ), 0))

Z

t

0

e−M (σ (T ), tk )ω(tk )

t≤tk <σ (T )

! X

e−M (t, σ (s))h(s)1s +

0 t

X

e−M (t, tk )ω(tk )

0
e−M (t, σ (s))h(s)1s +

Z t

σ (T )

! e−M (σ (T ), 0)e−M (t, σ (s))h(s)1s

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! X X 1 e−M (σ (T ), 0)e−M (t, tk )ω(tk ) e−M (t, tk )ω(tk ) + + (1 − e−M (σ (T ), 0)) 0
k=1

In the above, we have utilized the fact that tk is right-dense, which means σ (tk ) = tk . Conversely, assume Z σ (T ) m X y(t) = G(t, s)h(s)1s + G(t, tk )ω(tk ). 0

Notice that Z σ (T ) 0

(3.11)

k=1

"Z t 1 1 G (t, s)h(s)1s = e−M (t, σ (s))h(s)1s + e−M (σ (t), σ (t))h(t) 1 − e−M (σ (T ), 0) 0 # Z σ (T ) 1 + e−M (t, σ (s))h(s)1s − e−M (σ (T ), 0)e−M (σ (t), σ (t))h(t) 1

t

Z

σ (T )

G(t, s)h(s)1s + h(s).

= −M 0

Similarly, m X

G 1 (t, tk )ω(tk ) = −M

k=1

m X

G(t, tk )ω(tk ).

k=1

As a result, 1-differentiation of (3.11) yields Z σ (T ) m X 1 G 1 (t, s)h(s)1s + G 1 (t, tk )ω(tk ) y (t) = 0

k=1 σ (T )

(Z

G(t, s)h(s)1s +

= −M 0

m X

) G(t, tk )ω(tk ) + h(t),

k=1

i.e. y 1 (t) + M y(t) = h(t),

t ∈ J, t 6= tk .

Obviously y(tk+ ) − y(tk− ) = ω(tk ),

k = 1, 2, . . . , m,

and

y(0) = y(σ (T )).

This completes the proof of (i). To prove (ii), define an operator F by Z σ (T ) m X (F y)(t) := G(t, s)h(s)1s + G(t, tk )[−L k y(tk ) + Ik (η(tk )) + L k η(tk )], 0

t ∈ [0, σ (T )],

k=1

where G is defined as before. Clearly, the fixed point of F is equivalent to the solution of PBVP (3.1). In view of 1 maxt∈[0,σ (T )] {G(t, s)} = 1−e−M (σ (T ),0) , for any x, y ∈ PC, we derive m X |(F x)(t) − (F y)(t)| ≤ sup G(t, tk )(−L k )(x(tk ) − y(tk )) t∈[0,σ (T )] k=1 ≤

m X 1 L k kx − yk PC . 1 − e−M (σ (T ), 0) k=1

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Then m X 1 L k kx − yk PC . 1 − e−M (σ (T ), 0) k=1 Pm 1 Based on the fact 1−e−M (σ k=1 L k < 1 and the Banach Contraction fixed-point theorem, F has a unique fixed (T ),0) ∗ point y , which exactly is the solution of PBVP (3.1). The proof is thus finished. 

k(F x)(t) − (F y)(t)k PC ≤

Remark. Noticing that if T = R, then e−M (t, t0 ) = e−M(t−t0 ) and σ (t) = t for any t ∈ R. Therefore, the Green’s function obtained in the above lemma is in accordance with that in [19,20] when T = R. Lemma 3.2. Assume the following conditions: (C0 ) the sequence {tk }m k=1 satisfies 0 ≤ t0 < t1 < · · · < tm ; (C1 ) M satisfies µ < M1 and for k = 1, 2, . . . , t ≥ t0 , there are y 1 (t) ≤ −M y(t), y(tk+ )

(3.12)

