Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 416173, 13 pages
doi:10.1155/2011/416173
Research Article
Results for Twin Singular Nonlinear Problems
with Damping Term
Youwei Zhang
Department of Mathematics, Hexi University, Gansu 734000, China
Correspondence should be addressed to Youwei Zhang, ywzhang0288@163.com
Received 6 December 2010; Accepted 7 March 2011
Academic Editor: Wolfgang Castell
Copyright q 2011 Youwei Zhang. 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.
By using the functional type cone expansion and compression fixed-point theorem in cones, some
new and general results on the existence of positive solution for twin singular boundary value
problems with damping term are obtained. An example is given to illustrate our results.
1. Introduction
The study of multipoint boundary value problem for linear second-order ordinary differential
equation was initiated by I1’in and Moiseev 1 motivated by the work of Bitsadze and
Samarskiı̆ 2–4 on nonlocal linear elliptic boundary value problem, which is a new area of
still fairly theoretical exploration in mathematics. Several authors have expounded on various
aspects of this theory; see the survey paper by Gupta et al. 5–7 and the references cited
therein. Thereamong, the study of singular boundary value problem for ordinary differential
equation has led to several important applications in applied mathematics and physical
science, such as the Thomas-Fermi problem
x t − t−1/2 x3/2 0,
0 < t < 1,
x0 0 x1,
1.1
which appears in determining the electrical potential in an atom. For other application
results, we refer to 8–10. With regards to this, increasing attention is paid to question of
singular boundary value problem and has obtained many excellent results of the existence
of the positive solution for two multiple points nonlinear singular boundary value problem
11–15. The main techniques are the upper and lower solutions method 16, the LeraySchauder continuation theory 17, and the fixed-point theory in cones 18 et al. We would
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like to mention some results of Ma and O’Regan 13 and Yao 15, which motivated us
to consider the singular boundary value problems. In 13, the authors have studied a
multipoint boundary value problem
x t f t, xt, x t et,
x 0 0,
x1 0 < t < 1,
1.2
m−2
ai xξi ,
i1
where ξi ∈0, 1, ai ∈ R, i 1, 2, . . . , m − 2, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, f : 0, 1 × R2 → R is a
function satisfying Carathéodory’s conditions and 1 − tet ∈ L1 0, 1; the Leray-Schauder
continuation theorem leads to the existence of single C1 0, 1 solution. The literature 15 has
discussed a second-order boundary value problem
x t htfxt 0,
ax0 − bx 0 0,
0 < t < 1,
ax1 bx 1 0,
1.3
where ht is symmetric on 0, 1 and may be singular at both end points t 0 and t 1. The author has proved the existence of n symmetric positive solutions and established a
corresponding iterative scheme, the main tool being the monotone iterative technique.
A powerful tool for proving existence of solution to boundary value problem is the
fixed-point theory. In many cases, it is possible to find single, double, or multiple solutions
for boundary value problem, and for the same problem, using the various methods, one can
obtain different results under some appropriate conditions. To my best knowledge, very little
work has been done on the existence of positive solution for boundary value problem by
using the functional type cone expansion and compression fixed-point theorem. The aim of
this paper is to establish some new and general results on the existence of positive solution
to singular boundary value problems with damping term
u t − λu t htft, ut 0,
a < t < b,
n
ai uti ,
γub δu b u a − λua 0,
γua δu a n
1.4
1.5
i1
ai uti ,
u b − λub 0,
1.6
i1
where λ, γ, δ, ai , ti , i 1, 2, . . . , n, h, and f satisfy
H1 −∞ < a < t1 < · · · < tn < b < ∞, λ, γ, δ ∈0, ∞, ai ∈0, ∞, i 1, 2, . . . , n,
H2 d1 γλδ expλb− ni1 ai expλti > 0, d2 ni1 ai expλti −γλδ expλa > 0,
H3 h :a, b → 0, ∞ is continuous function, ht may be singular at t a and/or t b,
b
and 0 < a hsds < ∞,
H4 f : a, b × 0, ∞ → 0, ∞ satisfies the Carathéodory condition, that is, for each
x ∈ 0, ∞, the mapping t → ft, x is Lebesgue measurable on a, b and for a.e.
t ∈ a, b, the mapping x → ft, x is continuous on 0, ∞, and for each r > 0, there
exists φr ∈ L1 a, b such that ft, x ≤ φr t for all u ∈ 0, r and for a.e. t ∈ a, b.
