EQUATIONS ON TIME SCALES
ELVAN AKIN-BOHNER AND YOUSSEF N. RAFFOUL
Received 1 February 2006; Revised 25 March 2006; Accepted 27 March 2006
Using nonnegative definite Lyapunov functionals, we prove general theorems for the boundedness of all solutions of a functional dynamic equation on time scales. We ap-ply our obtained results to linear and nonlinear Volterra integro-dynamic equations on time scales by displaying suitable Lyapunov functionals.
Copyright © 2006 E. Akin-Bohner and Y. N. Raffoul. This is an open access article dis-tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-erly cited.
1. Introduction
In this paper, we consider the boundedness of solutions of equations of the form
xΔ(t)=Gt,x(s); 0≤s≤t:=Gt,x(·) (1.1)
on a time scale T(a nonempty closed subset of real numbers), wherex∈Rn andG: [0,∞)×Rn→Rnis a given nonlinear continuous function intandx. For a vectorx∈Rn, we takexto be the Euclidean norm ofx. We refer the reader to [8] for the continuous case, that is,T=R.
In [6], the boundedness of solutions of
xΔ(t)=Gt,x(t), xt0
=x0, t0≥0,x0∈R (1.2)
is considered by using a type I Lyapunov function. Then, in [5], the authors considered nonnegative definite Lyapunov functions and obtained sufficient conditions for the ex-ponential stability of the zero solution. However, the results in either [5] or [6] do not apply to the equations similar to
xΔ=a(t)x+
t
0B(t,s)f
x(s)Δs, (1.3)
which is the Volterra integro-dynamic equation. In particular, we are interested in ap-plying our results to (1.3) with f(x)=xn, wherenis positive and rational. The authors are confident that there is nothing in the literature that deals with the qualitative analysis of Volterra integro-dynamic equations on time scales. Thus, this paper is going to play a major role in any future research that is related to Volterra integro-dynamic equations.
Letφ: [0,t0]→Rnbe continuous, we define|φ| =sup{φ(t): 0≤t≤t0}.
where C is a constant and depends on t0. Moreover, solutions of (1.1) are uniformly boundedifCis independent oft0. Throughout this paper, we assume 0∈Tand [0,∞)= {t∈T: 0≤t <∞}.
Next, we generalize a “type I Lyapunov function” which is defined by Peterson and Tisdell [6] to Lyapunov functionals. We sayV: [0,∞)×Rn→[0,∞) is atype I Lyapunov tend the definition of the derivative of a type I Lyapunov function to type I Lyapunov functionals. IfV is a type I Lyapunov functional andxis a solution of (1.1), then (2.11) gives
Continuing in the spirit of [6], we have
˙
Example 1.1. Assumeφ(t,s) is right-dense continuous (rd-continuous) and let
V(t,x)=x2+
t
0φ(t,s)W
x(s)Δs. (1.9)
Ifxis a solution of (1.1), then we have by using (2.10) andTheorem 2.2that
˙
V(t,x)=2x·Gt,x(·)+μ(t)G2t,x(·)
+
t
0φ
Δ(t,s)Wx(s)Δs+φσ(t),tWx(t), (1.10)
whereφΔ(t,s) denotes the derivative ofφwith respect to the first variable.
We say that a type I Lyapunov functionalV: [0,∞)×Rn→[0,∞) isnegative definite ifV(t,x)>0 forx=0,x∈Rn,V(t,x)=0 forx=0 and along the solutions of (1.1), we have ˙V(t,x)≤0. If the condition ˙V(t,x)≤0 does not hold for all (t,x)∈T×Rn, then the Lyapunov functional is said to benonnegative definite.
In the case of differential equations or difference equations, it is known that if one can display a negative definite Lyapunov function, or functionals, for (1.1), then bounded-ness of all solutions follows. In [8], the second author displayed nonnegative Lyapunov functionals and proved boundedness of all solutions of (1.1), in the caseT=R.
