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Volume 2009, Article ID 937064, 19 pagesdoi:10.1155/2009/937064 Research Article Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales Liang-Gen Hu,

Volume 2009, Article ID 937064, 19 pages

doi:10.1155/2009/937064

Research Article

Positive Solutions to Singular and

Delay Higher-Order Differential Equations on

Time Scales

Liang-Gen Hu,1 Ti-Jun Xiao,2 and Jin Liang3

1 Department of Mathematics, University of Science and Technology of China, Hefei 230026, China

2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

3 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Correspondence should be addressed to Jin Liang,jinliang@sjtu.edu.cn

Received 21 March 2009; Accepted 1 July 2009

Recommended by Juan Jos´e Nieto

We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales Theorems on the existence of positive solutions are obtained

by utilizing the fixed point theorem of cone expansion and compression type An example is given

to illustrate our main result

Copyrightq 2009 Liang-Gen Hu 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

In this paper, we are concerned with the following singular three-point boundary value problemBVP for short for delay higher-order dynamic equations on time scales:

−1nuΔ2n t  wtft, ut − c, t ∈ a, b,

u t  ψt, t ∈ a − c, a,

uΔ2i a − βi1uΔ2i1 a  αi1uΔ2i ,

γ i1uΔ2i   uΔ2i

b, 0 ≤ i ≤ n − 1,

1.1

where c ∈ 0, b − a/2,  ∈ a, b, βi ≥ 0, 1 < γi < b − a  βi/ − a  βi, 0 ≤ αi <

b − γi   γi − 1a − βi/b − , i  1, 2, , n and ψ ∈ Ca − c, a The functional

w : a, b → 0, ∞ is continuous and f : a, b × 0, ∞ → 0, ∞ is continuous Our

nonlinearity w may have singularity at t  a and/or t  b, and f may have singularity at

To understand the notations used in1.1, we recall the following definitions which can be found in1,2

a A time scale T is a nonempty closed subset of the real numbers R T has the topology that it inherits from the real numbers with the standard topology It follows that the

jump operators σ, ρ :T → T,

σ t  inf{τ ∈ T : τ > t}, ρ t  sup{τ ∈ T : τ < t} 1.2

supplemented by inf ∅ : sup T and sup ∅ : inf T are well defined The point

t ∈ T is left-dense, left-scattered, right-dense, right-scattered if ρt  t, ρt < t,

σ t  t, σt < t, respectively If T has a left-scattered maximum t1right-scattered

minimum t2, define Tk  T − {t1} Tk  T − {t2}; otherwise, set Tk T Tk  T

By an intervala, b we always mean the intersection of the real interval a, b with

the given time scale, that is,a, b ∩ T Other types of intervals are defined similarly.

b For a function f : T → R and t ∈ T k, theΔ-derivative of f at t, denoted by fΔt,

is the numberprovided it exists with the property that, given any ε > 0, there is a neighborhood U ⊂ T of t such that



fσt − fs − fΔtσt − s ≤ ε|σt − s|, ∀s ∈ U. 1.3

c For a function f : T → R and t ∈ Tk, the ∇-derivative of f at t, denoted by f∇t,

is the numberprovided it exists with the property that, given any ε > 0, there is a neighborhood U ⊂ T of t such that



fρ t− fs − f∇tρ t − s ≤ ερ t − s, ∀s ∈ U. 1.4

d If FΔt  ftΦ∇t  gt, then we define the integral

t

a

f Δ  Ft − Fa

t

a

g ∇  Φt − Φa

. 1.5

Theoretically, dynamic equations on time scales can build bridges between continuous and discrete mathematics Practically, dynamic equations have been proposed as models in the study of insect population models, neural networks, and many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical of biological problems 2 Hence, two-point and multipoint boundary value problems for dynamic equations on time scales have attracted many researchers’ attentionsee, e.g., 1 19 and references therein Moreover, singular boundary value problems have also been treated

in many paperssee, e.g., 4,5,12–14,18 and references therein

In 2004, J J DaCunha et al 13 considered singular second-order three-point boundary value problems on time scales

uΔΔt  ft, ut  0, 0, 1 ∩ T,

u 0  0, up

 u σ21 1.6

and obtained the existence of positive solutions by using a fixed point theorem due to Gatica

et al.14, where f : 0, 1 × 0, ∞ → 0, ∞ is decreasing in u for every t ∈ 0, 1 and may have singularity at u 0

