Smooth Positive Solutions to a Class of Singular m-Point Boundary Value Problems

Volume 2009, Article ID 191627,13pages doi:10.1155/2009/191627

Research Article

Existence and Uniqueness of

Smooth Positive Solutions to a Class of Singular m-Point Boundary Value Problems

Xinsheng Du and Zengqin Zhao

School of Mathematics Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Xinsheng Du,[email protected] Received 2 April 2009; Revised 15 September 2009; Accepted 23 November 2009 Recommended by Donal O’Regan

This paper investigates the existence and uniqueness of smooth positive solutions to a class of singular m-point boundary value problems of second-order ordinary differential equations. A necessary and sufficient condition for the existence and uniqueness of smooth positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Our nonlinearity ft, u, vmay be singular atv, t0 and/ort1.

Copyrightq2009 X. Du and Z. Zhao. 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 and the Main Results

In this paper, we will consider the existence and uniqueness of positive solutions to a class of second-order singular m-point boundary value problems of the following differential equation:

−ut ft, ut, ut, t∈0,1, 1.1

with

u0 ^m−2

i1

α_iu η_i

, u1 0, 1.2

where 0 < αi <1, i 1,2, . . . , m−2, 0 < η1 < η2 <· · · < η_m−2 <1,are constants,_m−2

i1 αi <

1, m≥3,andfsatisfies the following hypothesis:

Hft, u, v:0,1×0,∞×0,∞ → 0,∞is continuous, nondecreasing onu, and nonincreasing onvfor each fixedt∈0,1,there exists a real numberb∈R such that for anyr ∈0,1,

ft, u, rv≤r^−bft, u, v, ∀t, u, v∈0,1×0,∞×0,∞, 1.3

there exists a functiong :1,∞ → 0,∞, gl< l, andgl/l² is integrable on 1,∞such that

ft, lu, v≤glft, u, v, ∀t, u, v∈0,1×0,∞×0,∞, l∈1,∞. 1.4

Remark 1.1. iInequality1.3implies

ft, u, cv≥c^−bft, u, v, ifc≥1. 1.5

Conversely,1.5implies1.3.

iiInequality1.4implies

ft, cu, v≥ g

c⁻¹₋₁

ft, u, v, if 0< c <1. 1.6

Conversely,1.6implies1.4.

Remark 1.2. It follows from1.3,1.4that

ft, u, u≤

⎧⎪

⎪⎨

⎪⎪

⎩ gu

v

ft, v, v, if u≥v >0, v

u b

ft, v, v, if v≥u >0.

1.7

When ft, u is increasing with respect to u, singular nonlinear m-point boundary value problems have been extensively studied in the literature, see1–3 . However, when ft, u, vis increasing onu, and is decreasing onv, the study on it has proceeded very slowly.

The purpose of this paper is to fill this gap. In addition, it is valuable to point out that the nonlinearityft, u, vmay be singular att0, t1 and/orv0.

When referring to singularity we mean that the functionsfin1.1are allowed to be unbounded at the pointsv0, t0, and/ort1. A functionut∈C0,1 ∩C²0,1is called aC0,1 positivesolution to1.1and 1.2if it satisfies1.1and1.2 ut > 0,fort ∈ 0,1. AC0,1 positivesolution to1.1and1.2is called a smoothpositivesolution if u0andu1⁻both existut>0 fort∈0,1. Sometimes, we also call a smooth solution aC¹0,1 solution. It is worth stating here that a nontrivialC0,1 nonnegative solution to the problem1.1,1.2must be a positive solution. In fact, it is a nontrivial concave function satisfying1.2which, of course, cannot be equal to zero at any pointt∈0,1.

To seek necessary and sufficient conditions for the existence of solutions to the above problems is important and interesting, but difficult. Thus, researches in this respect are rare

up to now. In this paper, we will study the existence and uniqueness of smooth positive solutions to the second-order singularm-point boundary value problem1.1and 1.2. A necessary and sufficient condition for the existence of smooth positive solutions is given by constructing lower and upper solutions and with the maximal principle. Also, the uniqueness of the smooth positive solutions is studied.