≤ dk y(tk ),

where dk are constants. Then Y y(t) ≤ y(t0 ) dk e−M (t, t0 ). t0
Proof. Noticing that µ < M1 , so e (−M) (t, t0 ) > 0. Let t ∈ [t0 , t1 ], then we get from (3.12) that
1 1 e (−M) (t, t0 )y(t) = e (−M) (t, t0 )y(t) + e (−M) (σ (t), t0 )y 1 (t) = (−M)e (−M) (t, t0 )y(t) + (1 + µ(t) (−M))e (−M) (t, t0 )y 1 (t) M 1 = e (−M) (t, t0 )y(t) + e (−M) (t, t0 )y 1 (t) 1 − µ(t)M 1 − µ(t)M
e (−M)(t,t0 ) = M y(t) + y 1 (t) ≤ 0. 1 − µ(t)M It then follows that y(t) ≤ y(t0 )e (−M) (t0 , t) = y(t0 )e−M (t, t0 ), For t ∈ (t1 , t2 ], we have y 1 (t) ≤ −M y(t),

t 6= t1 ,

y(t1+ ) ≤ d1 y(t1 ). Thus, owing to (3.13), we acquire y(t) ≤ y(t1+ )e−M (t, t1 ) ≤ d1 y(t1 )e−M (t, t1 ) ≤ d1 y(t0 )e−M (t1 , t0 )e−M (t, t1 ) = d1 y(t0 )e−M (t, t0 ). Repeatedly, we achieve for t ∈ (tk , tk+1 ]: y(t) ≤ y(tk+ )e−M (t, t0 ), y(tk+ ) ≤ dk y(tk ) ≤ d1 d2 . . . dk y(t0 )e−M (t, t0 ). Hence y(t) ≤ y(t0 )

Y t0
dk e−M (t, t0 ).



t ∈ [t0 , t1 ].

(3.13)

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Lemma 3.3. Suppose that y ∈ PC satisfies  1 y (t) ≤ −M y(t) − r y(t) , t ∈ J, t 6= tk , Imp(y)(tk ) ≤ −L k y(tk ) − d yk , k = 1, 2, . . . , m, Q for µ < M1 , L k < 1, k = 1, 2, . . . , m, m k=1 (1 − L k )e−M (σ (T ), 0) < 1 and  0 y(0) ≤ y(σ (T )); r y(t) = Mt + 1 [y(0) − y(σ (T ))] y(0) > y(σ (T )),  σ (T )  0 y(0) ≤ y(σ (T )); d yk = L k tk [y(0) − y(σ (T ))] y(0) > y(σ (T )).  σ (T )

(3.14)

Then y(t) ≤ 0 for t ∈ J . Proof. Case 1: y(0) ≤ y(σ (T )), then r y(t) = 0 and d yk = 0 hold. It follows from (3.14) and Lemma 3.2 that Y y(t) ≤ y(0) (1 − L k )e−M (t, 0). (3.15) 0
Q It is easy to see that m k=1 (1 − L k )e−M (t, 0) > 0 due to the fact L k < 1 and µ < Let t = σ (T ) in (3.15), thus we achieve Y y(0) ≤ y(σ (T )) ≤ y(0) (1 − L k )e−M (σ (T ), 0),

1 M.

So it suffices to verify y(0) ≤ 0.

0
which gives rise to " # m Y y(0) 1 − (1 − L k )e−M (σ (T ), 0) ≤ 0.

(3.16)

k=1

Q Noticing that m k=1 (1 − L k )e−M (σ (T ), 0) < 1, (3.16) thus leads to y(0) ≤ 0. From (3.15) we see that y(t) ≤ 0 for t ∈ J. Now turning to the case y(0) > y(σ (T )). Set y¯ (t) = y(t) + g(t),

t ∈ J,

where g(t) = − y(σ (T ))], t ∈ J . Clearly, g(0) = 0, g(σ (T )) = y(0) − y(σ (T )) > 0 and g(t) ≥ 0 on J . Consequently, we have t σ (T ) [y(0)

y¯ (0) = y(0) = y(σ (T )) + g(σ (T )) = y¯ (σ (T )), and by means of (3.14), we obtain y¯ 1 (t) = y 1 (t) + g 1 (t) Mt + 1 1 ≤ −M y(t) − [y(0) − y(σ (T ))] + [y(0) − y(σ (T ))] σ (T ) σ (T ) = −M y¯ (t), t ∈ J, t 6= tk , Imp( y¯ )(tk ) = Imp(y)(tk ) + Imp(g)(tk ) L k tk ≤ −L k y(tk ) − [y(0) − y(σ (T ))] σ (T ) = −L k y¯ (tk ), k = 1, 2, . . . , m. In view of case 1, we may easily get y(t) ≤ 0 for t ∈ J . This completes the proof.



