Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 620459, 10 pages
doi:10.1155/2010/620459
Research Article
Monotone Iterative Technique for First-Order
Nonlinear Periodic Boundary Value Problems on
Time Scales
Ya-Hong Zhao and Jian-Ping Sun
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
Correspondence should be addressed to Jian-Ping Sun, jpsun@lut.cn
Received 12 February 2010; Revised 17 May 2010; Accepted 26 June 2010
Academic Editor: Paul Eloe
Copyright q 2010 Y.-H. Zhao and J.-P. Sun. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We investigate the following nonlinear first-order periodic boundary value problem on time scales:
xΔ t ptxσt ft, xt, t ∈ 0, T T , x0 xσT . Some new existence criteria of positive
solutions are established by using the monotone iterative technique.
1. Introduction
Recently, periodic boundary value problems PBVPs for short for dynamic equations on
time scales have been studied by several authors by using the method of lower and upper
solutions, fixed point theorems, and the theory of fixed point index. We refer the reader to
1–10 for some recent results.
In this paper we are interested in the existence of positive solutions for the following
first-order PBVP on time scales:
xΔ t ptxσt ft, xt,
x0 xσT ,
t ∈ 0, T T ,
1.1
where σ will be defined in Section 2, T is a time scale, T > 0 is fixed and 0, T ∈ T. For each
interval I of R, we denote by IT I ∩ T. By applying the monotone iterative technique, we
obtain not only the existence of positive solution for the PBVP 1.1, but also give an iterative
scheme, which approximates the solution. It is worth mentioning that the initial term of our
iterative scheme is a constant function, which implies that the iterative scheme is significant
and feasible. For abstract monotone iterative technique, see 11 and the references therein.
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2. Some Results on Time Scales
Let us recall some basic definitions and relevant results of calculus on time scales 12–15.
Definition 2.1. For t ∈ T, we define the forward jump operator σ : T → T by
σt inf{τ ∈ T : τ > t},
2.1
while the backward jump operator ρ : T → T is defined by
ρt sup{τ ∈ T : τ < t}.
2.2
In this definition we put inf ∅ sup T and sup ∅ inf T, where ∅ denotes the empty set. If
σt > t, we say that t is right scattered, while if ρt < t, we say that t is left scattered. Also, if
t < sup T and σt t, then t is called right dense, and if t > inf T and ρt t, then t is called
left dense. We also need below the set Tk which is derived from the time scale T as follows. If
T has a left-scattered maximum m, then Tk T − {m}. Otherwise, Tk T.
Definition 2.2. Assume that x : T → R is a function and let t ∈ Tk . Then x is called
differentiable at t ∈ T if there exists a θ ∈ R such that, for any given > 0, there is an
open neighborhood U of t such that
|xσt − xs − θ|σt − s|| ≤ |σt − s|,
s ∈ U.
2.3
In this case, θ is called the delta derivative of x at t ∈ T and we denote it by θ xΔ t. If
F Δ t ft, then we define the integral by
t
a
fsΔs Ft − Fa.
2.4
Definition 2.3. A function f : T → R is called rd-continuous provided that it is continuous
at right-dense points in T and its left-sided limits exist at left-dense points in T. The set of
rd-continuous functions f : T → R will be denoted by Crd .
Lemma 2.4 see 13. If f ∈ Crd and t ∈ Tk , then
σt
t
fsΔs μtft,
where μt σt − t is the graininess function.
Lemma 2.5 see 13. If f Δ > 0, then f is increasing.
2.5
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Definition 2.6. For h > 0, we define the Hilger complex numbers as
Ch 1
,
z ∈ C : z
−
/
h
2.6
and for h 0, let C0 C.
Definition 2.7. For h > 0, let Zh be the strip
Zh z∈C:−
π
π
< Imz ≤
,
h
h
2.7
and for h 0, let Z0 C.
Definition 2.8. For h > 0, we define the cylinder transformation ξh : Ch → Zh by
ξh z 1
Log1 zh,
h
2.8
where Log is the principal logarithm function. For h 0, we define ξ0 z z for all z ∈ C.
Definition 2.9. A function p : T → R is regressive provided that
1 μtpt / 0,
∀t ∈ Tk .
2.9
The set of all regressive and rd-continuous functions will be denoted by R.