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3
We will impose some advisable conditions on the nonlinearity f to ensure the existence
of at least one positive solution for the above problems. In order to obtain our results, we
construct special operator which is the base for further discussion and provide two crucial
functionals on cones. Applying the functional type cone expansion and compression fixedpoint theorem to the operator and functionals, we obtain some new and general results on
the existence of at least one positive solution for the twin singular problems 1.4, 1.5 and
1.4, 1.6. Our results improve and generalize those in 15, 19.
Let α and β be nonnegative continuous functionals on a cone P in real Banach space
B. For positive numbers r and L, we define the sets
Pα, r {x ∈ P : r < αx},
P β, L x ∈ P : βx < L ,
1.7
P β, α, r, L x ∈ P : r < αx, βx < L .
We state the functional type cone expansion and compression fixed-point theorem 20.
Lemma 1.1. Let P be a cone in a real Banach space B, and let α and β be nonnegative continuous
functionals on P. Let Pβ, α, r, L be a nonempty bounded subset of P,
T : P β, α, r, L −→ P
1.8
is a completely continuous operator with
Pα, r ⊆ P β, L ,
inf
Tx > 0,
x∈∂Pβ,α,r,L
1.9
for Pβ, L. If αTx ≥ r for all x ∈ ∂Pα, r, βTx ≤ L for each x ∈ ∂Pβ, L, and for each
y ∈ ∂Pα, r, z ∈ ∂Pβ, L, θ ∈0, 1, and μ ∈ 1, ∞, the functionals satisfy the properties
α θy ≤ θα y ,
β μz ≥ μβz,
β0 0,
1.10
then T has at least one positive fixed-point x such that
r ≤ αx,
βx ≤ L.
1.11
2. Main Results
Let B be the Banach space C1 a, b with the norm u max{u0 , u 1 }, where u0 supt∈a,b |ut| and u 1 supt∈a,b |u t|, and let a cone in B
P1 v ∈ B : v is nondecreasing on a, b, va 0 .
2.1
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For v ∈ P1 , define the operator T1 by
T1 vt t
hsf
s, expλs c1 v b
a
vω exp−λωdω
ds,
t ∈ a, b,
2.2
s
where c1 v 1/d1 δvb n
i1
ai expλti b
ti
vω exp−λωdω.
Lemma 2.1. If v ∈ P1 is a fixed-point T1 , then
ut : expλt c1 v b
2.3
vω exp−λωdω
t
is one solution of the problem 1.4, 1.5.
Proof. Suppose that v ∈ P1 is a fixed-point T1 and ut expλtc1 v
thus we have
u t − λut −vt −T1 vt −
t
hsf s, expλs c1v a
b
t
vω exp−λωdω,
b
vωexp−λωdω
ds.
s
2.4
Further,
u t − λu t −htf
t, expλt c1 v tn
vω exp−λωdω
−htft, ut.
t
2.5
The boundary condition 1.5 is satisfied due to the relation between u, v, and c1 v.
For v ∈ P1 , we define the nonnegative continuous functionals α and β on P1 by
αv b
vω exp−λωdω,
βv vt1 .
2.6
t1
Lemma 2.2. Let r1 > 0. If v ∈ ∂P1 α, r1 , then
vt1 ≤
b
a
expλb
r1 ,
b − t1
vω exp−λωdω ≥ r1 .