2. Calculus on time scales
In this section, we introduce a calculus on time scales including preliminary results. An introduction with applications and advances in dynamic equations are given in [2,3]. Our aim is not only to unify some results whenT=RandT=Zbut also to extend them for other time scales such ashZ, whereh >0,qN0, whereq >1 and so on. We define the forward jump operatorσonTby
σ(t) :=inf{s > t:s∈T} ∈T (2.1)
for allt∈T. In this definition, we put inf(∅)=supT. Thebackward jump operatorρon Tis defined by
ρ(t) :=sup{s < t:s∈T} ∈T (2.2)
for allt∈T. Ifσ(t)> t, we saytisright-scattered, while ifρ(t)< t, we saytisleft-scattered. Ifσ(t)=t, we saytisright-dense, while ifρ(t)=t, we saytisleft-dense. Thegraininess functionμ:T→[0,∞) is defined by
μ(t) :=σ(t)−t. (2.3)
Thas left-scattered maximum pointm, thenTκ=T− {m}. Otherwise,Tκ=T. Assume
x:T→Rn. Then we definexΔ(t) to be the vector (provided it exists) with the property that given any>0, there is a neighborhoodUoftsuch that
xi
σ(t)−xi(s)
for alls∈Uand for eachi=1, 2,. . .,n. We callxΔ(t) thedelta derivativeofx(t) att, and it turns out thatxΔ(t)=x(t) ifT=RandxΔ(t)=x(t+ 1)−x(t) ifT=Z. IfGΔ(t)=g(t), then the Cauchy integral is defined by
t
ag(s)Δs=G(t)−G(a). (2.5) It can be shown that if f :T→Rnis continuous att∈Tandtis right-scattered, then
fΔ(t)= f
σ(t)−f(t)
μ(t) , (2.6)
while iftis right-dense, then
fΔ(t)=lim s→t
f(t)−f(s)
t−s , (2.7)
if the limit exists. If f,g:T→Rnare differentiable att∈T, then the product and quotient rules are as follows:
(f g)Δ(t)=fΔ(t)g(t) +fσ(t)gΔ(t), (2.8)
f
g Δ
(t)= fΔ(t)g(t)−f(t)gΔ(t)
g(t)gσ(t) ifg(t)g
σ(t)=0. (2.9)
Iff is differentiable att, then
fσ(t)=f(t) +μ(t)fΔ(t), where fσ=f ◦σ. (2.10)
We say f :T→Risrd-continuous provided f is continuous at each right-dense point
t∈Tand whenevert∈Tis left-dense, lims→t− f(s) exists as a finite number. We say that
p:T→Risregressiveprovided 1 +μ(t)p(t)=0 for allt∈T. We define the setof all regressive and rd-continuous functions. We define the set+of all positively regressive elements ofby+= {p∈: 1 +μ(t)p(t)>0 for allt∈T}.
The following chain rule is due to Poetzsche and the proof can be found in [2, Theorem 1.90].
Theorem2.1. Let f :R→Rbe continuously differentiable and supposeg:T→Ris delta differentiable. Then f ◦g:T→Ris delta differentiable and the formula
(f◦g)Δ(t)=
1
0 f
g(t) +hμ(t)gΔ(t)dhgΔ(t) (2.11)
holds.
We use the following result [2, Theorem 1.117] to calculate the derivative of the Lya-punov function in further sections.
there exists a neighborhood oft, independentUofτ∈[t0,σ(t)], such that
k
σ(t),τ−k(s,τ)−kΔ(t,τ)σ(t)−s≤σ(t)−s ∀s∈U, (2.12)
wherekΔdenotes the derivative ofkwith respect to the first variable. Then
g(t) :=
t
t0k(t,τ)Δτ implies g
Δ(t)=t t0k
Δ(t,τ)Δτ+kσ(t),t;
h(t) :=
b
t k(t,τ)Δτ implies k
Δ(t)=b t k
Δ(t,τ)Δτ−kσ(t),t.
(2.13)
We apply the following Cauchy-Schwarz inequality in [2, Theorem 6.15] to prove
Theorem 4.1.
Theorem2.3. Leta,b∈T. For rd-continuous f,g: [a,b]→R,
b
a
f(t)g(t)Δt≤b a
f(t)2 Δt
b
a
g(t)2 Δt
. (2.14)
Ifp:T→Ris rd-continuous and regressive, then theexponential functionep(t,t0) is for each fixedt0∈Tthe unique solution of the initial value problem
xΔ=p(t)x, xt0=1 (2.15)
onT. Under the addition ondefined by
(p⊕q)(t)=p(t) +q(t) +μ(t)p(t)q(t), t∈T, (2.16)
is an Abelian group (see [2]), where the additive inverse ofp, denoted byp, is defined by
(p)(t)= −p(t)
1 +μ(t)p(t), t∈T. (2.17)
We use the following properties of the exponential functionep(t,s) which are proved in Bohner and Peterson [2].