A functionαtis called a lower solution to the problem1.1,1.2, ifαt∈ C0,1 ∩ C²0,1and satisfies

αt ft, αt, αt≥0, t∈0,1, α0−^m−2

i1

α_iα η_i

≤0, α1≤0. 1.8

Upper solution is defined by reversing the above inequality signs. If there exist a lower solutionαtand an upper solutionβtto problem 1.1,1.2such thatαt ≤ βt, then αt, βtis called a couple of upper and lower solution to problem1.1,1.2.

To prove the main result, we need the following maximal principle.

Lemma 1.3maximal principle. Suppose that 0< η₁ < η₂ <· · ·< η_m−2 < b_n <1, n1,2, . . ., andFn {ut∈C0, bn ∩C²0, bn, u0−_m−2

i1 αiuηi≥0, ubn≥0}. Ifu∈Fnsuch that

−ut≥0, t∈0, bnthenut≥0, t∈0, bn . Proof. Let

−ut δt, t∈0, bn, 1.9

u0−^m−2

i1

αiu ηi

r1, ubn r2, 1.10

thenr₁≥0, r₂≥0, δt≥0, t∈0, bn.

By integrating1.9twice and noting1.10, we have

ut 1

b_n

1−_m−2

i1 α_i

_m−2

i1 α_iη_i

1−^m−2

i1

αi

t^m−2

i1

αiηi

r2 bn−tr1

_bn

0

Gnt, sδsds bn−t b_n

1−_m−2

i1 α_i

_m−2

i1 α_iη_i

m−2

i1

αi

_bn

0

Gn

ηi, s δsds,

1.11 where

G_nt, s 1 b_n

⎧⎨

⎩

tbn−s, 0≤t≤s≤bn,

sbn−t, 0≤s≤t≤bn. 1.12

In view of 1.11 and the definition ofGnt, s, we can obtain ut ≥ 0, t ∈ 0, bn . This completes the proof ofLemma 1.3.

Now we state the main results of this paper as follows.

Theorem 1.4. Suppose thatHholds, then a necessary and sufficient condition for the problem1.1 and1.2to have smooth positive solution is that

0<

₁

0

fs,1−s,1−sds <∞. 1.13

Theorem 1.5. Suppose thatHand1.13hold, then the smooth positive solution to problem1.1 and1.2is also the uniqueC0,1 positive solution.

2. Proof of Theorem 1.4

Suppose thatwt is a smooth positive solution to the boundary value problem1.1and 1.2. We will show that1.13holds.

It follows from

wt −ft, wt, wt≤0, t∈0,1, 2.1

thatwtis nonincreasing on0,1 .Thus, by the Lebesgue theorem, we have ₁

0

ft, wt, wtdt− ₁

0

wtdtw0−w1−<∞. 2.2

It is well known thatwtcan be stated as

wt ₁

0

Gt, sfs, ws, wsds 1−t

1−_m−2

i1 α_{i m−2}

i1 α_iη_i

m−2

i1

αi

₁

0

G ηi, s

fs, ws, wsds,

2.3

where

Gt, s

⎧⎨

⎩

t1−s, 0≤t≤s≤1,

s1−t, 0≤s≤t≤1. 2.4

By2.3and1.2we have 1

1−_m−2

i1 α_{i m−2}

i1 α_iη_i

m−2

i1

α_{i 1}

0

G η_i, s

fs, ws, wsds^m−2

i1

α_i w

η_i

, 2.5

therefore because of2.3and2.5,

wt≥1−t^m−2

i1

αi

w ηi

, t∈0,1 . 2.6

Sincewtis a smooth positive solution to1.1and1.2, we have

wt ₁

t

−ws

ds≤ max

t∈0,1 wt1−t, t∈0,1 . 2.7

Letm_m−2

i1 α_iwηi , Mmax_t∈0,1 |wt|.From2.6,2.7it follows that

m1−t≤wt≤M1−t, t∈0,1 . 2.8

Without loss of generality we may assume that 0< m <1< M.This together with the conditionHimplies

₁

0

fs,1−s,1−sds≤ ₁

0

f

s, 1

mws, 1

Mws

ds

≤g 1

m

M^b ₁

0

fs, ws, wsds <∞.