2.7
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5
Proof. For v ∈ ∂P1 α, r1 , that is, αv r1 . Since vt nondecreasing on a, t1 , we have
r1 αv b
vω exp−λωdω ≤
b
t1
vb exp−λωdω ≤ vb exp−λt1 b − t1 ,
2.8
t1
so we get 2.7. Furthermore,
b
vω exp−λωdω a
t1
vω exp−λωdω b
vω exp−λωdω ≥ r1 .
2.9
t1
a
Lemma 2.3. Let L1 > 0. If v ∈ ∂P1 β, L1 , then
b
vω exp−λωdω ≥ L1 exp−λbb − t1 .
2.10
t1
Proof. For v ∈ ∂P1 β, L1 , that is to say, βv L1 . In view of vt is nondecreasing on a, b,
for ω ∈ t1 , b, we have
vω ≥ vt1 βv L1 .
2.11
Hence,
b
vω exp−λωdω ≥ L1
t1
b
exp−λωdω ≥ L1 exp−λbb − t1 .
2.12
t1
Lemma 2.4. Let (H1 )–(H4 ) hold, then T1 : P1 → P1 is completely continuous.
Proof. By H1 –H4 , we observe that T1 v ∈ C1 a, b, T1 vt is nondecreasing on a, b, and
T1 va 0, so T1 : P1 → P1 . Since ht may be singular at t a and/or t b, we take the
arguments to show that T1 is completely continuous.
Assume that vn , v0 ∈ P1 . In view of f satisfying the Carathéodory condition, it is easy
to see that
vn − v0 0 −→ 0 implies that
b
supf s, expλs c1 vn vn ω exp−λωdω
s
s∈Ω b
−f s, expλs c1 v0 v0 ω exp−λωdω
ds −→ 0
s
2.13
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as n → ∞, where Ω a, b or a, b. Thus, we have
T1 vn − T1 v0 0 sup |T1 vn t − T1 v0 t|
t∈a,b
b
≤
hsf s, expλs c1 vn vn ω exp−λωdω
a
s
b
−f
s, expλs c1 v0 b
s
v0 ω exp−λωdω
ds,
T1 vn − T1 v0 sup T1 vn t − T1 v0 t
1
t∈a,b
b
htf t, expλt c1 vn vn ω exp−λωdω
s
≤ sup
t∈a,b
−f
t, expλt c1 v0 b
s
v0 ω exp−λωdω
.
2.14
Therefore,
T1 vn − T1 v0 −→ 0 n −→ ∞.
2.15
This means that the operator T1 : P1 → P1 is continuous.
∞
Choose two sequences {ϕn }∞
n1 , {ψn }n1 ⊂a, b satisfying ϕn ≤ ψn for any n ≥ 1, such
that ϕn → a and ψn → b as n → ∞, respectively. Define
⎧
⎪
inf ht,
⎪
⎪
a≤t≤ϕn
⎪
⎪
⎪
⎨
ϕn < t < ψn
hn t ht,
⎪
⎪
⎪
⎪
⎪
⎪
⎩ inf ht,
2.16
ψn ≤t≤b
and an operator sequence {T1n }∞
n1 by
T1n v t
a
hn sf
s, expλs c1 v b
vω exp−λωdω
ds.
2.17
s
Clearly, hn : a, b → 0, ∞ is a piecewise continuous function, and the operator T1n : P1 →
P1 is well defined. Further, we can see that T1n : P1 → P1 is completely continuous.
Let R > 0, BR : {v ∈ P1 : v0 ≤ R}, and MR sup{ft, u : t, u ∈ a, b × 0, R},
where R R expλb/d1 δ 1/λ ni1 ai 1 − exp−λb − ti R/λexp λb − a − 1 > 0.
