Multiple Positive Solutions of the Singular Boundary Value Problem for Second-Order

Volume 2010, Article ID 281908,13pages doi:10.1155/2010/281908

Research Article

Multiple Positive Solutions of the Singular Boundary Value Problem for Second-Order

Impulsive Differential Equations on the Half-Line

Jing Xiao,

Juan J. Nieto,

and Zhiguo Luo

1Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China

2Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Correspondence should be addressed to Zhiguo Luo,[email protected] Received 17 November 2009; Revised 22 January 2010; Accepted 21 February 2010 Academic Editor: Claudianor Alves

Copyrightq2010 Jing Xiao 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.

This paper uses a fixed point theorem in cones to investigate the multiple positive solutions of a boundary value problem for second-order impulsive singular differential equations on the half- line. The conditions for the existence of multiple positive solutions are established.

1. Introduction

Consider the following nonlinear singular Sturm-Liouville boundary value problems for second-order impulsive differential equation on the half-line:

ptut

ft, u 0, ∀t∈J, Δutk I_kutk, k1,2, . . . , n,

αu0−βlim

t→0ptut 0, γu∞ δ lim

t→∞ptut 0,

1.1

whereJ 0,∞, 0 < t₁ < · · · < t_n,J 0,∞,J J \ {t1, . . . , t_n},f ∈CJ ×J, J , p ∈ CJ, J ∩C¹J, J with p > 0 on J,and _∞

0 1/psds < ∞; α, β, γ, δ ≥ 0 with ρ βγ αδαγB0,∞ > 0, in whichBt, s _s

t1/pσdσ.Δutk ut_k−ut⁻_k,

whereut⁻_kand ut_kare, respectively, the left and right limits ofutat tk,k 1, . . . , n, 1≤n <∞.

The theory of singular impulsive differential equations has been emerging as an important area of investigation in recent years. For the theory and classical results, we refer the monographs to1,2 and the papers3–19 to readers. We point out that in a second-order differential equationu ft, u, u, one usually considers impulses in the positionuand the velocityu. However, in the motion of spacecraft one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position20 . The impulses only on the velocity occur also in impulsive mechanics21 .

In recent paper 3 , by using the Krasnoselskii’s fixed point theorem, Kaufmann has discussed the existence of solutions for some second-order boundary value problem with impulsive effects on an unbounded domain. In 22 Sun et al. and 23 Liu et al., respectively, discussed the existence and multiple positive solutions for singular Sturm- Liouville boundary value problems for second-order differential equation on the half-line.

But the Multiple positive solutions of this case with both singularity and impulses are not to be studied. The aim of this paper is to fill up this gap.

The rest of the paper is organized as follows. InSection 2, we give several important lemmas. The main theorems are formulated and proved inSection 3. And inSection 4, we give an example to demonstrate the application of our results.

2. Several Lemmas

Lemma 2.1see23 . If conditions_∞

0 1/psds <∞andρ >0 are satisfied, then the boundary value problem

ptut

νt 0, ∀t∈J, αu0−βlim

t→0ptut 0, γu∞ δ lim

t→∞ptut 0

2.1

has a unique solution for anyν ∈ LJ, R . Moreover, this unique solution can be expressed in the form

ut _∞

0

Gt, sνsds, 2.2

whereGt, sis defined by

Gt, s 1 ρ

⎧⎨

⎩

βαB0, s

δγBt,∞

, 0≤s≤t <∞, βαB0, t

δγBs,∞

, 0≤t≤s <∞. 2.3

Remark 2.2. It is easy to prove thatGt, shas the following properties:

1Gt, sis continuous onJ×J,

2Gt, sis continuous differentiable onJ×J, exceptts, 3∂_tGt, s|_ts−∂_tGt, s|_ts⁻ ps⁻¹,

4Gt, s≤Gs, s≤ρ⁻¹βαB0, sδγBs,∞<∞, 5Gs lim_t→∞Gt, s<∞,

6for allt∈a, b ⊂0,∞,s∈0,∞,Gt, s≥ωGs, s, where

wmin βαBb,∞

βαB0,∞,δγBb,∞ δγB0,∞

. 2.4

Obviously, 0< ω <1.

For the intervala, b , 0< a < t1, tn < b <∞, and the correspondingωinRemark 2.2, we defineP C¹J, R {u∈CJ, R :u∈CJ, R , ut⁻_kandut_kexist, andutk ut⁻_k}.