Definition 2.10. We define the set R of all positively regressive elements of R by
R p ∈ R : 1 μtpt > 0, ∀t ∈ T .
2.10
Definition 2.11. If p ∈ R, then the generalized exponential function is given by
t
ξμτ pτ Δτ ,
ep t, s exp
s
for s, t ∈ T,
where the cylinder transformation ξh z is defined as in Definition 2.8.
Lemma 2.12 see 13. If p ∈ R, then
i ep t, t ≡ 1,
ii ep t, s 1/ep s, t,
iii ep t, uep u, s ep t, s,
iv epΔ t, t0 ptep t, t0 , for t ∈ Tk and t0 ∈ T.
2.11
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Lemma 2.13 see 13. If p ∈ R and t0 ∈ T, then
ep t, t0 > 0,
∀t ∈ T.
2.12
3. Main Results
For the forthcoming analysis, we assume that the following two conditions are satisfied.
H1 p : 0, T T → 0, ∞ is rd-continuous, which implies that p ∈ R .
H2 f : 0, T T × 0, ∞ → 0, ∞ is continuous and ft, x is nondecreasing on x.
If we denote that
1
A
,
ep σT , 0 − 1
δ
A
1A
2
3.1
,
then we may claim that A > 0, which implies that 0 < δ < 1.
In fact, in view of H1 and Lemmas 2.12 and 2.13, we have
epΔ t, 0 ptep t, 0 > 0,
t ∈ 0, T T ,
3.2
which together with Lemma 2.5 shows that ep t, 0 is increasing on 0, σT T . And so,
ep σT , 0 > ep 0, 0 1.
3.3
E {x | x : 0, σT T −→ R is continuous}
3.4
This indicates that A > 0.
Let
be equipped with the norm x
maxt∈0,σT T |xt|. Then E is a Banach space.
First, we define two cones K and P in E as follows:
K {x ∈ E | xt ≥ 0, t ∈ 0, σT T },
3.5
P {x ∈ K | xt ≥ δ
x
, t ∈ 0, σT T },
and then we define an operator Φ : K → K :
1
Φxt ep t, 0
t
0
ep s, 0fs, xsΔs A
σT ep s, 0fs, xsΔs ,
0
t ∈ 0, σT T .
3.6
It is obvious that fixed points of Φ are solutions of the PBVP 1.1.
Since ft, x is nondecreasing on x, we have the following lemma.
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Lemma 3.1. Φ : K → K is nondecreasing.
Lemma 3.2. Φ : P → P is completely continuous.
Proof. Suppose that x ∈ P. Then
1
0 ≤ Φxt ep t, 0
≤
t
t
ep s, 0fs, xsΔs A
σT 0
ep s, 0fs, xsΔs
0
ep s, 0fs, xsΔs A
σT 0
3.7
ep s, 0fs, xsΔs
0
≤ 1 A
σT ep s, 0fs, xsΔs,
0
t ∈ 0, σT T ,
so,
Φx
≤ 1 A
σT 3.8
ep s, 0fs, xsΔs.
0
Therefore,
1
Φxt ep t, 0
t
0
A
≥
ep σT , 0
≥ δ
Φx
,
ep s, 0fs, xsΔs A
σT ep s, 0fs, xsΔs
0
σT 3.9
ep s, 0fs, xsΔs
0
t ∈ 0, σT T .
This shows that Φ : P → P . Furthermore, with similar arguments as in 7, we can prove that
Φ : P → P is completely continuous by Arzela-Ascoli theorem.
Theorem 3.3. Assume that there exist two positive numbers R1 < R2 such that
inf ft, δR1 ≥
t∈0,T T
1 AR1
,
A2 σT sup ft, R2 ≤
t∈0,T T
AR2
1 A2 σT .
3.10
Then the PBVP 1.1 has positive solutions x∗ and y∗ , which may coincide with
δR1 ≤ x∗ t ≤ R2 ,
for t ∈ 0, σT T ,
δR1 ≤ y∗ t ≤ R2 ,
for t ∈ 0, σT T ,
where x0 t ≡ R2 and y0 t ≡ R1 for t ∈ 0, σT T .
lim Φn x0 x∗ ,
n → ∞
lim Φn y0 y∗ ,
n → ∞
3.11
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Proof. First, we define
PR1 ,R2 {x ∈ P : R1 ≤ x
≤ R2 }.