Theorem2.4. Ifp,q∈, then fort,s,r,t0∈T, (i)ep(t,t)≡1ande0(t,s)≡1;
(ii)ep(σ(t),s)=(1 +μ(t)p(t))ep(t,s); (iii) 1/ep(t,s)=ep(t,s)=ep(s,t); (iv)ep(t,s)/eq(t,s)=epq(t,s);
(v)ep(t,s)eq(t,s)=ep⊕q(t,s).
Moreover, the following can be found in [1].
Theorem2.5. Lett0∈T.
(i)Ifp∈+, thene
p(t,t0)>0for allt∈T.
3. Boundedness of solutions
In this section, we use a nonnegative definite type I Lyapunov functional and establish sufficient conditions to obtain boundedness of solutions of (1.1).
Theorem3.1. LetD⊂Rn. Suppose that there exists a type I Lyapunov functionalV: [0,∞) ×D→[0,∞)such that for all(t,x)∈[0,∞)×D,
where we usedTheorem 2.5(i). Integrating both sides fromt0tot, we have
From inequality (3.1), we have
where we used the factTheorem 2.5(ii). Therefore, we obtain
|x| ≤W−1
In the next theorem, we give sufficient conditions to show that solutions of (1.1) are bounded.
Theorem 3.2. Let D⊂Rn. Suppose that there exists a type I Lyapunov functional V:
[0,∞)×D→[0,∞)such that for all(t,x)∈[0,∞)×D,
because of M≤λ3(t)/λ2(t), λ3(t)≤N, for t∈[0,∞) and Theorem 2.5(i). Integrating both sides fromt0tot, we obtain
Vt,x(t)eMt,t0≤Vt0,φ+ K
MeM
t,t0. (3.12)
This implies fromTheorem 2.4(iii) that for allt≥t0,
Vt,x(t)≤Vt0,φ
eM
t,t0
+ K
M. (3.13)
From inequality (3.1), we have
W1
|x|≤ 1
λ1
t0
λ2
t0
W2
|φ|+λ2
t0
W3
|φ|t0 0 φ1
t0,s
Δs+ K
M
(3.14)
for allt≥t0, where we used the factTheorem 2.5(ii) andλ1is nondecreasing.
The following theorem is the special case of [8, Theorem 2.6].
Theorem3.3. Suppose there exists a continuously differentiable type I Lyapunov functional
V: [0,∞)×Rn→[0,∞)that satisfies
λ1xp≤V(t,x), V(t,x)=0 ifx=0, (3.15)
V(t,x)Δ≤ −λ2(t)V(t,x)Vσ(t,x) (3.16)
for some positive constantsλ1 andpare positive constants, andλ2is a positive continuous function such that
c1= inf
0≤t0≤tλ2(t). (3.17)
Then all solutions of (1.1) satisfy
x ≤ 1
λ11/ p
1
1/Vt0,φ+c1t−t0
1/ p
. (3.18)
Proof. For anyt0≥0, let x be the solution of (1.1) withx(t0)=φ(t0). By inequalities (3.16) and (3.17), we have
V(t,x)Δ≤ −c1V(t,x)Vσ(t,x). (3.19)
Letu(t)=V(t,x(t)) so that we have
uΔ(t)
Since (1/u(t))Δ= −uΔ/u(t)u(σ(t)), we obtain
1
u(t)
Δ
≥c1. (3.21)
Integrating the above inequality fromt0tot, we have
u(t)≤ 1
1/ut0
+c1
t−t0
(3.22)
or
Vt,x(t)≤ 1 1/V(t0,φ) +c1
t−t0
. (3.23)
Using (3.15), we obtain
x ≤ 1
λ11/ p
1 1/Vt0,φ
+c1
t−t0
1/ p. (3.24)
The next theorem is an extension of [7, Theorem 2.6].
Theorem3.4. AssumeD⊂Rnand there exists a type I Lyapunov functionalV: [0,∞)×
D→[0,∞)such that for all(t,x)∈[0,∞)×D,
λ1xp≤V(t,x), (3.25)
˙
V(t,x)≤−λ2V(x) +L
1 +εμ(t) , (3.26)
whereλ1,λ2,p >0,L≥0are constants and0< ε < λ2. Then all solutions of (1.1) staying in
Dare bounded.