2.9

On the other hand, notice thatwtis a smooth positive solution to 1.1,1.2, we have

ft, wt, wt −wt/≡0, t∈0,1, 2.10

therefore, there exists a positive numbert₀∈0,1such thatft0, wt0, wt0>0.Obviously, wt0>0 and 1−t0 >0.It follows from1.7that

0< ft0, wt0, wt0≤

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ g

wt0 1−t₀

ft0,1−t0,1−t0, ifwt0≥1−t0, 1−t0

wt0 b

ft0,1−t0,1−t0, ifwt0≤1−t0.

2.11

Consequentlyft0,1−t0,1−t0>0,which implies that ₁

0

fs,1−s,1−sds >0. 2.12

From2.9and2.12it follows that

0<

₁

0

fs,1−s,1−s<∞, 2.13

which is the required inequality.

2.2. The Existence of Lower and Upper Solutions Sincegl/l²is integrable on1,∞,thus

llim→∞infgl

l 0. 2.14

Otherwise, if liml→∞infgl/l m0 > 0,then there exists a real number X > 0,such that gl/l² ≥ m₀/2lwhenl > X,this contradicts with the condition thatgl/l²is integrable on 1,∞.By conditionHand2.14we have

ft, ru, v≥hrft, u, v, r∈0,1, 2.15

rlim→0sup r

hr lim

p→∞sup p⁻¹ h

p⁻¹ lim

p→∞infg p

p 0, 2.16

wherehr gr⁻¹⁻¹, r∈0,1.

Suppose that1.13holds. Let bt

₁

0

Gt, sfs,1−s,1−sds

1−t 1−_m−2

i1 α_im−2

i1 α_iη_i

m−2

i1

α_{i 1}

0

G η_i, s

fs,1−s,1−sds.

2.17

Since by1.13,2.17we obviously have

bt∈C¹0,1 ∩C²0,1, bt −ft,1−t,1−t, t∈0,1, 2.18 and there exists a positive numberk <1 such that

k1−t≤bt≤ 1

k1−t, t∈0,1 . 2.19

By2.14and2.16we see, if 0< l < kis sufficiently small, then

hlk−l≥0, g 1

lk

−1

l ≤0. 2.20

Let

Ht lbt, Qt 1

lbt, t∈0,1 . 2.21

Then from2.19and2.21we have

lk1−t≤Ht≤1−t≤Qt≤ 1

lk1−t, t∈0,1 . 2.22

Consequently, with the aid of2.20,2.22and the conditionHwe have Ht ft, Ht, Ht≥ft, lk1−t,1−t−lft,1−t,1−t

≥hlk−l ft,1−t,1−t≥0, 2.23

Qt ft, Qt, Qt≤f

t, 1

lk1−t,1−t

−1

lft,1−t,1−t

≤

g 1

lk

− 1 l

ft,1−t,1−t≤0.

2.24

From2.17,2.21it follows that

H0 ^m−2

i1

α_iH η_i

, H1 0, 2.25

Q0 ^m−2

i1

αiQ ηi

, Q1 0, 2.26

therefore,2.23–2.26imply thatHt, Qtare lower and upper solutions to the problem 1.1and1.2, respectively.

2.3. The Sufficient Condition

First of all, we define a partial ordering inC0,1 ∩C²0,1byu≤v,if and only if

ut≤vt, ∀t∈0,1 . 2.27

Then, we will define an auxiliary function. For allut∈C0,1 ∩C²0,1,

gt, ut

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ft, Ht, Ht, ifut≤Ht,

ft, ut, ut, ifHt≤ut≤Qt,

ft, Qt, Qt, ifut≥Qt.

2.28

By the assumption ofTheorem 1.4, we have thatg :0,1×−∞,∞ → 0,∞is continuous.

