WITH IMPULSE
F. M. ATICI AND D. C. BILESReceived 3 December 2004 and in revised form 11 February 2005
We present existence results for discontinuous first- and continuous second-order dy-namic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions.
1. Introduction
We first briefly survey the recent results for existence of solutions to first-order problems with fixed-time impulses. Periodic boundary conditions using upper and lower solutions were considered in [19], using degree theory. A nonlinear alternative of Leray-Schauder type was used in [15] for initial conditions or periodic boundary conditions. The mono-tone iterative technique was employed in [14] for antiperiodic and nonlinear boundary conditions. Lower and upper solutions and periodic boundary conditions were studied in [20]. Semilinear damped initial value problems in a Banach space using fixed point theory were investigated in [6]. In [9], existence of solutions for the differential equa-tionu(t)=q(u(t))g(t,u(t)) subject to a general boundary condition is proven, in which
g is Carath´eodory andq∈L∞, and existence of lower and upper solutions is assumed. Schauder’s fixed point theorem was used there. This generalized an earlier result found in [18]. It appears that little has been done concerning dynamic equations with impulses on time scales (see [4,5,16] for earlier results). InSection 2, the present paper uses ideas from [9] to prove an existence result for discontinuous dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions.
The study of boundary value problems for nonlinear second-order differential equa-tions with impulses has appeared in many papers (see [10,11,13] and the references therein). InSection 3, we use ideas from [12,16] to prove an existence result for second-order dynamic equations on a time scale subject to fixed-time impulses and nonlinear boundary conditions. Nonlinear boundary conditions cover, among others, the periodic and the Dirichlet conditions, and have been introduced for ordinary differential equa-tions by Adje in [1]. Assuming the existence of a lower and an upper solution, we prove that the solution of the boundary value problem stays between them.
In [2], it was shown that the upper and lower solution method will not work for first-order dynamic equations involving∆-derivatives, unless restrictive assumptions are
made. Hence, inSection 2, we work with the∇-derivative. InSection 3, we can use the more conventional∆-derivative.
The monographs [17, 21] are good general references on impulsive differential equations—discussion of applications may be found in these books. Applications of the results given in this paper could involve those typically modelled on time scales which are subjected to sudden major influences, for example, an insect population sprayed with an insecticide or a financial market affected by a major terrorist attack.
For our purposes, we letT be a time scale (a closed subset ofR), let [a,b] be the closed and bounded interval inT, that is, [a,b] := {t∈T:a≤t≤b}anda,b∈T. For the readers’ convenience, we state a few basic definitions on a time scaleT[7,8].
Obviously a time scaleTmay or may not be connected. Therefore, we have the concept offorwardandbackward jump operatorsas follows: defineσ,ρ:T→Tby
σ(t)=inf{s∈T:s > t}, ρ(t)=sup{s∈T:s < t}. (1.1)
Ifσ(t)=t,σ(t)> t,ρ(t)=t,ρ(t)< t, thent∈Tis calledright dense(rd),right scattered, left dense,left scattered, respectively. We also define thegraininess functionµ:T→[0,∞) as µ(t)=σ(t)−t. The setsTκ,T
κ which are derived from T are as follows: if T has a left-scattered maximumt1, then Tκ=T− {t1}, otherwise Tκ=T. If Thas a
right-scattered minimumt2, thenTκ=T− {t2}, otherwiseTκ=T. If f :T→Ris a function, we define the functions fσ:Tκ→Rby fσ(t)= f(σ(t)) for allt∈Tκ, fρ:T
κ→Rby
fρ(t)= f(ρ(t)) for allt∈T
κandσ0(t)=ρ0(t)=t.
Iff :T→Ris a function andt∈Tκ, then thedelta derivativeof f at a pointtis defined to be the number f∆(t) (provided it exists) with the property that, for eachε >0, there is a neighborhood ofU1oftsuch that
f
σ(t)−f(s)−f∆(t)σ(t)−s≤εσ(t)−s, (1.2)
for alls∈U1. Ift∈Tκ, then we define thenabla derivativeof f at a pointtto be the number f∇(t) (provided it exists) with the property that, for eachε >0, there is a neigh-borhood ofU2oftsuch that
f
ρ(t)−f(s)−f∇(t)ρ(t)−s≤ερ(t)−s, (1.3)
for alls∈U2.