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We will prove that T1n approach T1 uniformly on BR . From the absolute continuity of integral,
we obtain
hsds 0,
lim
n→∞
ln
2.18
where ln a, ϕn ∪ ψn , b. For each v ∈ BR , t ∈ a, ϕn , we have
b
t
ds
T1n v − T1 v0 sup hn s − hsf s, expλs c1 v vω exp−λωdω
s
t∈a,ϕn a
≤ MR
ϕn
|hn s − hs|ds −→ 0
n −→ ∞.
a
2.19
For each v ∈ BR , t ∈ ψn , b, we have
b
t
ds
T1n v − T1 v0 sup hn s − hsf s, expλs c1 v vω exp−λωdω
s
t∈ψn ,b a
≤ MR
b
|hn s − hs|ds −→ 0
n −→ ∞.
a
2.20
It is easy to see that, for each v ∈ BR and ϕn < t < ψn , there is T1n v − T1 v0 → 0 as n → ∞.
Similarly, for any v ∈ BR , and t ∈ a, ϕn , ϕn , ψn , ψn , b, respectively, we can obtain that
b
T1n v − T1 v suphn t − htf t, expλt c1 v vω exp−λωdω
1
t t
≤ MR |hn t − ht| −→ 0 n −→ ∞.
2.21
From the above argument, we obtain
T1n v − T1 v max T1n v − T1 v0 , T1n v − T1 v 1 −→ 0 n −→ ∞.
2.22
That is to say, the sequence T1n is uniformly approximate T1 on any bounded subset of P1 .
Therefore, T1 : P1 → P1 is completely continuous.
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For convenience, we set
m1 t
b
exp−λt
hsds dt,
t1
M1 a
δ expλa
1 ,
u1 r1 expλa
d1 b − a
b
hsds,
a
u2 L1 expλt1 δ Σni1 ai b − ti expλti − b
.
d1
2.23
We are now ready to apply a functional type cone expansion and compression fixedpoint theorem to the operator T1 to give the sufficient conditions for the existence of at least
one positive solution to the problem 1.4, 1.5.
Theorem 2.5. Suppose that (H1 )–(H4 ) hold. Assume that there exist positive numbers k1 , r1 , and L1
with expλb/b − t1 r1 < L1 such that
A1 ft, w ≥ k1 , t, w ∈ a, b × r1 , ∞,
A2 ft, w ≥ r1 /m1 , t, w ∈ a, t1 × u1 , ∞,
A3 ft, w ≤ L1 /M1 , t, w ∈ t1 , b × u2 , ∞,
then the operator T1 has at least one fixed-point v such that r1 ≤ αv and βv ≤ L1 , and the problem
1.4, 1.5 has at least one positive solution u such that
ut expλt c1 v b
2.24
vω exp−λωdω .
t
Proof. The cone P1 and operator T1 are defined by 2.1 and 2.2, respectively. By the
properties of operator T1 , it suffices to show that the conditions of Lemma 1.1 hold with
respect to T1 . In view of Lemma 2.1, it is not difficult to prove that a fixed point of T1 is
coincident with the solution of the boundary value problem 1.4, 1.5, so we concentrate on
the existence of the fixed point of the operator T1 . Set P1 β, α, r1 , L1 is a nonempty bounded
subset of P1 . From Lemma 2.4, it can be shown that
T1 : P1 β, α, r1 , L1 −→ P1
2.25
is completely continuous by the Arzela-Ascoli lemma. For v ∈ ∂P1 β, α, r1 , L1 , the
assumption A1 implies that
T1 v0 b
hsf
s, expλs c1 v a
b
vω exp−λωdω
ds ≥ k1
s
T1 v sup htf
1
t∈a,b
t, expλt c1 v b
hsds,
a
vω exp−λωdω
t
2.26
b
≥ k1 sup ht.
t∈a,b
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the hypotheses of H3 lead to
T1 v ≥ k1 min
inf
sup ht,
v∈∂P1 β,α,r,L
b
hsds
2.27
> 0.