For α, β ∈ PC, we write α ≤ β if α(t) ≤ β(t) for all t ∈ J . In such a case, we denote [α, β] = {y ∈ PC : α(t) ≤ y(t) ≤ β(t), t ∈ J }.

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Theorem 3.1. Suppose the following conditions: (H1 ) the functions α, β are lower and upper solutions of PBVP (1.2) respectively, such that α(t) ≤ β(t) for t ∈ J ; (H2 ) f (t, y) : T × R → R is rd-continuous in the first variable and continuous in the second variable, such that f (t, y) − f (t, x) ≥ −M(y − x) (H3 ) the constant M fulfils µ <

1 M

m X 1 Lk < 1 1 − e−M (σ (T ), 0) k=1

for

α(t) ≤ x(t) ≤ y(t) ≤ β(t),

t ∈ J;

and 0 ≤ L k < 1, such that and

m Y

(1 − L k )e−M (σ (T ), 0) < 1;

k=1

(H4 ) for k = 1, 2, . . . , m all functions Ik satisfy Ik (y(tk )) − Ik (x(tk )) ≥ −L k (y(tk ) − x(tk ))

for α(tk ) ≤ x(tk ) ≤ y(tk ) ≤ β(tk ).

Then there exist monotone sequences {αn (t)} and {βn (t)} with α0 = α, β0 = β, such that lim αn (t) = y∗ (t),

n→∞

lim βn (t) = y ∗ (t)

n→∞

uniformly on J , and y∗ (t), y ∗ (t) are the minimal and the maximal solutions of PBVP (1.2), respectively, such that α = α0 ≤ α1 ≤ α2 ≤ · · · ≤ αn ≤ y∗ ≤ y ≤ y ∗ ≤ βn ≤ · · · ≤ β2 ≤ β1 ≤ β0 = β

on J,

where y is any solution of PBVP (1.2) satisfying α(t) ≤ y(t) ≤ β(t) on J . Proof. For any rd-continuous η(t) ∈ [α, β], consider the linear PBVP (3.1) with h(t) = f (t, η(t))+Mη(t). Condition (H3 ) and Lemma 3.1 mean that PBVP (3.1) has exactly one solution y(t). Define the operator A : [α, β] → PC, so that y(t) = Aη(t), then A is an operator from [α, β] to PC. We complete the proof by three claims. Claim 1. The operator A has the following properties: (i) α ≤ Aα, Aβ ≤ β; (ii) A is monotonically nondecreasing in [α, β], i.e. for any η1 , η2 ∈ [α, β], η1 ≤ η2 implies Aη1 ≤ Aη2 . To prove (i), set y¯ = α − α1 , where α1 = Aα. Then we obtain y¯ (0) − y¯ (σ (T )) = α(0) − α(σ (T )) since α1 (0) = α1 (σ (T )). Owing to (3.1) and (H1 ), we acquire y¯ 1 (t) = α 1 (t) − α11 (t) ≤ f (t, α(t)) − rα(t) − [−Mα1 (t) + f (t, α(t)) + Mα(t)] = −M y¯ (t) − r y¯ (t) , t ∈ J, t 6= tk , Imp( y¯ )(tk ) = Imp(α)(tk ) − Imp(α1 )(tk ) ≤ Ik (α(tk )) − dαk − [−L k α1 (tk ) + Ik (α(tk )) + L k α(tk )] = −L k y¯ (tk ) − d y¯ k , k = 1, 2, . . . , m, where rα(t) , r y¯ (t) , dαk and d y¯ k (k = 1, 2, . . . , m) are given by  0 y¯ (0) ≤ y¯ (σ (T )); rα(t) = r y¯ (t) = Mt + 1 [ y¯ (0) − y¯ (σ (T ))] y¯ (0) > y¯ (σ (T )),  σ (T ) and dαk = d y¯ k

 0 = L k tk [ y¯ (0) − y¯ (σ (T ))]  σ (T )

y¯ (0) ≤ y¯ (σ (T )); y¯ (0) > y¯ (σ (T )).