Remark 1.1. IfT=R, then f∆(t)= f∇(t)= f(t), and ifT=Z, then f∆(t)=∆f(t)= f(t+ 1)−f(t) and f∇(t)= ∇f(t)= f(t)−f(t−1).
A functionF:T→Ris called a∆-antiderivative of f :T→RprovidedF∆(t)=f(t) holds for allt∈Tk. Then the Cauchy∆-integral fromatotof f is defined by
t
A functionΦ:T→Ris called a∇-antiderivative of f :T→RprovidedΦ∇(t)=f(t) for allt∈Tk. We then define the Cauchy∇-integral fromatotof f by
t
a f(s)∇s=Φ(t)−Φ(a) ∀t∈T. (1.5)
Note that, in the caseT=R, we have b
a f(t)∆t= b
a f(t)∇t= b
a f(t)dt, (1.6)
and, in the caseT=Z, we have
b
a f(t)∆t= b−1
k=a
f(k), b
a f(t)∇t= b
k=a+1
f(k), (1.7)
wherea,b∈Twitha≤b.
There are two types of impulse effects that are studied in the literature. The first is the “fixed-time impulse”: a set of times 0< t1< t2<···< tn< Tis specified, and the solution is required to satisfy
ut+k=Ik
utk
(1.8)
fork=1, 2,. . .,n, where the functionsIkprovide the “impulse.” Also studied are “variable-time impulses,” in which a set of curvest=τ1(x),t=τ2(x),. . .,t=τn(x) is given, and the solution satisfiesu(t+)=I
k(u(t)) fort=τk(u(t)),k=1, 2,. . .,n.Impulses of both types introduce discontinuities in the solution. As mentioned in [17] and other works in the reference list, applications involving impulse effects can be found in biology, medicine, physics, economics, pharmacokinetics, and engineering. In this paper, we consider fixed-time impulses. Without loss of generality, we investigate systems with a single impulse.
2. First order
Let 0,t1,T∈Twith 0< t1< Tandt1right dense. LetJ=[0,T]∩T,
u∇(t)=gt,u(t), t∈J\t1 , (2.1)
ut+ 1
=Iut1
, (2.2)
Bu(0),u(T)=0. (2.3)
Note that (2.3) covers as special cases many initial and boundary conditions found in the literature. LetJ1=[0,t1]∩J,J2=(t1,T]∩J. Define
b
a y(s)∇s=
(a,b]y(s)∇s, where
Letuibe the restriction ofu:J→RtoJi,i=1, 2, then
ᏯJ1
=u:J1−→R:uis continuous onJ1 ,
ᏯJ2
=u:J2−→R:uis continuous onJ2andu
t+ 1
exists ,
A=u:J−→R:u1∈Ꮿ
J1
andu2∈Ꮿ
J2 .
(2.4)
Foru∈A, letu =sup{|u(t)|:t∈J}. (A, · ) is a Banach space. Foru,v∈A,
[u,v]≡w∈A:u(t)≤w(t)≤v(t)∀t∈J . (2.5)
Definition 2.1. u:T→Ris a solution of (2.1)–(2.3) if (i)u∈A,
(ii)
u(t)=u(0) + t
0g
s,u(s)∇s, t∈J1,
u(t)=Iut1
+
t
t1
gs,u(s)∇s, t∈J2,
(2.6)
(iii)B(u(0),u(T))=0.
We callα:J→Ra lower solution of (2.1)–(2.3) if (i)α∈A,
(ii)α(b)−α(a)≤abg(s,α(s))∇sfora≤banda,b∈J1, ora,b∈J2,
(iii)α(t+1)≤I(α(t1)),
(iv)B(α(0),α(T))≤0.
We callβ:J→Ran upper solution of (2.1)–(2.3) if it satisfies the same assumptions, but replace≤with≥.