BP C¹J, R {u∈ P C¹J, R : lim_{t→ ∞}ut exists}.K {u ∈ BP C¹J, R :ut > 0,t ∈ J and min_t∈a,b ut≥ωu}. It is easy to see thatBP C¹J, R is a Banach space with the norm u sup_t∈J|ut|, andKis a positive cone inBP C¹J, R . For details of the cone theory, see 1 .u∈P C¹J, R ∩C²JR is called a positive solution of BVP1.1ifut >0 for allt ∈J andutsatisfies1.1.

As we know that the Ascoli-Arzela Theorem does not hold in infinite intervalJ, we need the following compactness criterion:

Lemma 2.3see22 . LetM ⊂ BP C¹J, R . ThenMis relatively compact inBP C¹J, R if the following conditions hold.

iMis uniformly bounded inBP C¹J, R .

iiThe functions fromMare equicontinuous on any compact interval of0,∞.

iiiThe functions from M are equiconvergent, that is, for any given ε > 0, there exists a T Tε>0 such that|ft−f∞|< ε, for anyt > T,f ∈M.

The main tool of this work is a fixed point theorem in cones.

Lemma 2.4see4 . Let X be a Banach space andKis a positive cone inX. Assume thatΩ1,Ω2

are open subsets ofXwith 0∈Ω1,Ω1⊂Ω2. LetT:K∩Ω2\Ω1 → Kbe a completely continuous operator such that

iTu ≤ ufor allu∈K∩∂Ω1.

iithere exists aΦ∈Ksuch thatu /TuλΦ, for allu∈K∩∂Ω2andλ >0.

ThenT has a fixed point inK∩Ω2\Ω1.

Remark 2.5. Ifi is satisfied for u ∈ K ∩∂Ω2 and ii is satisfied for u ∈ K∩∂Ω1, then Lemma 2.4is still true.

Lemma 2.6see3 . The functionu∈K∩C²J, R is a solution of the BVP1.1if and only if u∈Ksatisfies the equation

ut _∞

0

Gt, sfs, usdsⁿ

k1

Gt, tkptkIkutk, t∈J. 2.5

The proof of this result is based on the properties of the Green function, so we omit it as elementary.

Define

Tut _∞

0

Gt, sfs, usdsⁿ

k1

Gt, tkptkIkutk, t∈J. 2.6

Obviously, the BVP1.1has a solutionuif and only ifu∈Kis a fixed point of the operator Tdefined by2.6.

Let us list some conditions as follows.

A1There exist two nonnegative functions:a∈CJ, J ,g ∈CJ, J such thatft, u≤ atgu. ft, u, at may be singular at t 0. Ik : J → J, k 1, . . . , n, are continuous.

A20<_∞

0 Gs,sasds <∞, 0< Gtk, tkptk<∞, k1, . . . , n.

Lemma 2.7. If A1andA2are satisfied, then for any bounded open setΩ ⊂ BP C¹J, R ,T : Ω∩K → Kis a completely continuous operator.

Proof. For any bounded open setΩ ⊂ BP C¹J, R , there exists a constantM > 0 such that u ≤Mfor anyu∈Ω.

First, we show thatT : Ω∩K → K is well defined. Letu ∈ Ω∩K. FromA1, we haveS_M max{S1, S₂}, where S₁ sup{gu : 0≤ u ≤ M},S₂ sup{Iku : 0 ≤ u≤ M, k1, . . . , n},and

_∞

0

Gt, sfs, usdsⁿ

k1

Gt, tkptkIkutk

≤S_{M ∞}

0

Gs, sasdsⁿ

k1

Gtk, t_kptk

<∞.

2.7

Hence,Tis well defined. For anyt₁, t₂∈J, we have _∞

0

|Gt1, s−Gt2, s|asds≤2 _∞

0

Gs, sasds <∞. 2.8

Thus, by the Lebesgue dominated convergence theorem and the fact thatGs, tis continuous ont, we have, for anyt₁, t₂∈J,u∈Ω∩K,

|Tut1−Tut2|

≤ _∞

0

|Gt1, s−Gt2, s|fs, usds

ⁿ

k1

|Gt1, t_k−Gt2, t_k|ptkIkutk

≤SM

_∞

0

|Gt1, s−Gt2, s|asdsⁿ

k1

|Gt1, tk−Gt2, tk|ptk

−→0, t1 −→t₂.