3.12
Φ PR1 ,R2 ⊂ PR1 ,R2 .
3.13
Then we may assert that
In fact, if x ∈ PR1 ,R2 , then
δR1 ≤ δ
x
≤ xt ≤ x
≤ R2 ,
for t ∈ 0, σT T ,
3.14
which together with H2 and 3.10 implies that
1
Φxt ep t, 0
≥
≥
≥
t
0
A
ep σT , 0
A
ep σT , 0
A
ep σT , 0
≥ R1 ,
≤
ep s, 0fs, xsΔs
0
σT ep s, 0fs, xsΔs
0
σT fs, xsΔs
0
σT fs, δR1 Δs
0
t ∈ 0, σT T ,
t
1
Φxt ep t, 0
t
ep s, 0fs, xsΔs A
σT ep s, 0fs, xsΔs A
σT 0
ep s, 0fs, xsΔs
3.15
0
ep s, 0fs, xsΔs A
0
σT ep s, 0fs, xsΔs
0
≤ 1 A
σT ep s, 0fs, xsΔs
0
≤ 1 Aep σT , 0
σT fs, xsΔs
0
≤ 1 Aep σT , 0
σT fs, R2 Δs
0
≤ R2 ,
t ∈ 0, σT T ,
which shows that
Φ PR1 ,R2 ⊂ PR1 ,R2 .
3.16
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Now, if we denote that x0 t ≡ R2 for t ∈ 0, σT T , then x0 ∈ PR1 ,R2 . Let
xn1 Φxn ,
n 0, 1, 2, . . . .
3.17
In view of ΦPR1 ,R2 ⊂ PR1 ,R2 , we have xn ∈ PR1 ,R2 , n 0, 1, 2, . . . . Since the set {xn }∞
n0 is
bounded and the operator Φ is compact, we know that the set {xn }∞
n1 is relatively compact,
∞
which implies that there exists a subsequence {xnk }∞
k1 ⊂ {xn }n1 such that
lim xnk x∗ ∈ PR1 ,R2 .
k → ∞
3.18
Moreover, since
0 ≤ x1 t ≤ x1 ≤ R2 x0 t,
for t ∈ 0, σT T ,
3.19
it follows from Lemma 3.1 that Φx1 ≤ Φx0 ; that is, x2 ≤ x1 . By induction, it is easy to know
that
xn1 ≤ xn ,
n 1, 2, . . . ,
3.20
which together with 3.18 implies that
lim xn x∗ ∈ PR1 ,R2 .
n → ∞
3.21
Since Φ is continuous, it follows from 3.17 and 3.21 that
Φx∗ x∗ ,
3.22
which shows that x∗ is a solution of the PBVP 1.1. Furthermore, we get from x∗ ∈ PR1 ,R2 that
δR1 ≤ δ
x∗ ≤ x∗ t ≤ x∗ ≤ R2 ,
for t ∈ 0, σT T .
3.23
On the other hand, if we denote that y0 t ≡ R1 for t ∈ 0, σT T and that yn1 Φyn , n 0, 1, 2, . . . , then we can obtain similarly that yn ∈ PR1 ,R2 , n 0, 1, 2, . . . , and there
∞
exists a subsequence {ynk }∞
k1 ⊂ {yn }n1 such that
lim ynk y∗ ∈ PR1 ,R2 .
k → ∞
3.24
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Moreover, since
y1 t Φy0 t
1
ep t, 0
t
ep s, 0f s, y0 s Δs A
σT 0
A
≥
ep σT , 0
A
≥
ep σT , 0
≥ R1 y0 t,
ep s, 0f s, y0 s Δs
0
σT ep s, 0f s, y0 s Δs
3.25
0
σT fs, δR1 Δs
0
t ∈ 0, σT T ,
it is also easy to know that
yn ≤ yn1 ,
n 1, 2, . . . .
3.26
With the similar arguments as above, we can prove that y∗ is a solution of the PBVP 1.1 and
satisfies
δR1 ≤ y∗ t ≤ R2 ,
for t ∈ 0, σT T .