Proof. For anyt0≥0, letxbe the solution of (1.1) withx(t0)=φ. Sinceε∈+,eε(t, 0) is well defined and positive. By (3.26), we obtain
Vt,x(t)eε(t, 0)Δ=V˙t,x(t)eσ
ε(t, 0) +εV
t,x(t)eε(t, 0),
≤−λ2V
t,x(t)+Leε(t, 0) +εVt,x(t)eε(t, 0),
=eε(t, 0)εVt,x(t)−λ2V
t,x(t)+L≤Leε(t, 0).
(3.27)
Integrating both sides fromt0tot, we obtain
Vt,x(t)eε(t, 0)≤Vt0,φ
+L
Dividing both sides of the above inequality byeε(t, 0) and then using (3.25) andTheorem 2.5, we obtain
x ≤
1
λ1
1/ p
Vt0,φ
+L
ε 1/ p
for allt≥t0. (3.29)
This completes the proof.
Remark 3.5. InTheorem 3.4, ifV(t0,φ) is uniformly bounded, then one concludes that all solutions of (1.1) that stay inDare uniformly bounded.
4. Applications to Volterra integro-dynamic equations
In this section, we apply our theorems from the previous section and obtain sufficient conditions that insure the boundedness and uniform boundedness of solutions of Volter-ra integro-dynamic equations. We begin with the following theorem.
Theorem4.1. SupposeB(t,s)is rd-continuous and consider the scalar nonlinear Volterra integro-dynamic equation
xΔ=a(t)x(t) +
t
0B(t,s)x
2/3(s)Δs, t≥0,x(t)=φ(t)for0≤t≤t
0, (4.1)
whereφis a given bounded continuous initial function on[0,∞), andais a continuous function on[0,∞). Suppose there are positive constantsν,β1,β2, withν∈(0, 1), andλ3= min{β1,β2}such that
2a(t) +μ(t)a2(t) +μ(t)a(t)
t
0
B(t,s)Δs+t 0
B(t,s)Δs
+ν
∞
σ(t)
B(u,t)Δu1 +μ(t)λ3
≤ −β1,
(4.2)
2
3
1 +μ(t)a(t)+μ(t)
t
0
B(t,s)Δs−ν1 +μ(t)λ3
≤ −β2, (4.3)
t
0
∞
t
B(u,s)ΔuΔs <∞, t 0
B(t,s)Δs <∞,
B(t,s)≥ν∞ t
B(u,s)Δu, (4.4)
then all solutions of (4.1) are uniformly bounded. Proof. Let
V(t,x)=x2(t) +ν
t
0
∞
t
UsingTheorem 2.2, we have along the solutions of (4.1) that
Also, usingTheorem 2.3, one obtains
t
A substitution of the above two inequalities into (4.6) yields
To further simplify (4.9), we make use of Young’s inequality, which says that for any two nonnegative real numberswandz, we have
wz≤we
By substituting the above inequality into (4.9), we arrive at
˙
Multiplying and dividing the above inequality by 1 +μ(t)λ3, and then applying conditions (4.2) and (4.3), ˙V(t,x) reduces to 3.1are satisfied. Next we make sure that condition (3.3) holds. Use (4.4) to obtain
W2
Thus condition (3.3) is satisfied withγ=0. An application ofTheorem 3.1yields the
results.
in the Lyapunov functional, otherwise, condition (4.5) may only hold ifB(t,s)=0 for all
t∈Twith 0≤s≤t <∞for a particular time scale. For example, if we takeT=Z, then condition (4.5) reduces to|B(t,s)| ≥ν∞
u=t|B(u,s)|, which can only hold ifB(t,s)=0 for
ν=1.
Remark 4.3. IfT=R, thenμ(t)=0 for alltand henceTheorem 4.1reduces to [8, Exam-ple 2.3].
Remark 4.4. We assert thatTheorem 4.1can be easily generalized to handle scalar non-linear Volterra integro-dynamic equations of the form
xΔ=a(t)x(t) +
t
0B(t,s)f
s,x(s)Δs, (4.15)
where|f(t,x(t))| ≤x2/3(t) +Mfor some positive constantM.
For the next theorem, we consider the scalar Volterra integro-dynamic equation
xΔ(t)=a(t)x(t) +
t
0B(t,s)f
s,x(s)Δs+gt,x(t), (4.16)
wheret≥0,x(t)=φ(t) for 0≤t≤t0,φis a given bounded continuous initial function,
a(t) is continuous fort≥0, andB(t,s) is right-dense continuous for 0≤s≤t <∞. We assume f(t,x) andg(t,x) are continuous inxandtand satisfy
g(t,x)≤γ1(t) +γ2(t)x(t), f(t,x)≤γ(t)x(t), (4.17)
whereγandγ2are positive and bounded, andγ1is nonnegative and bounded. For the next theorem, we need the identity
x(t)Δ
= x(t) +xσ(t)
x(t)+xσ(t)xΔ(t). (4.18)
Its proof can be found in [4].