2.9

Therefore,Tu∈CJ, R . By the property3ofGs, t, it is easy to getTu∈P C¹J, R . On the other hand, by2.6we have, for anyu∈Ω∩Kandt∈J,

Tut− _∞

0

Gsfs, usds

≤ _∞

0

Gt, s−Gsfs, usdsⁿ

k1

Gt, tk−GtkptkIkutk

≤S_{M ∞}

0

Gt, s−Gsasdsⁿ

k1

Gt, tk−Gtkptk

.

2.10

Then by A2, the property 5 of Remark 2.2 and the Lebesgue dominated convergence theorem, we have

t→lim∞Tut _∞

0

Gsfs, usdsⁿ

k1

GtkptkIkutk<∞. 2.11

ThusTu∈BP C¹J, R .

For anyu∈Ω∩K, we get

Tut _∞

0

Gt, sfs, usdsⁿ

k1

Gt, tkptkIkutk

≤ _∞

0

Gs, sfs, usdsⁿ

k1

Gtk, tkptkIkutk.

2.12

So

Tu ≤ _∞

0

Gs, sfs, usdsⁿ

k1

Gtk, tkptkIkutk. 2.13

On the other hand, fort∈a, b we obtain

Tut≥ω _∞

0

Gs, sfs, usdsⁿ

k1

Gtk, tkptkIkutk

≥ωTu. 2.14

ThusT :Ω∩K → K.

Next, we prove thatT is continuous. Let u_n → u₀ inΩ ∩ K, thenun ≤ Mn 1,2, . . ..We prove thatTun → Tu0. For anyε >0, byA2, there exists a constantA0>0 such that

S_{M ∞}

A0

Gs, sasds≤ ε

6. 2.15

On the other hand, by the continuities offt, uon0, A0 ×0, M and the continuities ofI_k onJ, for the aboveε >0, there exists aδ >0 such that, for anyu, v∈0, M ,|u−v|< δ,

ft, u−ft, v< ε 3

_A0

0

Gs, sds ₋₁

, t∈0, A0 ,

Gtk, tkptk|Ikutk−Ikvtk|< ε 3n.

2.16

Fromun−u₀ → 0, for the aboveδ, there exists a sufficiently large numberN such that, whenn > N, we have

|unt−u₀t| ≤ un−u₀< δ, t∈0, A0 ,

|untk−u0tk| ≤ un−u0< δ.

2.17

Therefore, by2.15–2.17, we have, forn > N,

Tun−Tu0 ≤ _∞

0

Gs, sfs, uns−fs, u0sds

ⁿ

k1

Gtk, t_kptk|Ikuntk−I_ku0tk|

≤2SM

_∞

A0

Gs, sasds

_A0

0

Gs, sfs, uns−fs, u0sds

ⁿ

k1

Gtk, tkptk|Ikutk−Iku0tk|

≤ ε 3 ε

3 ε 3 ε.

2.18

This implies that the operatorT is continuous.

Finally we show thatT :Ω∩K → Kis a compact operator. In fact for any bounded setD ⊂Ω, there exists a constantM1 >0 such thatu ≤M1for anyu∈D ∩ K. Hence, we obtain

Tu ≤SM1

_∞

0

Gs, sasdsⁿ

k1

Gtk, tkptk

<∞. 2.19

Therefore,TD ∩ Kis uniformly bounded inBP C¹J, R .

Givenr >0, for anyu∈D∩K, as the proof of2.9, we can get that{Tu:u∈D∩K}are equicontinuous on0, r . Sincer >0 is arbitrary,{Tu:u∈D∩K}are locally equicontinuous onJ. By2.6,A1,A2, and the Lebesgue dominated convergence theorem, we have

|Tut−Tu∞| ≤S_{M1 ∞}

0

Gt, s−Gsasdsⁿ

k1

Gt, tk−Gtkptk

−→0, t−→∞.

2.20

Hence, the functions from{Tu:u∈D∩K}are equiconvergent. ByLemma 2.3, we have that {Tu:u∈D∩K}is relatively compact inBP C¹J, R . Therefore,T :Ω∩K → Kis completely continuous. This completed the proof ofLemma 2.7.

3. Main Results

For convenience and simplicity in the following discussion, we use the following notations:

f0lim inf

u→0 min

t∈a,b

ft, u

u , g0lim inf

u→0

gu

u , I0k lim inf

u→0

ptkIku

u ,

f_∞lim inf

u→ ∞ min

t∈a,b

ft, u

u , g_∞lim inf

u→ ∞

gu

u , I_∞k lim inf

u→ ∞

ptkIku

u ,

I^qk lim sup

u→q

ptkIku

u , g^∞lim sup

u→ ∞

gu

u , I^∞k lim sup

u→ ∞

ptkIku

u ,

g^qlim sup

u→q

gu

u , g⁰lim sup

u→0

gu

u , I⁰k lim sup

u→0

ptkIku

u ,

3.1

Theorem 3.1. Let A1andA2 hold. Then the BVP 1.1 has at least two positive solutions satisfying 0<u1< q <u2if the following conditions hold:

H1ωf0

_b

aGs, sds_n

k1Gtk, tkI0k>1, ωf∞_b

aGs, sds_n

k1Gtk, tk·I∞k>

1,

H2there exists a q > 0 such that g^q_∞

0 Gs, sasds_n

k1Gtk, tkI^qk < 1,for all ωq≤u≤q, a.e.t∈0,∞.

Proof. By the definition off₀andI₀, for anyε >0, there existr ∈0, qsuch that

ft, u≥1−εf0u, ∀u ≤r, t∈a, b , ptkIku≥1−εI0ku, 1−εω

f₀

_b

a

Gs, sdsⁿ

k1

Gtk, t_kI0k

≥1, ∀u ≤r.

3.2

Define the open sets

Ωr

u∈BP C¹J, R :u< r

. 3.3

LetΦ≡1, thenΦ∈K. Now we prove that

u /TuλΦ, ∀u∈K∩∂Ωr, λ >0. 3.4

If not, then there existu0 ∈K∩∂Ωrandλ0>0 such thatu0Tu0λ0Φ. Letμmin_t∈a,b u0t, then for anyt∈a, b ,we have

u₀t Tu0t λ₀

_∞

0

Gt, sfs, u0sdsⁿ

k1

Gt, tkptkIku0tk λ₀

≥ω _∞

0

Gs, sfs, u0sdsω n k1

Gtk, tkptkIku0tk λ0

>1−εμω

f_{0 b}

a

Gs, sdsⁿ

k1

Gtk, t_kI0k

λ₀

≥μλ0.

3.5

This impliesμ > μλ₀, a contradiction. Therefore,3.4holds.

That by the definition off_∞andI_∞, for anyε >0 there existR > qsuch that ft, u≥1−εf_∞u, ∀u ≥R, t∈a, b ,

ptkIku≥1−εI∞ku, 1−εω

f_∞

_b

a

Gs, sdsⁿ

k1

Gtk, t_kI_∞k

≥1, ∀u ≥R.

3.6

Define the open sets:

ΩR

u∈BP C¹J, R :u< R

. 3.7

As the proof of3.4, we can get that

u /TuλΦ, ∀x∈K∩∂ΩR, λ >0. 3.8

On the other hand, for anyε >0, chooseqinH2such that

1ε

g^q _∞

0

Gs, sasdsⁿ

k1

Gtk, t_kI^qk

≤1, ωq≤u≤q. 3.9

By the definition ofg^q,I^q, for the aboveε >0, there existsδ >0, whenu∈q−δ, qδ; thus, we have

gu≤1εg^qu,

ptkIku≤1εI^qku. 3.10

Define

Ωq

u∈BP C¹J, R :u< q

. 3.11

Then, for anyu∈K∩∂Ωqandt∈0,∞, we can obtain

Tut _∞

0

Gt, sfs, usdsⁿ

k1

Gt, tkptkIkutk

≤ _∞

0

Gs, sasgusdsⁿ

k1

Gtk, t_kptkIkutk

≤1ε

g^q _∞

0

Gs, sasdsⁿ

k1

Gtk, tkI^qk

u

≤ u.

3.12

Therefore,Tu ≤ u.

Thus, we can obtain the existence of two positive solutionsu₁ andu₂ satisfying 0 <

u1< q <u2by usingLemma 2.4andRemark 2.5, respectively.

Using a similar proof ofTheorem 3.1, we can get the following conclusions.

Theorem 3.2. Let A1and A2 hold. Then the BVP 1.1 has at least two positive solutions satisfying 0<u1< q <u2if the following conditions hold:

H3g⁰_∞

0 Gs, sasds_n

k1Gtk, tkI⁰k<1, g^∞_∞

0 Gs, sasds_n

k1GtkI^∞k<

1,

H4there existsq >0 such thatωfq

_b

aGs, sds_n

k1Gtk, t_kIqk>1, for allωq≤u≤ q, a.e.t∈0,∞.

Corollary 3.3. In Theorems3.1and3.2, if conditionsH1andH3are replaced byH₁^∗andH₃^∗, respectively, then the conclusions also hold.

H₁^∗f0 ∞,or_n

k1I0k ∞;f_∞ ∞or_n

k1I_∞k ∞, H₃^∗g^∞0,_n

k1I^∞k 0,g⁰0,_n

k1I⁰k 0.

Remark 3.4. Notice that, in the above conclusions, we suppose that the singularity only exist inft, u, that is,ft, u → ∞ast → 0. If we permit ft, u → ∞ast → 0 or u → 0 andIkuk → ∞asuk → 0, then the discussion will be much more complex.

Now we state the corresponding results.

Let us define the following.

A^∗₁There exist four nonnegative functions a, g ∈ CJ, J , b, h ∈ CJ, J such that bthu ≤ ft, u ≤ atgu, and hu is nondecreasing on J. I_k : J → J, k1, . . . , n, are continuous.

A^∗₂0 < _∞

0 Gs, sasds < ∞, _∞

0 Gs, sbsds ≥ u^∗/ωh^∗, 0 < Gtk, tkptk <

∞, k1, . . . , n,whereu^∗∈K,h^∗h0.

Theorem 3.5. SupposeA^∗₁andA^∗₂hold, then the BVP1.1has at least two positive solutions satisfyingu^∗<u1< q <u2ifH1andH2hold.

Proof. DefineQ{u∈K:ut≥u^∗, for allt∈J}. We only need to prooveT :Ω∩Q → Qis a completely continuous operator. Then the rest of the proof is the same as thatTheorem 3.1.

Notice that

Tut≥ω _∞

0

Gs, sfs, usds≥ωh^∗ _∞

0

Gs, sbsds≥u^∗, 3.13

and changeS₁, S₂toS₁sup{gu:u^∗≤u≤M},S₂sup{Iku:u^∗≤u≤M,k1. . . , n}, then the same as the proof ofLemma 2.7, it is easy to compute thatT : Ω∩Q → Qis a completely continuous operator.

Corresponding to Theorem 3.2 and Corollary 3.3, there are Theorem 3.6 and Corollary 3.7. We just list here without proof.

Theorem 3.6. SupposeA^∗₁andA^∗₂hold, then the BVP1.1has at least two positive solutions satisfyingu^∗<u1< q <u2, ifH3and H4hold.

Corollary 3.7. In Theorems3.5and3.6, if conditionsH1andH3are replaced byH₁^∗andH₃^∗, respectively, then the conclusions also hold.

4. Example

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

Example 4.1. Consider the following boundary value problem:

e^tut|lnt|0, ∀t∈J, t /1, Δu

t11u²1,

u0 0, u∞ 0.

4.1

Conclusion 1. BVP4.1has at least two positive solutionsu1,u2satisfying 0<u1 <1/2 <

u2.

Proof. Letpt e^t,gu 1,ft, u at |lnt|,Iu u². Then by simple computation we have

Gt, s

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ _s

0

e^−σdσ _∞

t

e^−σdσ, 0≤s≤t <∞, _t

0

e^−σdσ _∞

s

e^−σdσ, 0≤t≤s <∞,

4.2

whereρ1. Furthermore,_∞

0 1/pσdσ_∞

0 e^−σdσ1<∞and 0<

_∞

0

Gs, sasds

_∞

0

1−e^−s

e^−s|lns|ds <∞, 0< Gt1, t₁pt1 1−e⁻¹<∞.

4.3

Leta, b 1,2 ⊂ 0,∞. Thenω e⁻². ThusA1andA2are satisfied. It is easy to get thatf₀ ∞, I∞1 ∞. Letq1/2. Then

g^q _∞

0

Gs, sasdsⁿ

k1

Gtk, t_kI^qk<1. 4.4

Hence,H₁^∗andH2are satisfied. Therefore, byCorollary 3.3, problem4.1has at least two positive solutionsu1,u2satisfying 0<u1<1/2<u2. The proof is completed.

Acknowledgment

This work is supported by the National Nature Science Foundation of P. R.China10871063 and Scientific Research Fund of Hunan Provincial Education Department07A038, partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project no.PGIDIT06PXIB207023PR.

References

1 D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.

2 R. P. Agarwal and D. O’Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.

3 E. R. Kaufmann, N. Kosmatov, and Y. N. Raffoul, “A second-order boundary value problem with impulsive effects on an unbounded domain,” Nonlinear Analysis: Theory, Methods & Applications, vol.

69, no. 9, pp. 2924–2929, 2008.

4 X. Zhang, X. Li, D. Jiang, and K. Wang, “Multiplicity positive solutions to periodic problems for first- order impulsive differential equations,” Journal of Computers & Mathematics with Applications, vol. 52, no. 6-7, pp. 953–966, 2006.

5 D. Guo, “Existence of solutions fornth order impulsive integro-differential equations in a Banach space,” vol. 47, no. 2, pp. 741–752.

6 D. Guo, “Multiple positive solutions of a boundary value problem fornth-order impulsive integro- differential equations in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 7, pp. 985–1006, 2004.

7 H. Zhang, L. Liu, and Y. Wu, “A unique positive solution fornth-order nonlinear impulsive singular integro-differential equations on unbounded domains in Banach spaces,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 649–659, 2008.

8 D. Guo, “Multiple positive solutions of a boundary value problem fornth-order impulsive integro- differential equations in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 4, pp. 618–641, 2005.

9 J. Sun and H. Chen, “Variational method to the impulsive equation with Neumann boundary conditions,” Boundary Value Problems, vol. 2009, Article ID 316812, 17 pages, 2009.

10 D. Guo, “Existence of positive solutions for nth-order nonlinear impulsive singular integro- differential equations in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 9, pp. 2727–2740, 2008.

11 X. Xian, D. O’Regan, and R. P. Agarwal, “Multiplicity results via topological degree for impulsive boundary value problems under non-well-ordered upper and lower solution conditions,” Boundary Value Problems, vol. 2008, Article ID 197205, 21 pages, 2008.

12 D. Guo, “Positive solutions of an infinite boundary value problem fornth-order nonlinear impulsive singular integro-differential equations in Banach spaces,” Nonlinear Analysis: Theory, Methods &

Applications, vol. 70, no. 5, pp. 2078–2090, 2009.

13 Y. Li and H. Zhang, “Extremal solutions of periodic boundary value problems for first-order impulsive integrodifferential equations of mixed-type on time scales,” Boundary Value Problems, vol.

2007, Article ID 73176, 16 pages, 2007.

14 H. Zhang, L. Liu, and Y. Wu, “Positive solutions fornth-order nonlinear impulsive singular integro- differential equations on infinite intervals in Banach spaces,” Nonlinear Analysis: Theory, Methods &

Applications, vol. 70, no. 2, pp. 772–787, 2009.

15 X. Zhang, “Positive solutions of singular multipoint boundary value problems for systems of nonlinear second-order differential equations on infinite intervals in Banach spaces,” Boundary Value Problems, vol. 2009, Article ID 978605, 22 pages, 2009.

16 A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 580–586, 2010.

17 D. O’Regan, B. Yan, and R. P. Agarwal, “Solutions in weighted spaces of singular boundary value problems on the half-line,” Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 751–

763, 2007.

18 J. Chu and J. J. Nieto, “Recent existence results for second-order singular periodic differential equations,” Boundary Value Problems, vol. 2009, Article ID 540863, 20 pages, 2009.

19 J. Li and J. J. Nieto, “Existence of positive solutions for multipoint boundary value problem on the half-line with impulses,” Boundary Value Problems, vol. 2009, Article ID 834158, 12 pages, 2009.

20 X. Liu and A. R. Willms, “Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft,” Mathematical Problems in Engineering, vol. 2, no. 4, pp. 277–299, 1996.

21 S. Pasquero, “Ideality criterion for unilateral constraints in time-dependent impulsive mechanics,”

Journal of Mathematical Physics, vol. 46, no. 11, Article ID 112904, 20 pages, 2005.

22 Y. Sun, Y. Sun, and L. Debnath, “On the existence of positive solutions for singular boundary value problems on the half-line,” Applied Mathematics Letters, vol. 22, no. 5, pp. 806–812, 2009.

23 L. Liu, Z. Wang, and Y. Wu, “Multiple positive solutions of the singular boundary value problems for second-order differential equations on the half-line,” Nonlinear Analysis: Theory, Methods &

Applications, vol. 71, no. 7-8, pp. 2564–2575, 2009.