Letp(t,x)=max{α(t), min{x,β(t)}}. We have the following assumptions throughout this section:
(1) for eachx∈R,g(·,x) is Lebesgue∇-measurable onJ, (2) for a.e. (∇)t∈J,g(t,·) is continuous,
(3) there is anh:J→[0,∞),0Th(s)∇s <∞such that|g(t,p(t,x))| ≤h(t) a.e. (∇) on
J, for allx∈R,
(4)I:R→Ris continuous and nondecreasing,
(5)B:R×R→Ris continuous and for eachx∈[α(0),β(0)],B(x,·) is nonincreas-ing.
Note. a.e. (∇) denotes the Lebesgue∇-measure.
Theorem2.2. Assume that conditions (1)–(5) are satisfied andα,βare lower and upper
Proof. Our proof follows that of Cabada and Liz [9]. Define an operatorG:A→Aby
Claim 2.3. Ifuis a fixed point of the operatorG, thenuis a solution of (2.1)–(2.3) such thatu∈[α,β].
Proof ofClaim 2.3. We assume thatu∈Asatisfies
u(t)=p0,u(0)+
From assumption (ii) of the definition of lower solution, we have
We then have
αt1≤ut1≤βt1, (2.14)
and using the fact thatIis nondecreasing, we have
αt+1
We may applySubclaim 1to (2.9) to verify thatusatisfies the first equation in property (ii) of a solution to (2.1)–(2.3), and applySubclaim 1to (2.10) to verify thatusatisfies the second equation in property (ii).
Subclaim 2. u(0)∈[α(0),β(0)].
Suppose thatα(0)> u(0)=u(0)−B(u(0),u(T)). Thus,u(0)=p(0,u(0))=α(0) and henceB(u(0),u(T))>0. Using assumption (5), we haveB(α(0),α(T))≥B(α(0),u(T))>
0, which contradictsαbeing a lower solution of (2.1)–(2.3). We then haveα(0)≤u(0) and, similarly,u(0)≤β(0), establishingSubclaim 2.
As a result ofSubclaim 2, we haveu(0)=p(0,u(0))=u(0)=u(0)−B(u(0),u(T)) and
henceB(u(0),u(T))=0, establishingClaim 2.3.
Claim 2.4. G:A→Ahas a fixed point.
Leta,b∈Jwith 0≤a≤b≤t1,
Gu(b)−Gu(a)=b ag
s,ps,u(s)∇s≤ b
a h(s)∇s=w(b)−w(a). (2.21) Similar results hold fort1< a≤b≤T.
Subclaim 4. G:S→Sis continuous.
Let un ∞n=1⊆S which converges to u∈S in the space (A, · ). Note that un→u
uniformly on compact subsets ofJ. Letn∈Nandt∈J1, then
Gu(t)−Gun(t)
=Gu(t)−p0,u(0)−Gun(t)−p
0,un(0)
+p0,u(0)−p0,un(0)
=
t
0g
s,ps,u(s)∇s−
t
0g
s,ps,un(s)∇s+p0,u(0)−p0,un(0)
(2.22)
and hence
Gu(t)−Gun(t)
≤
t1
0
g
s,ps,u(s)−gs,ps,un(s)∇s+p0,u(0)−p0,un(0).
(2.23)
Now take limn→∞ and apply the Lebesgue dominated convergence theorem and the continuity ofgin its second variable and ofpto conclude
lim
n→∞Gun(t)−Gu(t)=0. (2.24)
Note that (2.23) does not involvetin its right-hand side, so we can conclude that the convergence is uniform onJ1.
A similar argument shows thatGun→Guniformly on compact subsets ofJ2.
Hence, by Subclaims3and4, Schauder’s fixed point theorem applies toG, finishing
the proof ofClaim 2.4.
Claims2.3and2.4yield the desired result.
Example 2.5. LetT=[0, 1]∪[2, 3],t1=2,g(t,x)=t2+x2,I(x)=x+ 1,u(0)=0. (Note
thatIis not bounded, as required in [4].)
α(t)=0 is a lower solution .
To constructβon [0, 1], we solveβ=1 +β2(≥t2+β2),β(0)=0. Then this implies
thatβ(t)=tant. By considering boundary conditions, we have
β(2)=β(0) + 2
0β
∇(s)∇s=β(0) +1
0β
(s)ds+β∇(2),
β(2)=tan 1 +β(2)−β(1)
2−1 =⇒β(2)=β(2)=⇒β(2) is arbitrary, letβ(2)=1,
β(2+)=Iβ(2)=2.