3.27
Corollary 3.4. If the following conditions are fulfilled:
ft, x
∞,
t∈0,T T
x
lim inf
x→0
ft, x
0,
x → ∞
x
t∈0,T T
lim sup
3.28
then there exist two positive numbers R1 < R2 such that 3.10 is satisfied, which implies that the
PBVP 1.1 has positive solutions x∗ and y∗ , which may coincide with
δR1 ≤ x∗ t ≤ R2 ,
for t ∈ 0, σT T ,
δR1 ≤ y∗ t ≤ R2 ,
for t ∈ 0, σT T ,
lim Φn x0 x∗ ,
n → ∞
lim Φn y0 y∗ ,
3.29
n → ∞
where x0 t ≡ R2 and y0 t ≡ R1 for t ∈ 0, σT T .
Example 3.5. Let T 0, 1 ∪ 2, 3. We consider the following PBVP on T:
xΔ t xσt t 1 xt,
x0 x3.
t ∈ 0, 3T ,
3.30
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Since pt ≡ 1, T 0, 1 ∪ 2, 3 and T 3, we can obtain that
σT 3,
A
1
,
−1
2e2
δ
1
.
4e4
3.31
Thus, if we choose R1 9/16e8 2e2 − 12 and R2 2304e8 /2e2 − 12 , then all the conditions
of Theorem 3.3 are fulfilled. So, the PBVP 3.30 has positive solutions x∗ and y∗ , which may
coincide with
9
64e12 2e2 − 12
9
64e12 2e2 − 12
≤ x∗ t ≤
∗
≤ y t ≤
2304e8
2e2 − 1
2
2304e8
2e2 − 1
2
,
for t ∈ 0, 3T ,
lim Φn x0 x∗ ,
n → ∞
3.32
,
for t ∈ 0, 3T ,
n
∗
lim Φ y0 y ,
n → ∞
where x0 t ≡ 2304e8 /2e2 − 12 and y0 t ≡ 9/16e8 2e2 − 12 for t ∈ 0, 3T .
Acknowledgment
This work was supported by the National Natural Science Foundation of China 10801068.
References
1 A. Cabada, “Extremal solutions and Green’s functions of higher order periodic boundary value
problems in time scales,” Journal of Mathematical Analysis and Applications, vol. 290, no. 1, pp. 35–54,
2004.
2 Q. Dai and C. C. Tisdell, “Existence of solutions to first-order dynamic boundary value problems,”
International Journal of Difference Equations, vol. 1, no. 1, pp. 1–17, 2006.
3 P. W. Eloe, “The method of quasilinearization and dynamic equations on compact measure chains,”
Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 159–167, 2002.
4 S. G. Topal, “Second-order periodic boundary value problems on time scales,” Computers &
Mathematics with Applications, vol. 48, no. 3-4, pp. 637–648, 2004.
5 P. Stehlı́k, “Periodic boundary value problems on time scales,” Advances in Difference Equations, vol. 1,
pp. 81–92, 2005.
6 J.-P. Sun and W.-T. Li, “Positive solution for system of nonlinear first-order PBVPs on time scales,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 1, pp. 131–139, 2005.
7 J.-P. Sun and W.-T. Li, “Existence of solutions to nonlinear first-order PBVPs on time scales,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 67, no. 3, pp. 883–888, 2007.
8 J.-P. Sun and W.-T. Li, “Existence and multiplicity of positive solutions to nonlinear first-order PBVPs
on time scales,” Computers & Mathematics with Applications, vol. 54, no. 6, pp. 861–871, 2007.
9 J.-P. Sun and W.-T. Li, “Positive solutions to nonlinear first-order PBVPs with parameter on time
scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 3, pp. 1133–1145, 2009.
10 S.-T. Wu and L.-Y. Tsai, “Periodic solutions for dynamic equations on time scales,” Tamkang Journal of
Mathematics, vol. 40, no. 2, pp. 173–191, 2009.
11 B. Ahmad and J. J. Nieto, “The monotone iterative technique for three-point second-order
integrodifferential boundary value problems with p-Laplacian,” Boundary Value Problems, vol. 2007,
Article ID 57481, 9 pages, 2007.
12 R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in
Mathematics, vol. 35, no. 1-2, pp. 3–22, 1999.
10
Advances in Difference Equations
13 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications,
Birkhäuser, Boston, Mass, USA, 2001.
14 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
15 B. Kaymakcalan, V. Lakshmikantham, and S. Sivasundaram, Dynamic Systems on Measure Chains, vol.
370 of Mathematics and Its Applications, Kluwer Academic Publishers, Boston, Mass, USA, 1996.