Let{bn}be a sequence satisfyingη_m−2 < b1 <· · ·< bn< b_n1 <· · ·<1,andbn → 1 as n → ∞,and let{rn}be a sequence satisfying

Hbn≤r_n≤Qbn, n1,2, . . . . 2.29

For eachn,let us consider the following nonsingular problem:

−ut gt, ut, t∈0, bn , u0−^m−2

i1

α_iu η_i

0, ubn r_n. 2.30

Obviously, it follows from the proof ofLemma 1.3that problem2.30is equivalent to the integral equation

ut A_nut

1−_m−2

i1 α_i

t_m−2

i1 α_iη_i r_n bn

1−_m−2

i1 αi

_m−2

i1 αiηi

_bn

0

G_nt, sgs, usds

bn−t b_n

1−_m−2

i1 α_i

_m−2

i1 α_iη_i

m−2

i1

αi

_bn

0

Gn

ηi, s

gs, usds, t∈0, bn ,

2.31

where Gnt, s is defined in the proof ofLemma 1.3. It is easy to verify thatAn : Xn → X_n C0, bn is a completely continuous operator andA_nXnis a bounded set. Moreover, u ∈C0, bn is a solution to2.30if and only ifA_nu u.Using the Schauder’s fixed point theorem, we assert thatAnhas at least one fixed pointun∈C²0, bn .

We claim that

Ht≤u_nt≤Qt, t∈0, bn . 2.32

From this it follows that

−ut ft, ut, ut, t∈0, bn . 2.33

Suppose by contradiction thatunt≤Qtis not satisfied on0, bn . Let

zt Qt−u_nt, t∈0, bn , 2.34

therefore

zt^∗ min

t∈0,bn

zt<0. 2.35

Since by the definition ofQtand2.30we obviously havet^∗/0, t^∗/b_n. Let

cinf{t1 |zt<0, t∈t1, t^∗ },

dsup{t2 |zt<0, t∈t^∗, t₂}. 2.36

So, whent∈c, d, we haveQt< u_nt,and

gt, unt ft, Qt, Qt, u_nt gt, Qt 0,

Qt gt, Qt Qt ft, Qt, Qt≤0.

2.37

Thereforezt Qt−u_nt≤0, t∈c, d,that is,ztis an upper convex function inc, d.

By2.30and2.36, forc, dwe have the following two cases:

izc zd 0, iizc<0, zd 0.

For casei: it is clear thatzt≥0, t∈c, d,this is a contradiction.

For caseii: in this casec 0, zt^∗ 0. Sincezt is decreasing onc, d , thus, zt ≤ 0, t ∈ t^∗, d ,that is, ztis decreasing ont^∗, d .Fromzd 0,we see zt^∗ ≥ 0, which is in contradiction withzt^∗<0.

From this it follows thatunt≤Qt, t∈0, bn .

Similarly, we can verify thatHt≤u_nt, t∈0, bn .Consequently2.32holds.

Using the method of 4 and 5, Theorem 3.2 , we can obtain that there is aC0,1 positive solution wt to1.1,1.2 such thatHt ≤ wt ≤ Qt,and a subsequence of {unt}converges towton any compact subintervals of0,1.

3. Proof of Theorem 1.5

Suppose thatu₁tandu₂tareC0,1 positive solutions to1.1and1.2, and at least one of them is a smooth positive solution. Ifu₁t/≡u₂tfor anyt∈0,1 ,without loss of generality, we may assume thatu2t^∗> u1t^∗for somet^∗∈0,1.Let

T inf{t1|0≤t₁< t^∗, u₂t> u₁t, t∈t1, t^∗ }, Ssup{t2|t^∗≤t₂<1, u₂t> u₁t, t∈t^∗, t₂},

yt u1tu₂t−u2tu₁t, t∈0,1.

3.1

It follows from3.1that

0≤T < S≤1, u₂t≥u₁t, t∈T, S. 3.2

By1.2, it is easy to check that there exist the following two possible cases:

1u₁T u₂T, u1S u₂S, 2u1T< u2T, u1S u2S.