By Lemma 3.3, we get y¯ (t) ≤ 0 on J , i.e. α ≤ Aα. Similar arguments show that Aβ ≤ β. To prove (ii), take y1 = Aη1 , y2 = Aη2 , where η1 ≤ η2 on J and η1 , η2 ∈ [α, β]. Set y¯ = y1 − y2 , employing (3.1), (H2 ) and (H4 ), we achieve y¯ 1 (t) = = ≤ =

y11 (t) − y21 (t) [−M y1 (t) + f (t, η1 (t)) + Mη1 (t)] − [−M y2 (t) + f (t, η2 (t)) + Mη2 (t)] −M(y1 (t) − y2 (t)) −M y¯ (t), t ∈ J, t 6= tk ,

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Imp( y¯ )(tk ) = = ≤ =

Imp(y1 )(tk ) − Imp(y2 )(tk ) −L k y1 (tk ) + Ik (η1 (tk )) + L k η1 (tk ) − [−L k y2 (tk ) + Ik (η2 (tk )) + L k η2 (tk )] −L k (y1 (tk ) − y2 (tk )) −L k y¯ (tk ), k = 1, 2, . . . , m,

y¯ (0) = y¯ (σ (T )). By virtue of Lemma 3.3, there is y¯ (t) ≤ 0 on J , i.e. Aη1 ≤ Aη2 . Claim 2. There exist solutions of PBVP (1.2). Now, define the sequences {αn (t)}, {βn (t)} with α0 = α, β0 = β such that αn+1 = Aαn , βn+1 = Aβn . Note that αn , βn (n = 1, 2, . . .) satisfy Z σ (T ) m X αn (t) = G(t, s)h n−1 (s)1s + G(t, tk )[−L k αn (tk ) + Ik (αn−1 (tk )) + L k αn−1 (tk )], t ∈ [0, σ (T )], 0

βn (t) =

k=1 σ (T )

Z

G(t, s)h¯ n−1 (s)1s +

0

m X

G(t, tk )[−L k βn (tk ) + Ik (βn−1 (tk )) + L k βn−1 (tk )],

t ∈ [0, σ (T )],

k=1

where h n−1 = f (t, αn−1 (t)) + Mαn−1 (t), h¯ n−1 = f (t, βn−1 (t)) + Mβn−1 (t),

t ∈ [0, σ (T )], t ∈ [0, σ (T )].

From Claim 1, it is easy to see that the sequences {αn (t)}, {βn (t)} satisfy α = α0 ≤ α1 ≤ α2 ≤ · · · ≤ αn ≤ · · · ≤ βn ≤ · · · ≤ β2 ≤ β1 ≤ β0 = β

on J.

Consequently, there exist y∗ and y ∗ such that limn→∞ αn (t) = y∗ (t), limn→∞ βn (t) = y ∗ (t) uniformly on J . Clearly, y∗ , y ∗ satisfy PBVP (1.2). Claim 3. y∗ , y ∗ are extremal solutions of PBVP (1.2). Let y(t) be any solution of (1.2) such that y ∈ [α, β]. Assume that there exists a positive integer n such that αn (t) ≤ y(t) ≤ βn (t) on J . Based on the monotonically nondecreasing property of A, it then follows that αn+1 = Aαn ≤ Ay = y, i.e. αn+1 (t) ≤ y(t) on J . Similarly, one derives y(t) ≤ βn+1 (t) on J . Since α0 (t) ≤ y(t) ≤ β0 (t) on J , by induction we see that αn (t) ≤ y(t) ≤ βn (t) on J for every n. Therefore y∗ (t) ≤ y(t) ≤ y ∗ (t) on J as n → ∞. The proof is then finished.  4. An example Consider the following periodic boundary value problem on a time scale T:    1 1 \ 1 t  y (t) = sin y(t) − 4y(t) + e , t ∈ 0, T, t 6= ,    3 5     1 1 Imp(y) = ,  5  15       y(0) = y 1 . 3

(4.1)

Case 1. If T = R. In this case f (t, x) − f (t, y) = sin x − 4x + et − (sin y − 4y + et ) = sin x − sin y − 4(x − y) ≥ −5(x − y) for all t ∈ [0, 13 ], x, y ∈ R, x ≥ y. This shows condition (H2 ) is fulfilled for M = 5.