To constructβon [2, 3], we solveβ=9 +β2(≥t2+β2),β(2)=2, then we have
β=3 tan
tan−1 2
3
+ 3t−6
. (2.26)
ApplyingTheorem 2.2, we know there exists a solutionusuch that 0≤u(t)≤β(t) for
t∈T.
3. Second order
In this section, we are concerned with second-order dynamic equations with functional boundary conditions and impulse:
y(t)=ft,yσ(t), t∈Tκ2
≡[a,b]\t1 , (3.1)
L1
y(a),y∆(a),yσ2(b),y∆σb)=0, (3.2)
L2
y(a),yσ2b)=0, (3.3)
yt+ 1
−yt−1
=r1, (3.4)
y∆t+ 1
−y∆t1−
=Iyt1
,y∆t−1
, (3.5)
wheret1∈Twitha < t1< bandt1 right-dense,r1∈R,I is a real-valued function and
J=[a,b]. We set y∆(t1)=y∆(t+1) ift1 is left-scattered, and y∆(t1)=y∆(t1−) ift1 is
left-dense. We note that these impulses are different from those studied in [16]. (3.2) and (3.3) cover many conditions found in the literature such as separated and nonseparated boundary conditions, respectively,
L1(x,y,z,w)=x, L2(x,y)=y,
L1(x,y,z,w)=y−z, L2(x,y)=y−x, (3.6)
as in [7, Chapter 4].
LetJ1=[a,t1],J2=(t1,b]. We define the following spaces of functions.
Letyibe the restriction ofy:J→RtoJi,i=1, 2, then
ᏯJ
1
=y:J1−→R:yandy∆are continuous onJ1 ,
ᏯJ
2
=y:J2−→R:yandy∆are continuous onJ2andy
t+ 1
andy∆t+ 1
exist ,
A=y:J−→R:y1∈ᏯJ1andy2∈ᏯJ2 .
(3.7)
Fory∈A, lety =sup{|y(t)|:t∈J}. (A, · ) is a Banach space. Forx,y∈A,
[x,y]≡z∈A:x(t)≤z(t)≤y(t)∀t∈J . (3.8)
Definition 3.1. The functionsαandβare, respectively, a lower and an upper solution of problem (3.1)–(3.5) if the following properties hold:
(i)α,β∈A;
We assume the following conditions are satisfied for the functionsf,L1andL2, andI.
(F) The function f : [a,b]×R→Ris continuous.
(L)L1∈C(R4,R) is nondecreasing in the second variable, nonincreasing in the
fourth. Moreover,L2:R2→Ris a continuous function and it is nonincreasing
with respect to its first variable.
(I)I is continuous and strictly increasing with respect to the first variable and non-increasing in the second variable.
We consider the following modified truncated problem:
y(t)−yσ(t)=ft,pσ(t),yσ(t)−pσ(t),yσ(t), t∈Tκ2
Proof. It is not difficult to verify that the problem
y∆∆−yσ=0, t∈[a,b],
y(a)=yσ2(b)=0, (3.17)
has only the trivial solution.
By using [7, Theorem 4.67 and Corollary 4.74], we have that for everyh∈Crd[a,b] has a fixed point. Herey1(t),y2(t) are the solutions of the linear homogeneous equation
y∆∆−yσ=0,t∈[a,b] and satisfy the boundary conditionsy
1(a)=1,y1(σ2(b))=0 and
y2(a)=0,y2(σ2(b))=1.
Gis called the Green’s function of the Dirichlet problem. One can verify that (see [7, page 169]) it is continuous in [a,σ2(b)]×[a,σ2(b)] andG∆(·,s) is continuous att=s=
One can easily observe thatQhas a fixed pointyif and only ifyis a solution of (3.11)– (3.15).
SinceL1,L2,p, andGare bounded and continuous, it can be shown that there exists
R >0 such that the compact operatorQ:S→SwhereS= {y∈A:y ≤R}.
SinceSis a closed, bounded, and convex set, in view of the Tychonoff-Schauder fixed point theorem, there is at least one fixed point ofQ.