Assume that case1holds. Byu_it≤0 on0,1,it is easy to see thatu_iT0 i1,2 existfinite or∞, moreover, one of them must be finite. The same conclusion is also valid for u_iS−0 i1,2.It follows from3.2that

u2t−u₁t |_tT0≥0, 3.3

consequently

u₂T0≥u₁T0, u₁T0is finite. 3.4

Similarly

u₂S−0≤u₁S−0, u₁S−0is finite. 3.5

From3.1,3.4, and3.5we have lim inf

t−→T0yt≥0≥lim sup

t−→S−0yt. 3.6

On the other hand,3.2,1.7, and conditionHyield ft, u2t, u2t≤g

u2t u₁t

ft, u1t, u1t

≤ u2t

u₁tft, u1t, u1t, t∈T, S,

3.7

that is,

ft, u2t, u2t

u₂t ≤ ft, u1t, u1t

u₁t , t∈T, S, 3.8

therefore

u₁t

u1t ≤ u₂t

u2t, t∈T, S. 3.9

From this it follows that

yt u₁tu₂t−u₂tu₁t≥0, t∈T, S. 3.10 Ifyt ≡ 0 onT, S, then, by3.6 we haveyt ≡ 0,and then u2t/u1t ≡ 0, which imply that there exists a positive numbercsuch thatu2t cu1tonT, S.It follows from3.2thatc > 1,thereforeT 0, S1.Substitutingu₂t cu₁tinto1.1and using conditionH, we have

cft, u1t, u1t ft, cu1t, cu1t

≤gcft, u1t, u1t, t∈0,1. 3.11

Noticing3.11andft, u1t, u1t/≡0, t∈0,1,we have

c≤gc, 3.12

which contradicts with the condition thatgc< c.Therefore,yt≥0 andyt/≡0 onT, S.

Thus,yT0< yS−0, which contradicts with3.6. So case1is impossible.

By analogous methods, we can obtain a contradiction for case2. Sou1t≡u2tfor anyt∈0,1 ,which implies that the result ofTheorem 1.5holds.

4. Concerned Remarks and Applications

Remark 4.1. The typical function satisfyingHisft, u, u _n

i1aitu^λi _m

j1bjtu^−μj, wherea_i, b_j∈C0,1, 0< λ_i<1, μ_j>0, i1,2, . . . , n, j1,2, . . . , m.

Remark 4.2. ConditionHincludes e-concave functionsee6 as special case. For example, Liu and Yu7 consider the existence and uniqueness of positive solution to a class of singular boundary value problem under the following condition:

f t, λu,v

λ

≥λ^αft, u, v, ∀u, v >0, λ∈0,1, 4.1

whereα∈0,1andft, u, vis nondecreasing onu, nonincreasing onv.Clearly, condition His weaker than the above condition4.1.

In fact, for anyλ≥1,from4.1it follows that ft, λu, v≤f

t, λu,1

λv

≤λ^αft, u, v. 4.2

On the other hand, for any 0< λ <1,from4.1it follows that ft, u, v≥f

t, λu, λv λ

≥λ^αft, u, λv, 4.3

that is,ft, u, λv≤λ^−αft, u, v.

In what follows, by using the results obtained in this paper, we study the boundary value problem

ut μt^−γ1−t^−l

u^−αt u^βt A

0, 0< t <1, u0 ^m−2

i1

α_iu η_i

, u1 0,

4.4

whereμ >0, α >0, β <1, A≥0.We have the following theorem.

Theorem 4.3. A necessary and sufficient condition for problem4.4to have smooth positive solution is that

max

γα, lα, γ−β, l−β, γ, l

<1. 4.5

Moreover, when the positive solution exists, it is unique.

Remark 4.4. Consider1.1and the following singularm-point boundary value conditions:

u0 0, u1 ^m−2

i1

α_iu η_i

. 4.6

By analogous methods, we have the following results.

Assume thatutis aC0,1 positive solution to1.1and4.6, thenutcan be stated ut

₁

0

Gt, sfs, us, us t

1−_m−2

i1 α_iη_i

m−2

i1

α_{i 1}

0

G η_i, s

fs, us, usds, 4.7

whereGt, sis defined in2.4.

Theorem 4.5. Suppose thatHholds, then a necessary and sufficient condition for the problem1.1 and4.6to have smooth positive solution is that

0<

₁

0

fs, s, sds <∞. 4.8

Theorem 4.6. SupposeHand4.8hold, then the smooth positive solution to problem1.1and 4.6is also uniqueC0,1 positive solution.

Acknowledgment

Research supported by the National Natural Science Foundation of China 10871116, the Natural Science Foundation of Shandong ProvinceQ2008A03 and the Doctoral Program Foundation of Education Ministry of China200804460001.

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