In 2006, Boey and Wong11 were concerned with higher-order differential equation

on time scales of the form

−1n−1yΔn t  −1 p1F t, y σ n−1t , t ∈ a, b,

yΔi a  0, 0 ≤ i ≤ p − 1,

yΔi σb  0, p ≤ i ≤ n − 1,

1.7

where p, n are fixed integers satisfying n ≥ 2, 1 ≤ p ≤ n − 1 They obtained some existence

theorems of positive solutions by using Krasnosel’skii fixed point theorem

Recently, Anderson and Karaca8 studied higher-order three-point boundary value problems on time scales and obtained criteria for the existence of positive solutions

The purpose of this paper is to investigate further the singular BVP for delay higher-order dynamic equation 1.1 By the use of the fixed point theorem of cone expansion and compression type, results on the existence of positive solutions to the BVP 1.1 are established

The paper is organized as follows InSection 2, we give some lemmas, which will be required in the proof of our main theorem In Section 3, we prove some theorems on the existence of positive solutions for BVP1.1 Moreover, we give an example to illustrate our main result

2 Lemmas

For 1 ≤ i ≤ n, let Git, s be Green’s function of the following three-point boundary value

problem:

−uΔΔt  0, t ∈ a, b,

u a − βi uΔa  αi u , γ i u   ub, 2.1 where  ∈ a, b and αi , β i , γ isatisfy the following condition:

C

β i ≥ 0, 1 < γi < b − a  βi

 − a  βi , 0≤ αi <

b − γi γ i− 1a − βi

b −  . 2.2

Throughout the paper, we assume that σb  b.

From8, we know that for any t, s ∈ a, b × a, b and 1 ≤ i ≤ n,

G it, s 

⎧

⎨

⎩

G i1t, s, s ∈ a, ,

G i2t, s, s ∈ , b, 2.3

where

G i1t, s  1

d i

⎧

⎨

⎩



γ it −   b − tσ s  βi − a, σ s ≤ t,



γ iσs − ω  b − σst  βi − a αi − bt − σs, t ≤ s,

G i2t, s  1

d i

⎧

⎨

⎩



σ s1 − αi  αi   βi − ab − t  γi − a  βit − σs, σs ≤ t,



t 1 − αi  αi   βi − ab − σs, t ≤ s,

d iγ i− 1a − βi 1 − αib  α i − γi.

2.4 The following four lemmas can be found in8

Lemma 2.1 Suppose that the condition (C) holds Then the Green function of G it, s in 2.3 satisfies

G it, s > 0, t, s ∈ a, b × a, b. 2.5

Lemma 2.2 Assume that the condition (C) holds Then Green’s function G it, s in 2.3 satisfies

G it, s ≤ max{Gib, s, Giσs, s}, t, s ∈ a, b × a, b. 2.6

nonincreasing in t and

G ib, s

G iσs, s 

γ i



 − a  βib − σs



σ s1 − αi  αi   βi − ab − σs

≥ γ i



 − a  βi

b 1 − αi  αi   βi − a > 0.

2.7

Therefore, we have

G ib, s ≤ Git, s ≤ Giσs, s ≤ δi G ib, s, 2.8 where

δ i b 1 − αi  αi   βi − a

γ i



 − a  βi > 1. 2.9

2 If t and s satisfy the other cases, then we get that Git, s is nondecreasing in t and

G it, s ≤ Gib, s. 2.10

Lemma 2.4 Assume that (C) holds Then Green’s function G it, s in 2.3 verifies the following

inequality:

G it, s ≥ min



γ ib − a



G ib, s

≥ min



δ ib − a ,

γ ib − a

 max{Gib, s, Giσs, s}

2.11



γ i− 1a − βi 1 − αiσs  α i − γi< 0. 2.12

So there exists a misprint on8, Page 2431, line 23 From 2.3, it follows that

G it, s

G ib, s 



σ s1 − αi  αi   βi − ab − t  γi − a  βit − σs

γ i



 − a  βib − σs

≥



  βi − ab − t  γi − a  βit − σs

γ i

 − a  βib − a ≥

γ ib − a .

2.13

Consequently, we get

G it, s ≥ b − t

γ ib − a G ib, s. 2.14

If s ∈ γ i  − a  βi − αi  − βi  a/1 − αi, b, s ≤ t, then, from 2.8, we obtain

G it, s ≥ t − a

b − a G ib, s ≥

δ ib − a G iσs, s. 2.15

Remark 2.6 If we set h it : min{t − a/δib − a, b − t/γib − a}, then we have

G it, s ≥ hit max{Gib, s, Giσs, s}, t, s ∈ a, b × a, b. 2.16 Denote

Gi·, s  max

Thus we have

G it, s ≥ hitGi·, s, t, s ∈ a, b × a, b. 2.18

Lemma 2.7 Assume that condition (C) is satisfied For G it, s as in 2.3, put H1t, s : G1t, s

and recursively define

H jt, s 

b

a

H j−1t, rGjr, sΔr 2.19

−1n uΔ2n t  0, t ∈ a, b,

uΔ2i a − βi1uΔ2i1 a  αi1uΔ2i ,

γ i1uΔ2i   uΔ2i

b, 0 ≤ i ≤ n − 1.

