Volume 2010, Article ID 620459, 10 pagesdoi:10.1155/2010/620459 Research Article Monotone Iterative Technique for First-Order Nonlinear Periodic Boundary Value Problems on Time Scales Ya
Trang 1Volume 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
Copyrightq 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:
xΔt 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:
xΔt ptxσt ft, xt, t ∈ 0, T T,
whereσ will be defined inSection 2,T is a time scale, T > 0 is fixed and 0, T ∈ T For each
intervalI 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 PBVP1.1, 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, see11 and the references therein
Trang 22 Some Results on Time Scales
Let us recall some basic definitions and relevant results of calculus on time scales12–15
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 operatorρ : 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 setTkwhich is derived from the time scaleT as follows If
T has a left-scattered maximum m, then T k T − {m} Otherwise, T k T.
Definition 2.2 Assume that x : T → R is a function and let t ∈ T k Then x is called
differentiable at t ∈ T if there exists a θ ∈ R such that, for any given > 0, there is an open neighborhoodU 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 xΔt If
FΔt 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 inT and its left-sided limits exist at left-dense points in T The set of
rd-continuous functionsf : T → R will be denoted by Crd.
Lemma 2.4 see 13 If f ∈ Crdand t ∈ T k , then
σt
t fsΔs μtft, 2.5
where μt σt − t is the graininess function.
Lemma 2.5 see 13 If fΔ> 0, then f is increasing.
Trang 3Definition 2.6 For h > 0, we define the Hilger complex numbers as
Ch
z ∈ C : z / − h1
and forh 0, let C0 C.
Definition 2.7 For h > 0, let Z hbe the strip
Zh
z ∈ C : − π h < Imz ≤ π h
and forh 0, let Z0 C.
Definition 2.8 For h > 0, we define the cylinder transformation ξ h:Ch → Zhby
ξ h z h1Log1 zh, 2.8 where Log is the principal logarithm function Forh 0, we define ξ0z z for all z ∈ C Definition 2.9 A function p : T → R is regressive provided that
1 μtpt / 0, ∀t ∈ T k 2.9 The set of all regressive and rd-continuous functions will be denoted byR.
Definition 2.10 We define the setRof all positively regressive elements ofR by
Rp ∈ R : 1 μtpt > 0, ∀t ∈ T. 2.10
Definition 2.11 If p ∈ R, then the generalized exponential function is given by
e p t, s exp
t
s ξ μτpτ Δτ , for s, t ∈ T, 2.11 where the cylinder transformationξ h z is defined as inDefinition 2.8
i e p t, t ≡ 1,
ii e p t, s 1/e p s, t,
iii e p t, ue p u, s e p t, s,
iv eΔ
p t, t0 pte p t, t0, for t ∈ T k and t0∈ T.
Trang 4Lemma 2.13 see 13 If p ∈ Rand t0∈ T, then
e p 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
A e 1
p σT, 0 − 1 , δ
A
1 A
2
then we may claim thatA > 0, which implies that 0 < δ < 1.
In fact, in view ofH1 and Lemmas2.12and2.13, we have
eΔ
p t, 0 pte p t, 0 > 0, t ∈ 0, T T, 3.2 which together withLemma 2.5shows thate p t, 0 is increasing on 0, σT T And so,
e p σT, 0 > e p 0, 0 1. 3.3
This indicates thatA > 0.
Let
E {x | x : 0, σT T−→ R is continuous} 3.4
be equipped with the norm t∈0,σT T|xt| Then E is a Banach space.
First, we define two conesK and P in E as follows:
K {x ∈ E | xt ≥ 0, t ∈ 0, σT T},
T}, 3.5
and then we define an operatorΦ : K → K :
Φxt 1
e p t, 0
t
0
e p s, 0fs, xsΔs A
σT
0
e p s, 0fs, xsΔs
, t ∈ 0, σT T.
3.6
It is obvious that fixed points ofΦ are solutions of the PBVP 1.1
Sinceft, x is nondecreasing on x, we have the following lemma.
Trang 5Lemma 3.1 Φ : K → K is nondecreasing.
Lemma 3.2 Φ : P → P is completely continuous.
Proof Suppose that x ∈ P Then
0≤ Φxt e 1
p t, 0
t
0
e p s, 0fs, xsΔs A
σT
0
e p s, 0fs, xsΔs
≤
t
0
e p s, 0fs, xsΔs A
σT
0
e p s, 0fs, xsΔs
≤ 1 A
σT
0
e p s, 0fs, xsΔs, t ∈ 0, σT T,
3.7
so,
σT
0
e p s, 0fs, xsΔs. 3.8 Therefore,
Φxt e 1
p t, 0
t
0
e p s, 0fs, xsΔs A
σT
0
e p s, 0fs, xsΔs
≥ e A
p σT, 0
σT
0
e p s, 0fs, xsΔs
T.
3.9
This shows thatΦ : 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< R2such that
inf
t∈0,T Tft, δR1 ≥ 1 AR1
A2σT , t∈0,T supTft, R2 ≤ AR2
1 A2σT . 3.10 Then the PBVP1.1 has positive solutions x∗and y∗, which may coincide with
δR1≤ x∗t ≤ R2, for t ∈ 0, σT T, lim
n → ∞Φn x0 x∗,
δR1≤ y∗t ≤ R2, for t ∈ 0, σT T, lim
n → ∞Φn y0 y∗, 3.11 where x0t ≡ R2and y0t ≡ R1for t ∈ 0, σT T.