Proof. Let y be a solution of (3.11)–(3.15), by definition ofL∗1 andL∗2, we know that
y(a)∈[α(a),β(a)] andy(σ2(b))∈[α(σ2(b)),β(σ2(b))]. We will prove thaty∈[α,β] for
t∈(a,σ2(b)).
Considerz(t)=y(t)−β(t). By definition ofβ,zis continuous on [a,σ2(b)]. Suppose,
to the contrary, there is at∗∈(a,t1)∪(t1,σ2(b)) such that (y−β)(t∗)=maxt∈T{y(t)−
β(t)}>0.
Suppose thatt∗is left scattered. In this case, we have that
y∆t∗≤β∆t∗, y∆∆ρt∗≤β∆∆ρt∗. (3.22)
Consequently, by using conditionF, we arrive at the following contradiction:
0> y∆∆ρt∗−β∆∆ρt∗−y(t∗)−β(t∗)
≥fρ(t∗),β(t∗)−fρ(t∗),β(t∗)=0. (3.23)
Whent∗is left dense the contradiction holds in a similar way. Now supposet∗=t1.
Case 1. t1is left scattered.
Then we havez∆(t+
1)≤0 andz∆(t1+)=z∆(t1). Consequently, by using condition (I),
we arrive at the following contradiction:
0=z∆t+ 1
−z∆t1−
=y∆t+1
−y∆t−1
−β∆t1+
−β∆t1−
≥Iy(t1
,y∆t1
−Iβt1
,β∆t1
> Iβt1
,y∆t1
−Iβt1
,β∆t1
≥0.
(3.24)
Case 2. t1is left dense.
For sufficiently small>0, we have
z∆(s)≥0 fors∈t1−,t1
, z∆s∗≤0 fors∗∈t1,t1+
. (3.25)
Then the contradiction holds in a similar way.
Analogously, the fact thatα(t)≤y(t) for allt∈Tcan be shown.
Theorem3.4. Assume that(L)holds. If y∈[α,β]is a solution of (3.11)–(3.15), then y
satisfies equalities (3.1)–(3.5). Proof. If y(σ2(b))−L
2(y(a),y(σ2(b)))< α(σ2(b)), the definition of L∗2 gives us that
y(σ2(b))=α(σ2(b)).
Now using (L), we obtain a contradiction:
ασ2(b)> yσ2(b)−L 2
y(a),yσ2(b)
≥ασ2(b)−L 2
α(a),ασ2(b)=ασ2(b). (3.26)
Analogously, we arrive atα(σ2(b))≤y(σ2(b))−L
2(y(a),y(σ2(b)))≤β(σ2(b)) and (3.13)
To prove thatL1(y(a),y∆(a),y(σ2(b)),y∆(σ(b)))=0, it is enough using (3.12) to show
α(a)≤y(a) +L1y(a),y∆(a),yσ2(b),y∆σ(b)≤β(a). (3.27)
If y(a) +L1(y(a),y∆(a),y(σ2(b)),y∆(σ(b)))< α(a), then y(a)=α(a) implies that 0=
L2(y(a),y(σ2(b)))=L2(α(a),α(σ2(b))).
SinceL2is injective with respect to the second variable, we havey(σ2(b))=α(σ2(b)).
Using the definition ofL1, we obtain a contradiction:
α(a)> y(a) +L1
Example 3.5. Let Tbe any time scale and let the point 1/2 be a right-dense point in T∩[0, 1]. We define f,L1, andL2in the following way:
f(t,y)=ysinh(y−1)2, L1(x,y,z,w)=1−x,
L2(x,y)= −y, I(x,y)=x−1.
(3.29)
Next we consider the following boundary value problem:
y(t)=ft,yσ(t), t∈[0, 1]κ2\
One can easily verify that
α(t)=
is a lower solution and
β(t)=
Theorem 3.3assures that there exists a solutiony(t) of the problem (3.30)–(3.34) such thaty∈[α,β]. We note that
y(t)=
1, ift∈
0,1
2
,
0, ift∈
1
2, 1
,
(3.37)
is one such solution.
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F. M. Atici: Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA
E-mail address:[email protected]
D. C. Biles: Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA
