Positive Solutions to Nonlinear First-Order Nonlocal BVPs with Parameter on Time Scales

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Volume 2011, Article ID 198598,15pages doi:10.1155/2011/198598

Research Article

Positive Solutions to Nonlinear First-Order Nonlocal BVPs with Parameter on Time Scales

Chenghua Gao

¹

and Hua Luo

²

1Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

2School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China

Correspondence should be addressed to Chenghua Gao,[email protected] Received 4 May 2010; Accepted 3 June 2010

Academic Editor: Daniel Franco

Copyrightq2011 C. Gao and H. Luo. 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 discuss the existence of solutions for the first-order multipoint BVPs on time scaleT:u^Δt

ptuσt λft, uσt,t ∈0, T_T,u0−_m

i1αiuξi 0, whereλ >0 is a parameter,T >0 is a fixed number, 0, T ∈T,f : 0, T_T×0,∞ → 0,∞is continuous,pis regressive and rd- continuous,αi ≥ 0,ξi ∈ T, i 1,2, . . . , m, 0 ξ0 < ξ1 < ξ2 < · · · < ξm−1 < ξm σT, and 1−_m

i1αi/epξi,0>0. For suitableλ >0, some existence, multiplicity, and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.

1. Introduction

LetTbe a time scalea nonempty closed subset of the real lineR. We discuss the existence of positive solutions to the first-order multipoint BVPs on time scaleT:

u^Δt ptuσt λft, uσt, t∈0, T_T,

u0−^m

i1

α_iuξ_i 0, 1.1

whereT > 0 is a fixed number, 0, T ∈ T,f : 0, T_T×0,∞ → 0,∞is continuous, pis regressive and rd-continuous,αi ≥ 0, ξi ∈T, i 1,2, . . . , m, 0 ξ0 < ξ1 < ξ2 <· · · < ξm−1 <

ξm σT, and 1−_m

i1αi/epξi,0>0,ept, sis defined in its standard form; see1, page 59for details.

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The multipoint boundary value problems arise in a variety of diﬀerent areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross-section and composed of N parts of diﬀerent densities can be set up as a multipoint boundary value problem 2; also many problems in the theory of elastic stability can be handled by a multipoint problem 3. So, the existence of solutions to multipoint boundary value problems have been studied by many authors; see4–13and the reference therein. Especially, in recent years the existence of positive solutions to multipoint boundary value problems on time scales has caught considerable attention; see10–14. For other background on dynamic equations on time scales, one can see1,15–18.

Our ideas arise from13,16. In13, Tian and Ge discussed the existence of positive solutions to nonlinear first-order three-point boundary value problems on time scaleT:

u^Δt ptuσt ft, uσt, t∈0, T_T, u0−αu

η

βuσT, 1.2

wheref :0, T_T×0,∞ → 0,∞is continuous,pis regressive and rd-continuous,α, β≥0 andα/epξ,0 β/epσT,0 < 1. The existence results are based on Krasnoselskii’s fixed- point theorem in cones and Leggett-Williams’s theorem.

As we can see, if we takeλ1,ξ1 η,ξ_mσT, andα_i 0 fori2, . . . , m−1, then 1.1is reduced to1.2. Because of the influence of the parameterλ, it will be more diﬃcult to solve1.1than to solve1.2.

In 2009, by using the fixed-point index theory, Sun and Li16discussed the existence of positive solutions to the first-order PBVPs on time scaleT:

u^Δt ptuσt λfut, t∈0, T_T,

u0 uσT. 1.3

For suitableλ >0, they gave some existence, multiplicity, and nonexistence criteria of positive solutions.

Motivated by the above results, by using the well-known fixed-point index theory16, 19, we attempt to obtain some existence, multiplicity and nonexistence criteria of positive solutions to1.1for suitableλ >0.

The rest of this paper is arranged as follows. Some preliminary results including Green’s function are given inSection 2. InSection 3, we obtain some useful lemmas for the proof of the main result. InSection 4, some existence and multiplicity results are established.

At last, some nonexistence results are given inSection 5.

2. Preliminaries

Throughout the rest of this paper, we make the following assumptions:

H1f:0, T_T×0,∞ → 0,∞is continuous andf·, u>0 foru >0,

H2p:0, T_T → 0,∞is rd-continuous, which implies thatp∈ RwhereRis defined in16,18,20.

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Moreover, let

f0lim inf

u→0 inf

t∈0,T_T

ft, u

u , f∞lim inf

u→∞ inf

t∈0,T_T

ft, u

u ,

f⁰lim sup

u→0

sup

t∈0,TT

ft, u

u , f^∞lim sup

u→∞ sup

t∈0,TT

ft, u

u .

2.1

Our main tool is the well-known results from the fixed-point index, which we state here for the convenience of the reader.

Theorem 2.1 see19. LetX be a Banach space and P be a cone in X. For r > 0, we define Pr {x ∈ P | x < r}. Assume thatΦ : Pr → P is completely continuous such Φx /xfor x∈∂P_r {x∈P | xr}.

iIfΦx ≥ xforx∈∂P_r, then

iΦ, P_r, P 0. 2.2

iiIfΦx ≤ xforx∈∂Pr, then

iΦ, P_r, P 1. 2.3

LetE : {x|x : 0, σT_T → R is continuous}be equipped with the normx max_t∈0,σT_T|xt|. It is easy to see thatE, · is a Banach space.

Forh∈E, we consider the following linear BVP:

u^Δt ptuσt λhσt, t∈0, T_T, 2.4

u0−^m

i1

αiuξi 0. 2.5

Forξi, s∈0, σT_T, define

Hξis

⎧⎪

⎨

⎪⎩ αi

epξi,0, s≤ξi, 0, s > ξi.

2.6

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Lemma 2.2. Forh∈E, the linear BVP2.4-2.5has a solutionuif and only ifusatisfies

ut λ

ept,0

Γ^m

i1

α_i epξi,0

_ξ_i

0

eps,0hσsΔs _t

0

eps,0hσsΔs

, 2.7

where

Γ 1 1−_m

i1

α_i/e_pξ_i,0. 2.8

Proof. By2.4, we have

ute_pt,0_Δ

λe_pt,0hσt. 2.9

So,

utept,0 u0 λ _t

0

eps,0hσsΔs. 2.10

And so,

ut 1

ept,0

u0 λ

_t

0

e_ps,0hσsΔs

. 2.11

Combining this with2.5, we get

ut λ

ept,0

Γ^m

i1

α_i epξi,0

_ξ_i

0

e_ps,0hσsΔs _t

0

e_ps,0hσsΔs

. 2.12

Lemma 2.3. If the functionuis defined in2.7, thenumay be expressed by

ut λ

_T

0

Gt, shσsΔs, 2.13

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where

Gt, s 1

e_pt,0

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ Γ^m

ji1

α_j ep

ξj,0e_ps,0 e_ps,0, ξ_i≤s, σs≤t, σt≤ξ_i1, i0,1, . . . , m−1,

Γ

⎛

⎝ⁱ

j1

Hξjs ^m

ji1

α_j e_p

ξ_j,0

⎞

⎠

×eps,0 eps,0, σs≤ξi≤t, σt≤ξi1, i1,2, . . . , m−1, Γ^m

ji1

αj

e_p

ξ_j,0eps,0, ξi≤t, σt≤s, σs≤ξi1, i0,1, . . . , m−1,

Γ^m

ji2

H_ξ_jse_ps,0, ξ_i≤t, σt≤ξ_i1≤s, i0,1, . . . , m−2.

2.14

Proof. Whenσt≤ξ1,

ut λ

e_pt,0

Γ _m

i1

αi

e_pξ_i,0 _t

0

e_ps,0hσsΔs^m

i1

αi

e_pξ_i,0 _ξ₁

t e_ps,0hσsΔs

Γ^m

i2

α_i epξi,0

_ξ_i

ξ1

e_ps,0hσsΔs _t

0

e_ps,0hσsΔs

.

2.15

1Forσs≤t,

Gt, s 1

e_pt,0

Γ^m

i1

αi

e_pξ_i,0eps,0 eps,0

. 2.16

2Forσt≤s, σs≤ξ1,

Gt, s 1

e_pt,0

Γ^m

i1

αi

e_pξ_i,0eps,0

. 2.17

3Forσt≤ξ1 ≤s,

Gt, s 1

e_pt,0

Γ^m

i2

Hξiseps,0

. 2.18

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Whenξi≤t, σt≤ξi1, i1,2, . . . , m−2,

ut λ

ept,0

⎧⎨

⎩Γ

⎡

⎣ⁱ

j1

αj

ep ξj,0

_ξ_j

0

e_ps,0hσsΔs ^m

ji1

αj

ep ξj,0

_ξ_i

0

e_ps,0hσsΔs

⎤

⎦

Γ

⎡

⎣^m

ji1

αj

ep ξj,0

_ξ_j

ξi

eps,0hσsΔs

⎤

⎦ _t

0

eps,0hσsΔs

⎫⎬

⎭.

2.19 1Forσs≤ξi,

Gt, s eps,0 ept,0

⎡

⎣Γ

⎛

⎝ⁱ

j1

Hξjs ^m

ji1

αj

e_p ξ_j,0

⎞

⎠1

⎤

⎦. 2.20

2Forξi≤s, σs≤t,

Gt, s eps,0 ept,0

⎡

⎣Γ^m

ji1

αj

ep

ξj,0 1

⎤

⎦. 2.21

3Forσt≤s, σs≤ξi1,

Gt, s eps,0 e_pt,0Γ^m

ji1

αj

e_p

ξ_j,0. 2.22

4Forξi1≤s,

Gt, s e_ps,0 ept,0Γ^m

ji2

H_ξ_js. 2.23

Whent≥ξm−1,

ut λ

e_pt,0

⎧⎨

⎩Γ

⎡

⎣^m−1

j1

αj

e_p ξ_j,0

_ξ_j

0

eps,0hσsΔs αm

e_pξ_m,0 _ξ_m−1

0

eps,0hσsΔs

⎤

⎦ Γ α_m

epξm,0 _t

ξm−1

e_ps,0hσsΔs

Γ αm

e_pξ_m,0 _ξ_m

t e_ps,0hσsΔs _t

0

e_ps,0hσsΔs

⎫⎬

⎭.

2.24

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1Forσs≤ξ_m−1,

Gt, s e_ps,0 e_pt,0

Γ _m−1

i1

Hξis αm

e_pξ_m,0

1

. 2.25

2Forξ_m−1≤s, σs≤t,

Gt, s eps,0 ept,0

Γ α_m epξm,01

. 2.26

3Forσt≤s,

Gt, s eps,0 ept,0Γ αm

epξm,0. 2.27

Lemma 2.4. Green’s functionGt, shas the following properties.

iGt, s≥0, t, s∈0, σT_T×0, T_T, iim≤Gt, s≤M,wherem Γ_m

i1α_i/e_pσT,0²;M Γ_m

i1α_ie_pσT,0;

iiiGt, s≥m/Msup_t,s∈0,σT_T_×0,T_TGt, s,t, s∈0, σT_T×0, T_T. Proof. This proof is similar to13, Lemma 2.3, so we omit it.

Now, we define a conePinEas follows:

P{x∈E|xt≥0, xt≥δxon0, σT_T}, 2.28

whereδm/M. Forr >0, letP_r {u∈P | u< r}and∂P_r {u∈P | ur}.

Forλ >0, define an operatorΦλ:P → E:

Φλut λ _T

0

Gt, sfs, uσsΔs, t∈0, σT_T. 2.29

Similar to the proof of13, Lemma 2.4, we can see thatΦλ:P → Pis completely continuous.

By the above discussions, its not diﬃcult to see thatubeing a solution of BVP1.1equals the solution thatuis a fixed point of the operatorΦ_λ.

3. Some Lemmas

Lemma 3.1. Letη >0. Ifu∈Pandft, uσt≥ηuσt,t∈0, T_T, then

Φλx ≥ληδmTu. 3.1

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Proof. Sinceu∈Pandft, uσt≥ηuσt,t∈0, T_T, we have

Φλut λ _T

0

Gt, sfs, uσsΔs≥λη _T

0

Gt, suσsΔs≥ληδmTu. 3.2

Lemma 3.2. Letε >0. Ifu∈P andft, uσt≤εuσt,t∈0, T_T, then

Φ_λu ≤λεMTu. 3.3

Proof. Sinceu∈Pandft, uσt≤εuσt,t∈0, T_T, we have

Φλut λ _T

0

Gt, sfs, uσsΔs≤λε _T

0

Gt, suσsΔs≤λεMTu. 3.4

Lemma 3.3. Letr >0. Ifu∈∂P_r, then

λmrmT≤ Φλx ≤λMrMT, 3.5

wheremr min_t,u∈0,T_T_×δr,rft, u;Mr max_t,u∈0,T_T_×δr,rft, u.

Proof. Sinceu∈∂Pr, we haveδr≤uσt≤r,t∈0, T_T. So,

Φλut λ _T

0

Gt, sfs, uσsΔs≥λmr

_T

0

Gt, sΔs≥λmrmT,

Φλut λ _σT

0

Gt, sfs, uσsΔs≤λMr

_σT

0

Gt, sΔs≤λMrMT.

3.6

4. Some Existence and Multiplicity Results

Theorem 4.1. Assume that (H1) and (H2) hold and thatf⁰ ∈0,∞, f^∞ ∈0,∞. Then the BVP 1.1has at least two positive solutions for

1

m1mT < λ < 1 2MTmax

f⁰, f^∞. 4.1

Proof. Letr11. Then it follows from4.1andLemma 3.3that

Φ_λu ≥λm1mT >1u, foru∈∂P_r₁. 4.2

In view ofTheorem 2.1, we have

iΦλ, Pr1, P 0. 4.3

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Now, combined with the definition off⁰, we may choose 0< r2< r1such thatft, u≤ f⁰εuforu∈0, r2andt∈0, T_Tuniformly, whereε >0 satisfies

λεMT < 1

2. 4.4

So,

ft, uσt≤ f⁰ε!

uσt, foru∈∂P_r₂, t∈0, T_T. 4.5

In view of4.1,4.4,4.5, andLemma 3.2, we have Φλu ≤λ f⁰ε!

MTu<u, foru∈∂Pr2. 4.6

It follows fromTheorem 2.1that

iΦ_λ, P_r₂, P 1. 4.7

By4.3and4.7, we get

i Φλ, Pr1\Pr2, P!

−1. 4.8

This shows thatΦ_λhas a fixed point inP_r₁\P_r₂, which is a positive solution of the BVP1.1.

Now, by the definition off^∞, there exits anH >" 0 such thatft, u ≤ f^∞εufor u∈H," ∞andt∈0, T_T, whereε >0 is chosen so that

λεMT < 1

2. 4.9

Letr3max{2r1,H/δ}. Then for" u∈∂Pr3,ut≥δuδr3≥H," t∈0, σT_T. So, ft, uσt≤

f^∞ε

uσt, foru∈∂Pr3, t∈0, T_T. 4.10

In view of4.1,4.9, andLemma 3.2, we have Φλu ≤λ

f^∞ε

MTu<u, forx∈∂Pr3. 4.11

iΦλ, Pr3, P 1. 4.12

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By4.3and4.12, we get

1. 4.13

This shows thatΦ_λhas a fixed point inP_r₃\P_r₁, which is another positive solution of the BVP 1.1.

Similar to the proof ofTheorem 4.1, we have the following results.

Theorem 4.2. Suppose that (H1) and (H2) hold and λ > 1

m1mT. 4.14

Then,

iequation1.1has at least one positive solution iff⁰0, iiequation1.1has at least one positive solution iff^∞0, iiiequation1.1has at least two positive solutions iff⁰ f^∞0.

Theorem 4.3. Assume that (H1) and (H2) hold. Iff0∈0,∞, f∞∈0,∞, then the BVP1.1has at least two positive solutions for

2 δmT min

f0, f∞ < λ < 1

M1MT. 4.15

Proof. Letr11. Then it follows from4.15andLemma 3.3that

Φ_λu ≤λM1MT <1u, foru∈∂P_r₁. 4.16

In view ofTheorem 2.1, we have

iΦλ, Pr1, P 1. 4.17

Sincef0>0, we may choose 0< r2< r1such thatft, u≥f0−η1uforu∈0, r2and t∈0, T_T, where 0< η1< f0satisfiesλη1δmσT<1.So,

ft, uσt≥

f0−η1

uσt, foru∈∂P2, t∈0, T_T. 4.18

In view of4.15,4.18, andLemma 3.1, we have Φλu ≥λ

f0−η1

δmTu>u, foru∈∂P2. 4.19

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iΦ_λ, P_r₂, P 0. 4.20

1. 4.21

This shows thatΦλhas a fixed point inPr1\Pr2, which is a positive solution of the BVP1.1.

Now, by the definition off_∞, there exists anH >" 0 such thatft, u ≥f_∞−η2ufor u∈H," ∞andt∈0, T_T, where 0< η2< f_∞satisfies

λη2δmT <1. 4.22

Letr3max{2r1,H/δ}. Then for" u∈∂P_r₃,ut≥δuδr3≥H," t∈0, σT_T. So, ft, uσt≥

f_∞−η2

uσt, foru∈∂P_r₃, t∈0, T_T. 4.23

Combined with4.22andLemma 3.1, we have Φ_λu ≥λ

f_∞−η2

δmTu>u foru∈∂P_r₃. 4.24

iΦλ, Pr3, P 0, 4.25

−1, 4.26

This shows thatΦ_λhas a fixed point inP_r₃\P_r₁, which is another positive solution of the BVP 1.1.

Similar to the proof ofTheorem 4.3, we have the following results.

Theorem 4.4. Suppose that (H1) and (H2) hold and that λ < 1

M1MT. 4.27

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Then,

iequation1.1has at least one positive solution iff0 ∞, iiequation1.1has at least one positive solution iff_∞∞, iiiequation1.1has at least two positive solutions iff0f_∞∞.

Theorem 4.5. Suppose that (H1) and (H2) hold. Iff⁰ ∈ 0,∞, f0 ∈ 0,∞, f^∞ ∈ 0,∞, f∞ ∈ 0,∞, then the BVP1.1has at least one positive solution for

1 δmTmin

f∞, f0

< λ < 1 MTmax

f⁰, f^∞. 4.28

Proof. We only deal with the case that f_∞ ≤ f0, f⁰ ≥ f^∞. The other three cases can be discussed similarly.

Letλsatisfy4.28and letε >0 be chosen such that

f∞−ε >0, 1 δmT

f_∞−ε ≤λ≤ 1

MT

f⁰ε. 4.29

From the definition of f⁰, we know that there exists a constantr1 > 0 such that ft, u ≤ f⁰εuforu∈0, r1andt∈0, T_T. So,

ft, uσt≤ f⁰ε!

uσt, foru∈∂P_r₁, t∈0, T_T. 4.30

This combines with4.29andLemma 3.2, we have Φλu ≤λ f⁰ε!

MTu ≤ u, foru∈∂Pr1. 4.31

iΦλ, Pr1, P 1. 4.32

On the other hand, from the definition off_∞, there exists anH >" 0 such thatft, u≥f_∞−εu foru∈H," ∞andt∈0, T_T. Letr2 max{2r1,H/δ}. Then for" u∈∂Pr2,uσt≥ δu δr2≥H," t∈0, T_T. So,

ft, ut≥ f∞−ε

ut, foru∈∂Pr2, t∈0, T_T. 4.33

Combined with4.29andLemma 3.1, we have Φλu ≥λ

f∞−ε

δmTu ≥ u. 4.34

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iΦ_λ, P_r₂, P 0. 4.35

−1, 4.36

which implies that the BVP1.1has at least one positive solution inP_r₂\P_r₁.

Remark 4.6. By making some minor modifications to the proof ofTheorem 4.5, we can obtain the existence of at least one positive solution, if one of the following conditions is satisfied:

if⁰ 0,f∞∈0,∞andλ >1/δmTf∞. iif⁰ 0,f∞∞andλ∈0,∞.

iiif^∞0,f0∈0,∞andλ >1/δmTf0. ivf^∞0,f0∞andλ∈0,∞.

Remark 4.7. From Conditions ii and iv of Remark 4.6, we know that the conclusion in Theorem 4.5holds forλ1 in these two cases. Byf⁰0 andf_∞∞, there exist two positive constants 0< r < R <∞such that, fort∈0, T_T,

ft, u≤ u

MT foru∈0, r, ft, u≥ u

δmT foru∈R,∞. 4.37

This is the condition of Theorem 3.2 of13. Byf^∞ 0 andf0 ∞, there exist two positive constants 0< r < R <∞such that fort∈0, T_T,

ft, u≤ u

MT foru∈R,∞; ft, u≥ u

δmT foru∈0, r. 4.38

This is the condition of Theorem 3.3 of13. So, our conclusions extend and improve the results of13.

5. Some Nonexistence Results

Theorem 5.1. Assume that (H1) and (H2) hold. Iff⁰∈0,∞andf^∞∈0,∞, then the BVP1.1 has no positive solutions for suﬃciently smallλ >0.

Proof. In view of the definition of f⁰, f^∞, there exist positive constants ε1, ε2, r1 and r2

satisfyingr1< r2and

ft, u≤ f⁰ε1

!u, u∈0, r1, ft, u≤ f^∞ε2

u, u∈r2,∞. 5.1

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Let

c1max

#

f⁰ε1, f^∞ε2, max

t∈0,T_T,u∈r1,r2

ft, u u

$

. 5.2

Thenc1>0 and we have

ft, u≤c1u, u∈0,∞, t∈0, T_T. 5.3

We assert that the BVP1.1has no positive solutions for 0< λ <1/c1MT.

Suppose on the contrary that the BVP 1.1 has a positive solutionu for 0 < λ <

1/c1MT. Then from5.3andLemma 3.2, we get

uΦ_λu ≤λc1MTu<u, 5.4

which is a contradiction.

Theorem 5.2. Assume that (H1) and (H2) hold. Iff0∈0,∞andf_∞∈0,∞, then the BVP1.1 has no positive solutions for suﬃciently largeλ >0.

Proof. By the definition off0, f∞, there exist positive constants η1, η2, r1, andr2 satisfying f0> η1,f_∞> η2,r1< r2, and

ft, u≥ f0−η1

u, u∈0, r1, ft, u≥ f_∞−η2

u, u∈r2,∞. 5.5

Let

c2min

#

f0−η1, f∞−η2, min

t∈0,T_T,u∈r1,r2

ft, u u

$

. 5.6

Thenc2>0 and we have

ft, u≥c2u, u∈0,∞, t∈0, T_T. 5.7

We assert that the BVP1.1has no positive solutions forλ >1/c2δmT.

Suppose on the contrary that the BVP1.1has a positive solutionuforλ >1/c2δmT.

Then from5.7andLemma 3.1we get

uΦ_λu ≥λc2δmTu>u, 5.8

which is a contradiction.

Corollary 5.3. Assume that (H1) and (H2) hold. Iff0 ∞andf∞∞, then the BVP1.1has no positive solutions for suﬃciently largeλ >0.

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Acknowledgments

This work was supported by the NSFC Young Item no. 70901016, HSSF of Ministry of Education of Chinano. 09YJA790028, Program for Innovative Research Team of Liaoning Educational Committeeno. 2008T054, the NSF of Liaoning Provinceno. L09DJY065, and NWNU-LKQN-09-3

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