2.20

Lemma 2.8 Assume that (C) holds Denote

K :n−1

j1

j1

h1tLGn·, s ≤ Hnt, s ≤ KGn·, s, t, s ∈ a, b × a, b, 2.22

where

b

a

G j·, sΔs > 0, lj

b

a

G j·, sh j1sΔs, 1 ≤ j ≤ n − 1. 2.23

Proof We proceed by induction on n ≥ 2 We denote the statement by Pn FromLemma 2.7,

it follows that

H2t, s 





b

a

H1t, rG2r, sΔr





≤

b

a

G1·, r G2·, sΔr  k1G2·, s,

2.24

and from2.18, we have

H2t, s 

b

a

H1t, rG2r, sΔr

≥

b

a

h1tG1·, r × h2rG2·, sΔr

 h1tl1G2·, s

2.25

So P2 is true.

We now assume that Pm is true for some positive integer m ≥ 2 FromLemma 2.7, it follows that

Hm1t, s 





b

a

H mt, rGm1r, sΔr





≤

b

a

H mt, rGm1r, sΔr

≤

⎛

⎝b

a

m−1



j1

k j × Gm·, rΔr

⎞

⎠Gm1·, s

m

j1

k jGm1·, s,

H m1t, s 

b

a

H mt, rGm1r, sΔr

≥

b

a

h1 t × m−1

j1

l j G m·, rhm1rGm1·, sΔr

 h1tm

j1

l jGm1·, s

2.26

So Pm  1 holds Thus Pn is true by induction.

Lemma 2.9 see 20 Let E, ·  be a real Banach space and P ⊂ E a cone Assume that T :

P ζ,η → P is completely continuous operator such that

i Tu  u for u ∈ ∂Pζ and Tu  u for u ∈ ∂Pη ,

ii Tu  u for u ∈ ∂Pζ and Tu  u for u ∈ ∂Pη

3 Main Results

We assume that{am} m≥1and{bm} m≥1are strictly decreasing and strictly increasing sequences, respectively, with limm→ ∞a m  a, limm→ ∞b m  b and a1 < b1 A Banach space E  Ca, b is

the set of real-valued continuousin the topology of T functions ut defined on a, b with

the norm

u  max

Define a cone by



u ∈ E : ut ≥ h1tL

K u, t ∈ a, b



Set

P ξ  {u ∈ P : u < ξ}, ∂Pξ  {u ∈ P : u  ξ}, ξ > 0,

P ζ,ηu ∈ P : ζ < u < η, 0 < ζ < η,

Y1  {t ∈ a, b : t − c < a}, Y2  {t ∈ a, b : t − c ≥ a},

Y m  {t ∈ Y2 : t − c ∈ a, am ∪ bm , b }.

3.3

Assume that

C1 ψ : a − c, a → 0, ∞ is continuous;

C2 we have

0 < K

q

p Gn·, swsΔs, K

b

a Gn·, swsΔs < ∞, 3.4

for constants p and q with a  c < p < q < b;

C3 the function f : a, b × 0, ∞ → R is continuous and w : a, b → R is continuous satisfying

lim

m→ ∞sup

u ∈P K



Y Gn·, swsfs, us − cΔs  0, ∀0 < ζ < η. 3.5

We seek positive solutions u : a, b → R, satisfying1.1 For this end, we transform

1.1 into an integral equation involving the appropriate Green function and seek fixed points

of the following integral operator

Define an operator T : Ca, b → Ca, b by

Tut 

b

a

H nt, swsfs, us − cΔs, ∀u ∈ Ca, b, 3.6

where Ca, b  {u ∈ Ca, b | ut ≥ 0, t ∈ a, b}.

Proposition 3.1 Let (C1), (C2), and (C3) hold, and let ζ, η be fixed constants with 0 < ζ < η Then

Proof We separate the proof into four steps.

By conditionC3, there exists some positive integer m0satisfying

sup

u ∈P ζ,η

K



Y m0 Gn·, swsfs, us − cΔs ≤ 1, 3.7

where

Y m0 {t ∈ Y2: t − c ∈ a, am0 ∪ bm0, b}; 3.8

here, we used the fact that for each u ∈ Pζ,η and t ∈ am0 c, bm0 c ∩ a, b,

η ≥ ut − c ≥ h1t − cL

K u ≥ ζ min



h1b − cL

K



 ζh > 0, 3.9

where

h min

h

1am0L

h1bm0L

h1 b − cL

K



Set

D : maxf

t, ψ t − c: t ∈ Y1



,

Q : maxf t, ut − c : t ∈ Y2, ζh≤ ut − c ≤ η. 3.11

Then we obtain

Tu t ≤ sup

t ∈a,b

sup

u ∈P ζ,η

b

a

H nt, swsfs, us − cΔs

≤ K sup

u ∈P ζ,η



Y1

Gn·, swsfs, us − cΔs

 sup

u ∈P ζ,η

K



Y m0 Gn·, swsfs, us − cΔs

 sup

u ∈P ζ,η

K



Y2\Y m0 Gn·, swsfs, us − cΔs

≤ 1  max{D, Q}K

b

a Gn·, swsΔs < ∞.