(4.2)

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To verify (H3 ), take L 1 = 13 . Noticing that when T = R, µ(t) = 0, µ < we know that X 1 1 1 ≈ 0.41 < 1, Lk = 1 − e−M (σ (T ), 0) k=1 3 1 − e− 53

Y

1 M

is naturally satisfied. By computation,

(1 − L k )e−M (σ (T ), 0) =

k=1

2 −5 e 3 < 1, 3

(4.3)

which implies that (H3 ) is true. 1 , the condition (H4 ) holds as well. Since Ik ≡ 15 It remains to find lower and upper solutions. In fact,    1   , 1, for t ∈ 0, 3 5   α(t) = e−t − π, β(t) = 6 1 1  10   , for t ∈ , 5 5 3 are a pair of lower and upper solutions which satisfy α ≤ β for all t ∈ [0, 31 ]. 1

1

1

3 3 Due to the fact that α(0) = 1 − 10 π > e− 3 − 10 π = α( 31 ), then rα(t) = 3(5t + 1)(1 − e− 3 ), dαk = 15 (1 − e− 3 ). 3 3 0 −t −t One may see that α (t) = −e , f (t, α(t)) = sin(e − 10 π ) − 4(e−t − 10 π ) + et . By virtue of Mathematica, one 0 can easily get that mint∈[0, 1 ] ( f (t, α(t)) − rα(t) − α (t)) ≈ 0.52 > 0. Then we have 3

α (t) ≤ f (t, α(t)) − rα(t) , 0

  1 t ∈ 0, , 3

and    1 1 1 1 Imp(α) =0< − 1 − e− 3 ≈ 0.01, 5 15 5 which implies that α is a lower solution of PBVP (4.1). On the other hand, β(0) = 1 < 65 = β( 13 ) implies r¯β(t) = 3t + 35 , d¯βk = t ∈ [0, 13 ] \ { 15 }, therefore it follows that

1 25 .

By means of β 0 (t) = 0 for

     3 1 t 0   f (t, β(t)) + r ¯ = sin 1 − 4 + e + 3t + < −0.74 < 0 = β (t), t ∈ 0, , β(t)  5 5     6 3 6 1 1    f (t, β(t)) + r¯β(t) = sin − 4 × + et + 3t + , < −0.87 < 0 = β 0 (t), t ∈ , 5 5 5 5 3

(4.4)

and Imp(β)

  1 1 1 = 0.2 > + ≈ 0.11. 5 15 25

(4.5)

Consequently, β is an upper solution of PBVP (4.1). An application of Theorem 3.1 means that there exist monotone sequences that approximate the extremal solutions of PBVP (4.1). Obviously, since α(0) > α(T ) holds, the method of classical upper and lower solutions in [18] is not applicable. In fact, α must fulfil α(0) ≤ α(T ) in [18]. S Case 2. Let T = { 15 − ( 51 )N } [ 15 , 13 ], where N = {1, 2, . . .}. T In this case, (4.2) obviously holds for all t ∈ [0, 13 ] T, x, y ∈ R, x ≥ y. This implies that condition (H2 ) is true with M = 5. S Noticing that when T = { 15 − ( 15 )N } [ 51 , 13 ], then for t ∈ { 51 − ( 15 )N }, µ(t) = σ (t) − t = ( 15 ) N − ( 51 ) N +1 , for any T N ∈ N; while for t ∈ [ 15 , 13 ], we have µ(t) = 0. So, the inequality µ < M1 is always satisfied for all t ∈ [0, 13 ] T. Therefore let L 1 = 13 and by (4.3) the condition (H3 ) is fulfilled. Since Ik ≡

1 15 ,

condition (H4 ) is naturally true.

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Now, we shall find lower and upper solutions.    1 \   T, 1, for t ∈ 0, 3 5  \ α(t) = , β(t) = 1 1 6  10   , for t ∈ , T 5 5 3 are a pair of lower and upper solutions which satisfy α ≤ β for all t ∈ [0, 31 ]. T We may see that α(0) = α( 13 ), which means rα(t) = dαk = 0. Note that α 1 (t) = 0 for t ∈ [0, 13 ] T, 3 f (t, α(t)) = sin 10 − 65 + et . By a simple computation we obtain mint∈[0, 1 ] T T f (t, α(t)) ≈ 0.096 > 0, Then 3 we obtain that   1 \ α 1 (t) ≤ f (t, α(t)) − rα(t) , t ∈ 0, T, 3 and   1 1 Imp(α) =0< , 5 15 which implies α is a lower solution T of PBVP (4.1). Based on the fact that [0, 13 ] T ⊂ [0, 13 ] and (4.3) and (4.4), one derives that    1 \ 1   f (t, β(t)) + r ¯ < β (t), t ∈ 0, T, β(t)  5  \ 1 1    f (t, β(t)) + r¯β(t) < β 1 (t), t ∈ , T, 5 3 and Imp(β)