Trang 6Proof First, we define
P R1,R2 {x ∈ P : R1 2}. 3.12 Then we may assert that
ΦP R1,R2
⊂ P R1,R2 . 3.13
In fact, ifx ∈ P R1,R2 , then
δR1 2, for t ∈ 0, σT T, 3.14 which together withH2 and 3.10 implies that
Φxt e 1
p t, 0
t
0
e p s, 0fs, xsΔs A
σT
0
e p s, 0fs, xsΔs
≥ A
e p σT, 0
σT
0
e p s, 0fs, xsΔs
≥ e A
p σT, 0
σT
0 fs, xsΔs
≥ e A
p σT, 0
σT
0 fs, δR1Δs
≥ R1, t ∈ 0, σT T,
Φxt e 1
p t, 0
t
0
e p s, 0fs, xsΔs A
σT
0
e p s, 0fs, xsΔs
≤
t
0
e p s, 0fs, xsΔs A
σT
0
e p s, 0fs, xsΔs
≤ 1 A
σT
0
e p s, 0fs, xsΔs
≤ 1 Ae p σT, 0
σT
0
fs, xsΔs
≤ 1 Ae p σT, 0
σT
0
fs, R2Δs
≤ R2, t ∈ 0, σT T,
3.15
which shows that
ΦP R ,R
⊂ P R ,R . 3.16
Trang 7Now, if we denote thatx0t ≡ R2fort ∈ 0, σT T, thenx0∈ P R1,R2 Let
x n1 Φx n , n 0, 1, 2, 3.17
In view ofΦP R1,R2 ⊂ P R1,R2 , we havex n ∈ P R1,R2 , n 0, 1, 2, Since the set {x n}∞n0is bounded and the operatorΦ is compact, we know that the set {x n}∞
n1is relatively compact, which implies that there exists a subsequence{x n k}∞k1 ⊂ {x n}∞n1such that
lim
Moreover, since
0≤ x1 1 2 x0t, for t ∈ 0, σT T, 3.19
it follows fromLemma 3.1thatΦx1 ≤ Φx0; that is,x2 ≤ x1 By induction, it is easy to know that
x n1 ≤ x n , n 1, 2, , 3.20 which together with3.18 implies that
lim
SinceΦ is continuous, it follows from 3.17 and 3.21 that
Φx∗ x∗, 3.22
which shows thatx∗ is a solution of the PBVP1.1 Furthermore, we get from x∗ ∈ P R1,R2
that
2, for t ∈ 0, σT T. 3.23
On the other hand, if we denote that y0t ≡ R1 for t ∈ 0, σT T and thaty n1
Φy n , n 0, 1, 2, , then we can obtain similarly that y n ∈ P R1,R2 , n 0, 1, 2, , and there
exists a subsequence{y n k}∞k1 ⊂ {y n}∞n1such that
lim
Trang 8Moreover, since
y1t Φy0
t
e 1
p t, 0
t
0
e p s, 0fs, y0s Δs A
σT
0
e p s, 0fs, y0s Δs
≥ A
e p σT, 0
σT
0
e p s, 0fs, y0s Δs
≥ e A
p σT, 0
σT
0 fs, δR1Δs
≥ R1 y0t, t ∈ 0, σT T,
3.25
it is also easy to know that
y n ≤ y n1 , n 1, 2, 3.26
With the similar arguments as above, we can prove thaty∗is a solution of the PBVP1.1 and satisfies
δR1≤ y∗t ≤ R2, for t ∈ 0, σT T. 3.27
Corollary 3.4 If the following conditions are fulfilled:
lim
x → 0 inf
t∈0,T T
ft, x
x ∞, x → ∞lim sup
t∈0,T T
ft, x
x 0, 3.28
then there exist two positive numbers R1 < R2 such that3.10 is satisfied, which implies that the PBVP1.1 has positive solutions x∗and y∗, which may coincide with
δR1≤ x∗t ≤ R2, for t ∈ 0, σT T, lim
n → ∞Φn x0 x∗,
δR1≤ y∗t ≤ R2, for t ∈ 0, σT T, lim
n → ∞Φn y0 y∗, 3.29 where x0t ≡ R2and y0t ≡ R1for t ∈ 0, σT T.
Example 3.5 Let T 0, 1 ∪ 2, 3 We consider the following PBVP on T:
xΔt xσt t 1xt, t ∈ 0, 3 T,
x0 x3.
3.30
Trang 9Sincept ≡ 1, T 0, 1 ∪ 2, 3 and T 3, we can obtain that
σT 3, A 1
2e2− 1, δ
1 4e4. 3.31
Thus, if we chooseR1 9/16e82e2− 12andR2 2304e8/2e2− 12, then all the conditions
ofTheorem 3.3are fulfilled So, the PBVP3.30 has positive solutions x∗andy∗, which may coincide with
9 64e122e2− 12 ≤ x∗t ≤ 2304e8
2e2− 12, for t ∈ 0, 3 T, lim
n → ∞Φn x0 x∗,
9 64e122e2− 12 ≤ y∗t ≤ 2304e8
2e2− 12, for t ∈ 0, 3 T, lim
n → ∞Φn y0 y∗,
3.32
wherex0t ≡ 2304e8/2e2− 12andy0t ≡ 9/16e82e2− 12fort ∈ 0, 3 T
Acknowledgment
This work was supported by the National Natural Science Foundation of China10801068
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