3.12

Consequently, Tu is bounded and well defined.

Tu  sup

t ∈a,b

b

a

H nt, swsfs, us − cΔs

≤ K

b

a Gn·, swsfs, us − cΔs.

3.13

Then by the above inequality

Tut 

b

a

H nt, swsfs, us − cΔs

≥

b

a

h1 tLGn·, swsfs, us − cΔs

≥ h1 tL

K Tu

3.14

This leads to Tu ∈ P.

Step 3 We will show that T : P ζ,η → P is continuous Let {um} m≥1be any sequence in Pζ,η

such that limm→ ∞u m  u ∈ Pζ,η Notice also that as m → ∞,

φ ms f s, ums − c − fs, us − cw s −→ 0, for s ∈ a  c, b,

f s, ums − c − fs, us − cw s

f

s, ψ s − c− fs, ψ s − cw s  0, for s ∈ a, a  c,



Y2

H nt, sφmsΔs ≤ sup

x ∈P ζ,η

2K



Y2

Gn·, swsfs, xsΔs < ∞.

3.15

Now these together withC2 and the Lebesgue dominated convergence theorem 10 yield

that as m → ∞,

Tum − Tu  sup

t ∈a,b

b

a

H nt, swsf s, ums − c − fs, us − cΔs −→ 0. 3.16

Define

w mt 

⎧

⎪

⎪

⎪

⎪

min{wt, wam}, a ≤ t ≤ am,

w t, a m ≤ t ≤ bm ,

min{wt, wbm}, bm≤ t ≤ b,

f mt, ut − c 

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

t, ψ t − c, a ≤ t < a  c,

min

f t, ut − c, ft, uam, a  c ≤ t ≤ am  c,

f t, ut − c, t ∈ am  c, bm  c ∩ a, b,

min

f t, ut − c, ft, ubm, t ∈ bm  c, b ∩ a, b,

3.17

and an operator sequence{Tm} for a fixed m by

Tm u t 

b

a

H nt, swmsfms, us − cΔs, ∀t ∈ a, b. 3.18

Clearly, the operator sequence{Tm} is compact by using the Arzela-Ascoli theorem

3, for each m ∈ N We will prove that Tm converges uniformly to T on P ζ,η For any u ∈ Pζ,η,

we obtain

Tm u − Tu  sup

t ∈a,b







b

a

H nt, sw msfms, us − c − wsfs, us − cΔs





≤ K

b

a Gn·, sw msfms, us − c − wsfs, us − cΔs

≤ K



Y1

Gn·, s|wms − ws|fs, ψ s − cΔs

 K



Y2

Gn·, sw msfms, us − c − wsfs, us − cΔs.

3.19

From C1, C2, and the Lebesgue dominated convergence theorem 10, we see that the right-hand side3.19 can be sufficiently small for mbeing big enough Hence the sequence {Tm} of compact operators converges uniformly to T on Pζ,η so that operator T is compact Consequently, T : P ζ,η → P is completely continuous by using the Arzela-Ascoli theorem

3

Proposition 3.2 It holds that v ∈ P ζ,η is a solution of1.1 if and only if Tv  v.

−1n vΔ2n t  −1 n TvΔ2n t  wtft, vt − c, 3.20

and for any 0≤ i ≤ n − 1,

vΔ2i a − βi1vΔ2i1 a  αi1vΔ2i , γ i1vΔ2i   vΔ2i

b. 3.21

From8, Lemma 3.1, we know that vt ≥ 0 on a, b So we conclude that v is the solution

of BVP1.1

For convenience, we list the following notations and assumptions:



μK

q

p Gn·, swsΔs

−1

, μ min



h1

p

L

h1

q

L K



;

K

b

a Gn·, swsΔs

−1

;

3.22

the set of real-valued continuousin the topology of T functions ut defined on a,... ζ,η → P is completely continuous operator such that

i Tu  u for u ∈ ∂Pζ and Tu  u for u ∈ ∂Pη ,

ii Tu  u for u ∈ ∂Pζ and Tu  u for u ∈ ∂Pη

3... induction.