Theorem4.5. Suppose there exist constantsk >1andε,αwith0< ε < αsuch that
a(t) +γ2(t) +k
∞
σ(t)
B(u,t)Δuγ(t)1 +εμ(t)
≤ −α <0, (4.19)
wherek=1 +ζfor someζ >0. Suppose
1 +μ(t)εB(t,s)≥λ
∞
t
B(u,s)Δu, (4.20)
whereλ≥kα/ζ,0≤s < t≤u <∞,
t0
0
∞
t0
and for some positive constantL,
γ1(t)
1 +εμ(t)≤L. (4.22)
Then all solutions of (4.16) are uniformly bounded. Proof. Define
Along the solutions of (4.16), we have
˙
The results follow formTheorem 3.4andRemark 3.5.
In the next theorem, we establish sufficient conditions that guarantee the boundedness of all solutions of the vector Volterra integro-dynamic equation
xΔ=Ax(t) +
t
wheret≥0,x(t)=φ(t) for 0≤t≤t0,φis a given bounded continuous initialk×1 vector function. Also,AandC(t,s) arek×kmatrix withC(t,s) being continuous onT×T,g,x
arek×1 vector functions that are continuous fort∈T. IfDis a matrix, then|D|means the sum of the absolute values of the elements.
Theorem4.6. SupposeCT(t,s)=C(t,s). LetIbe thek×kidentity matrix. Assume there exist positive constantsL,ν,ξ,β1,β2,λ3, andk×k positive definite constant symmetric matrixBsuch that
ATB+BA+μ(t)ATBA≤ −ξI, (4.26)
−ξ+ATBg+|Bg|+
t
0|B|
C(t,s)Δs+μ(t)t 0
ATBC(t,s)Δs
+ν
∞
σ(t)
C(u,t)Δu1 +μ(t)λ3
≤ −β1,
(4.27)
|B| −ν+μ(t)gTB2+ 1 +ATB+
t
0
C(t,s)Δs1 +μ(t)λ3
≤ −β2, (4.28)
μ(t)gTg+|Bg|1 +μ(t)λ 3
+μ(t)ATBg=L, (4.29)
C(t,s)≥ν∞ σ(t)
C(u,s)Δu, (4.30)
t
0
∞
t
C(u,s)ΔuΔs <∞, t 0
C(t,s)Δs <∞. (4.31)
Then there exists anr1∈(0, 1]such that
r1xTx≤xTBx≤xTx. (4.32)
Proof. Let the matrixBbe defined by (4.26) and define
V(t,x)=xTBx+ν
t
0
∞
t
C(u,s)Δux2(s)Δs. (4.33)
HerexTx=x2=(x2
1+x22+···+xk2). Using the product rule given in (2.8), we have along the solutions of (4.25) that
˙
V(t,x)=xΔTBx+xσTBxΔ−νt 0
C(t,s)x2(s)Δs+ν
∞
σ(t)
C(u,t)Δux2
=xΔTBx+x+μ(t)xΔTBxΔ−νt 0
C(t,s)x2(s)Δs+ν
∞
σ(t)
C(u,t)Δux2
=xΔTBx+xTBxΔ+μ(t)xΔTBxΔ−νt 0
C(t,s)x2(s)Δs+ν
∞
σ(t)
Substituting the right-hand side of (4.25) forxΔinto (4.34) and making use of (4.26), we
By noting that the right side of (4.35) is scalar and by recalling thatBis a symmetric matrix, expression (4.35) simplifies to
˙
Next, we perform some calculations to simplify inequality (4.36),
Finally,
A substitution of the above inequalities into (4.36) yields
˙
Thus condition (3.3) is satisfied withγ=0. An application ofTheorem 3.1yields the
results.
Remark 4.7. It is worth mentioning thatTheorem 4.6is new whenT=R.
Acknowledgments
The first author acknowledges financial support through a University of Missouri Re-search Board grant and a travel grant from the Association of Women in Mathematics.
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Elvan Akin-Bohner: Department of Mathematics and Statistics, University of Missouri-Rolla, 310 Rolla Building, Rolla, MO 65409-0020, USA
E-mail address:[email protected]
Youssef N. Raffoul: Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA
