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doi:10.1155/2009/737461

Research Article

Necessary and Sufficient Conditions for

the Existence of Positive Solution for

Singular Boundary Value Problems on Time Scales

Meiqiang Feng,

1

Xuemei Zhang,

2, 3

Xianggui Li,

1

and Weigao Ge

3

1School of Science, Beijing Information Science & Technology University, Beijing 100192, China

2Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China 3Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Xuemei Zhang,zxm74@sina.com

Received 27 March 2009; Revised 3 July 2009; Accepted 15 September 2009

Recommended by Alberto Cabada

By constructing available upper and lower solutions and combining the Schauder’s fixed point theorem with maximum principle, this paper establishes sufficient and necessary conditions to guarantee the existence ofCld0,1Tas well asCΔld0,1Tpositive solutions for a class of singular boundary value problems on time scales. The results significantly extend and improve many known results for both the continuous case and more general time scales. We illustrate our results by one example.

Copyrightq2009 Meiqiang Feng et al. 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.

1. Introduction

Recently, there have been many papers working on the existence of positive solutions to boundary value problems for differential equations on time scales; see, for example, 1–

22. This has been mainly due to its unification of the theory of differential and difference equations. An introduction to this unification is given in 10, 14,23, 24. Now, this study is still a new area of fairly theoretical exploration in mathematics. However, it has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models; see, for example,9,10.

Motivated by works mentioned previously, we intend in this paper to establish sufficient and necessary conditions to guarantee the existence of positive solutions for the singular dynamic equation on time scales:

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subject to one of the following boundary conditions:

x0 x1 0, 1.2

or

x0 xΔ1 0, 1.3

where T is a time scale, 0,1 T 0,1 ∩T, where 0 is right dense and 1 is left dense. andH f : 0,1 T × 0,∞ → 0,∞ is continuous. Suppose further that ft, x is nonincreasing with respect tox, and there exists a functiongk :0,1 → 1,∞ such that

ft, kxgk ft, x ,t, x ∈0,1 T×0,. 1.4

A necessary and sufficient condition for the existence ofCld0,1Tas well asCΔld0,1T

positive solutions is given by constructing upper and lower solutions and with the maximum principle. The nonlinearityft, x may be singular att 0 and/ort 1. By singularity we mean that the functionsfin1.1 is allowed to be unbounded at the pointst0 and/ort1.

A functionxtCld0,1TCΔld∇0,1 Tis called aCld0,1Tpositive solution of1.1 if it

satisfies1.1 xt > 0,fort ∈ 0,1 T ; if evenxΔ0 , xΔ1− exist, we call it is aCΔld0,1T solution.

To the best of our knowledge, there is very few literature giving sufficient and necessary conditions to guarantee the existence of positive solutions for singular boundary value problem on time scales. So it is interesting and important to discuss these problems. Many difficulties occur when we deal with them. For example, basic tools from calculus such as Fermat’s theorem, Rolle’s theorem, and the intermediate value theorem may not necessarily hold. So we need to introduce some new tools and methods to investigate the existence of positive solutions for problem1.1 with one of the above boundary conditions.

The time scale related notations adopted in this paper can be found, if not explained specifically, in almost all literature related to time scales. The readers who are unfamiliar with this area can consult, for example,6,11–13,25,26for details.

The organization of this paper is as follows. InSection 2, we provide some necessary background. InSection 3, the main results of problem1.1 -1.2 will be stated and proved. InSection 4, the main results of problem1.1 –1.3 will be investigated. Finally, inSection 5, one example is also included to illustrate the main results.

2. Preliminaries

In this section we will introduce several definitions on time scales and give some lemmas which are useful in proving our main results.

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Definition 2.2. Define the forwardbackward jump operatorσt attfort <supTρt att

fort >infT by

σt inf{τ > t:τ∈T}ρt sup{τ < t:τ ∈T} 2.1

for allt∈T. We assume throughout thatThas the topology that it inherits from the standard topology onRand saytis right scattered, left scattered, right dense and left dense if σt > t, ρt < t, σt t,andρt t, respectively. Finally, we introduce the sets TkandT

kwhich

are derived from the time scaleTas follows. IfThas a left-scattered maximumt1, thenTk

T−t1, otherwiseTk T. IfThas a right-scattered minimumt2, thenTk T−t2, otherwise

TkT.

Definition 2.3. Fixt ∈Tand lety: T → R. DefineyΔt to be the numberif it exists with the property that givenε >0 there is a neighborhoodUoftwith

yσtysyΔt σts< ε|σts| 2.2

for allsU, whereyΔdenotes thedelta derivative ofywith respect to the first variable, then

gt :

t

a

ωt, τ Δτ 2.3

implies

gΔt

t

a

ωΔt, τ Δτωσt , τ . 2.4

Definition 2.4. Fixt ∈Tand lety: T → R. Defineyt to be the numberif it exists with the property that givenε >0 there is a neighborhoodUoftwith

yρtysyt ρts< ερts 2.5

for allsU. Callyt thenabla derivative ofyt at the pointt.

IfTRthenfΔt ft ft . IfTZthenfΔt ft1 −ft is the forward difference operator whileft ftft−1 is the backward difference operator.

Definition 2.5. A functionf :T → Ris called rd-continuous provided that it is continuous at all right-dense points ofTand its left-sided limit existsfinite at left-dense points ofT. We letC0rdT denote the set of rd-continuous functionsf:T → R.

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Definition 2.7. A functionF:Tk Ris called a delta-antiderivative off :Tk Rprovided

thatFΔt ft holds for allt∈Tk. In this case we define the delta integral offby t

a

fs ΔsFtFa, 2.6

for alla, t∈T.

Definition 2.8. A functionΦ:TkRis called a nabla-antiderivative off:TkRprovided

thatΦ∇t ft holds for allt∈Tk. In this case we define the delta integral offby t

a

fss Φt −Φa 2.7

for alla, t∈T.

Throughout this paper, we assume thatTis a closed subset ofRwith 0,1∈T. LetECld0,1T, equipped with the norm

x: sup

t∈0,1T

|xt |. 2.8

It is clear thatEis a real Banach space with the norm.

Lemma 2.9Maximum Principle . Leta, b ∈0,1Tanda < b. IfxCld0,1T∩CldΔ∇0,1 T,

xa ≥0, xb ≥0, andxΔ∇t ≤0, ta, b T.Thenxt ≥0, ta, bT.

3. Existence of Positive Solution to

1.1

-

1.2

In this section, by constructing upper and lower solutions and with the maximum principle

Lemma 2.9, we impose the growth conditions onfwhich allow us to establish necessary and sufficient condition for the existence of1.1 -1.2 .

We know that

Gt, s

⎧ ⎨ ⎩

s1−t , if 0≤st≤1,

t1−s , if 0≤ts≤1 3.1

is the Green’s function of corresponding homogeneous BVP of1.1 -1.2 . We can prove thatGt, s has the following properties.

Proposition 3.1. Fort, s ∈0,1T×0,1T, one has

Gt, s ≥0,

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To obtain positive solutions of problem1.1 -1.2, the following results ofLemma 3.2

Proof. Without loss of generality, we suppose that there is only one right-scattered pointt1 ∈

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that is,

t0

0

Δt

s

0

fs, us

t0

0

Δt

s

0

fs, us. 3.5

Similarly, we can prove

1

σt0

s

1

s

fs, u Δt

1

σt0

Δt

t

σt0

fs, us. 3.6

The proof is complete.

Theorem 3.3. Suppose thatH holds. Then problem1.1 -1.2 has aCld0,1Tpositive solution if and only if the following integral condition holds:

0<

1

0

es fs,1 ∇s <. 3.7

Proof.

(1) Necessity

ByH , there existsgk : 0,1 → 1,∞ such thatft, kxgk ft, x . Without loss of generality, we assume thatgk is nonincreasing on0,1withg1 ≥1.

Suppose thatuis a positive solution of problem1.1 -1.2, then

uΔ∇tft, ut ≤0, 3.8

which implies thatuis concave on0,1T. Combining this with the boundary conditions, we haveuΔ0 >0, uΔ1 <0.ThereforeuΔ0 uΔ1 <0. So by10, Theorem 1.115, there exists

t0∈0,1 Tsatisfying uΔt0 0 oruΔt0 uΔσt0 ≤0. AnduΔt >0 fort∈0, t0 ,uΔt <0,

fortσt0 ,1 . Denoteumax{ut0 , uσt0 }, thenumaxt∈0,1Tut .

First we prove 0<10es fs,1 ∇s. ByH , for any fixedu, v >0, we have

ft, u ft,u vv

gu v

ft, v , uv. 3.9

It follows that

ft, ug

2u uv|uv|

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Ifft,1 ≡0, then we have by3.10

0≤ft, ug

2u u1|u−1|

ft,1 ∀t∈0,1 T. 3.11

This meansft, ut ≡ 0, then ut ≡ 0, which is a contradiction withut being positive solution. Thusft,1 /≡0, then 0<01es fs,1 ∇s.

Second, we prove10es fs,1 ∇s <∞. IfuΔt0 0,then

t0

t

fs, uss

t0

t

uΔ∇ssuΔt0 uΔt uΔt fort∈0, t0 t

t0

fs, uss

t

t0

uΔ∇ssuΔt uΔt0 −uΔt fortt0,1 .

3.12

IfuΔt0 uΔσt0 <0, thenuΔt0 >0,uΔσt0 <0, and

t0

t

fs, uss

t0

t

uΔ∇ssuΔt0 uΔtuΔt fort∈0, t0 t

σt0

fs, uss

t

σt0

uΔ∇ssuΔt uΔσt0 ≤ −uΔt fortσt0 ,1 .

3.13

It follows that

t0

t

fs, us

t0

t

fs, ussuΔt for t∈0, t0 t

σt0

fs, us

t

σt0

fs, uss≤ −uΔt fortσt0 ,1 .

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By3.14 we have

Combining this with3.10 we obtain

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(2) Su

ciency

Let

at

1

0

Gt, s fs,1 ∇s, bt

1

0

Gt, s fs, ess. 3.20

Then

et

1

0

es fs,1 ∇satbt

1

0

es fs, ess,

aΔ∇tft,1 , bΔ∇tft, et .

3.21

Let

k1 1

0

es fs,1 ∇s, lmin1, k1−1, Lmax1, k1−1, k2 1

0

es fs, ess,

3.22

thenl≤1, L≥1.

LetHt lat , Qt Lbt ,then

latl

1

0

es fs,1 ∇s≤1, Lk1etLbtLk2ρ. 3.23

So, we have

HΔ∇t ft, Ht ft, latlft,1 ≥ft,1 −lft,1 ≥0,

QΔ∇t ft, Qt ft, LbtLft, et

ft, Lk1etLft, et

ft, etLft, et ≤0,

3.24

andH0 H1 Q0 Q1 0. HenceHt , Qt are lower and upper solutions of problem1.1 -1.2 , respectively. ObviouslyHt >0 fort∈0,1 T.

Now we prove that problem1.1 -1.2 has a positive solutionx∗ ∈ Cld0,1T with

0< Htx∗≤Qt .

Define a function

Ft, x

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ft, Ht , x < Ht ,

ft, x , HtxQt ,

ft, Qt , x > Qt .

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ThenF :0,1 T×RRis continuous. Consider BVP

xΔ∇t Ft, x ,

x0 x1 0.

3.26

Define mappingA:EEby

Axt

1

0

Gt, s Fs, xss. 3.27

Then problem 1.1 -1.2 has a positive solution if and only if A has a fixed point x∗ ∈ Cld0,1Twith 0< Htx∗≤Qt .

ObviouslyAis continuous. LetD{x| xρ, xE, ρ∗∈R}.By3.7 and3.16 , for allxD, we have

1

0

Gt, s Fs, xss

1

0

Gt, s fs, Hss

1

0

Gt, s fs,0 ∇s

g0

1

0

Gt, s ft,1 ∇s

g0

1

0

es ft,1 ∇s <.

3.28

ThenAD is bounded. By the continuity ofGt, s we can easily found that{Aut |utD}are equicontinuous. ThusAis completely continuous. By Schauder fixed point theorem we found thatAhas at least one fixed pointx∗∈D.

We prove 0< Htx∗≤Qt . If there existst∗∈0,1 Tsuch that

xt> Qt. 3.29

Letzt Qtx, cinf{t1|0≤t1< t, zt <0,tt1, t∗}, dsup{t2|t< t2≤1, zt <

0,tt, t2}thenQt < x∗fortc, d T.ThusFt, xft, Qt , tc, d T. By3.24 we

know that−zΔ∇t QΔ∇txΔ∇t ≤0.Andzc Qcxc ≥0, zd Qdxd ≥0. ByLemma 2.9we havezt ≥0, tc, dT, which is a contradiction. Thenx∗≤Qt . Similarly we can proveHtx∗. The proof is complete.

Theorem 3.4. Suppose that (H) holds. Then problem1.1 -1.2 has aCΔld0,1Tpositive solution if and only if the following integral condition holds:

0<

1

0

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Proof.

(1) Necessity

LetutCΔld0,1Tbe a positive solution of problem1.1 -1.2. ThenuΔt is decreasing on 0,1T. HenceuΔ∇t is integrable and

1

0

ft, utt

1

0

uΔ∇tt <. 3.31

By simple computation and using 10, Theorem 1.119, we obtain limt→0ut /et >

0,limt→1−ut /et > 0. So there existM > 1 > m > 0 such thatmetutMet .

ByH we obtain

gM−1−1ft, etft, Metft, ut ,

1

0

ft, ettgM−1

1

0

ft, utt <.

3.32

By

et ft,1 ≤ft, etget ft,1 , 3.33

we have0<10et ft,1 ∇t≤10ft, ett <.

(2) Su

ciency

Letrt 10Gt, s fs, ess, then

et

1

0

Gs, s fs, essrt

1

0

fs, ess. 3.34

Similar toTheorem 3.3, letl min{1, k−1

2 }, L max{1, k−21}, Ht lat , Qt Lrt , there

existsωt satisfyingHtωtQt , and

ft, ωtft, Htft, lk2et

glk2

ft, et , 3.35

then ω∗Δ∇t is integral and ω∗Δ1− , ω∗Δ0 exist, hence ωt is a positive solution in

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4. Existence of Positive Solution to

1.1

1.3

Now we deal with problem1.1–1.3. The method is just similar to what we have done in

Section 3, so we omit the proof of main result of this section. Let

G1t, s ⎧ ⎨ ⎩

s if 0≤st≤1,

t if 0≤ts≤1. 4.1

be the Green’s function of corresponding homogeneous BVP of1.1 –1.3 . We can prove thatG1t, s has the following properties.

Similar to3.2, we have

G1t, s ≥0, t, s ∈0,1T×0,1T,

e1t e1sG1t, sG1t, t te1t , t, s ∈0,1T×0,1T.

4.2

Theorem 4.1. Suppose thatH holds, then problem1.11.3 has aCld0,1Tpositive solution if and only if the following integral condition holds:

0<

1

0

e1s fs,1 ∇s <. 4.3

Theorem 4.2. Suppose thatH holds, then problem1.11.3 has aCΔld0,1Tpositive solution if and only if the following integral condition holds:

0<

1

0

fs, e1ss <. 4.4

5. Example

To illustrate how our main results can be used in practice we present an example.

Example 5.1. We have

xΔ∇t t−1/2ex, t∈0,1 T,

x0 x1 0,

(13)

where ft, x t−1/2ex

Remark 5.2. Example 5.1 implies that there is a large number of functions that satisfy the conditions ofTheorem 3.3. In addition, the conditions ofTheorem 3.3are also easy to check.

Acknowledgments

This work is sponsored by the National Natural Science Foundation of China 10671012, 10671023 and the Scientific Creative Platform Foundation of Beijing Municipal Commission of EducationPXM2008-014224-067420 .

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References

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