  1 1 1 = 0.2 > + ≈ 0.11. 5 15 25

Consequently, β is an upper solution of PBVP (4.1). Theorem 3.1 thus tells that there exist monotone sequences that approximate the extremal solutions of PBVP (4.1). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

R.P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999) 3–22. R.P. Agarwal, M. Bohner, D. O’Regan, A. Peterson, Dynamic equations on time scales: A survey, J. Comput. Appl. Math. 141 (2002) 1–26. F.M. Atici, D.C. Biles, First and second order dynamic equations with impulse, Adv. Difference Equ. 2 (2005) 119–132. A. Belarbi, M. Benchohra, A. Ouahab, Extremal solutions for impulsive dynamic equations on time scales, Comm. Appl. Nonlinear Anal. 12 (2005) 85–95. M. Benchohra, S.K. Ntouyas, A. Ouahab, Existence results for second-order boundary value problem of impulsive dynamic equations on time scales, J. Math. Anal. Appl. 296 (2004) 65–73. M. Benchohra, S.K. Ntouyas, A. Ouahab, Extremal solutions of second order impulsive dynamic equations on time scales, J. Math. Anal. Appl. 324 (2006) 425–434. M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhauser, Boston, 2001. M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003. L. Chen, J. Sun, Nonlinear boundary value problem for first order impulsive integro-differential equations of mixed type, J. Math. Anal. Appl. 325 (2007) 830–842. W. Ding, J. Mi, M. Han, Periodic boundary value problems for the first order impulsive functional differential equations, Appl. Math. Comput. 165 (2005) 433–446. S. Gao, L. Chen, J.J. Nieto, A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine 24 (2006) 6037–6045. W. Ge, The persistence of nonoscillatory solutions of difference equations under impulsive perturbations, Comput. Math. Appl. 50 (2005) 1579–1586. F. Geng, D. Zhu, Q. Lu, A New Existence result for impulsive dynamic equations on time scales, Appl. Math. Lett. 20 (2007) 206–212. Z. He, X. Zhang, Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions, Appl. Math. Comput. 156 (2004) 605–620.

F. Geng et al. / Nonlinear Analysis 69 (2008) 4074–4087

4087

[15] S. Heikkila, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Dikerential Equations, Marcel Dekker, Inc., New York, 1994. [16] J. Henderson, Double solutions of impulsive dynamic boundary value problems on a time scale, J. Difference Equ. Appl. 8 (2002) 345–356. [17] S. Hilger, Analysis on a measure chain—a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56. [18] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, Word Scientific, Singapore, 1989. [19] J. Li, J.J. Nieto, J. Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl. 325 (2007) 226–236. [20] J.J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl. 205 (1997) 423–433. [21] J.J. Nieto, R. Rodriguez-Lopez, Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary vlaue conditions, J. Comput. Appl. Math. 40 (2000) 433–442. [22] J.J. Nieto, R. Rodriguez-Lopez, Remarks on periodic boundary value problems for functional differential equations, J. Comput. Appl. Math. 158 (2003) 339–353. [23] J.J. Nieto, R. Rodriguez-Lopez, Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. Math. Anal. Appl. 318 (2006) 593–610. [24] J.J. Nieto, Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Anal. 51 (2002) 1223–1232. [25] J. Li, J. Shen, Periodic boundary value problems for delay differential equations with impulses, J. Comput. Appl. Math. 193 (2006) 563–573. [26] J.J. Nieto, R. Rodriguez-Lopez, New comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl. 328 (2007) 1343–1368. [27] S. Tang, R.A. Cheke, Y. Xiao, Optimal impulsive harvesting on non-autonomous BevertonHolt difference equations, Nonlinear Anal. 65 (2006) 2311–2341. [28] X. Yang, J. Shen, Periodic boundary value problems for second-order impulsive integro-differential equations, J. Comput. Appl. Math., in press (doi:10.1016/j.cam.2006.10.082).