a
t∈a,b
If v ∈ P1 α, r1 , then βv vt1 ≤ expλb/b − t1 r1 < L1 , and so v ∈ P1 β, L1 , that
is, P1 α, r1 ⊆ P1 β, L1 . It follows that the conditions of Lemma 1.1 hold with respect to
T1 . By the definition of functionals α and β, we can check that the functionals satisfy the
b
b
properties αθy t1 θyω exp−λωdω θ t1 yω exp−λωdω θαy for y ∈ ∂P1 α, r1 and θ ∈0, 1, βμz μzt1 μβz for z ∈ ∂P1 β, L1 and μ ∈ 1, ∞, β0 0.
We now prove that αT1 v ≥ r1 , in Lemma 1.1, holds. In fact, if v ∈ ∂P1 α, r1 , by the
properties of c1 v and Lemma 2.2, for each t ∈ a, t1 ,
ut expλt c1 v b
vω exp−λωdω
t
2.28
≥ expλac1 v r1 ≥ r1 expλa
δ expλa
1 .
d1 b − a
Hence, by the assumption A2 and 2.28, there is
αT1 v b
T1 vt exp−λtdt
t1
t
b
exp−λt
t1
r1
≥
m1
hsf
s, expλs c1 v a
exp−λt
t1
vω exp−λωdω
ds dt
s
t
b
b
hsds dt r1 .
a
2.29
Finally, we assert that βT1 v ≤ L1 , in Lemma 1.1, also holds. If v ∈ ∂P1 β, L1 , by Lemma 2.3,
for t ∈ t1 , b,
ut expλt c1 v b
vω exp−λωdω
t
≥ vt1 expλt1 L1 expλt1 δ Σni1 ai b − ti expλti − b
d1
δ Σni1 ai b − ti expλti − b
.
d1
2.30
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International Journal of Mathematics and Mathematical Sciences
The assumption A3 and 2.30 imply that
βT1 v T1 vb
b
s, expλs c1 v hsf
a
L1
≤
M1
b
vω exp−λωdω
ds
s
b
2.31
hsds L1 .
a
To sum up, the hypotheses of Lemma 1.1 are satisfied. Hence, the operator T1 has at
least one fixed point, that is, the problem 1.4, 1.5 has at least one positive solution.
Let the cone
P2 v ∈ B : v is nonincreasing on a, b, vb 0 .
2.32
Evidently, P2 ⊂ B. For v ∈ P2 , define the operator T2 by
T2 vt b
hsf s, expλs c2 v s
t
vω exp−λωdω
ds,
t ∈ a, b,
2.33
a
t
where c2 v 1/d2 δva − ni1 ai expλti ai vω exp−λωdω.
We only give the preliminary lemmas and result of the problem 1.4, 1.6, the proofs
are similar to the above argument.
Lemma 2.6. If v ∈ P2 is a fixed-point T2 , then
ut expλt c2 v t
vω exp−λωdω
2.34
a
is one solution of the problem 1.4, 1.6.
For v ∈ P2 , the nonnegative continuous functionals α and β on P2 are defined by
αv tn
vω exp−λωdω,
βv vtn .
2.35
a
Lemma 2.7. Let r2 > 0. If v ∈ ∂P2 α, r2 , then
expλtn r2 ,
vtn ≤
tn − a
b
a
vω exp−λωdω ≥ r2 .
2.36
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11
Lemma 2.8. Let L2 > 0. If v ∈ ∂P2 β, L2 , then
tn
vω exp−λωdω ≥ L2 exp−λtn tn − a.
2.37
a
Lemma 2.9. Let (H1 )–(H4 ) hold, then T2 : P2 → P2 is completely continuous.
For convenience, we set
m2 b
tn
exp−λt
M2 hsds dt > 0,
t
a
u3 r2 expλb,
b
hsds > 0,
a
2.38
δ − Σni1 ai ti − a expλti − a
u4 L2 expλa
.
d2
Theorem 2.10. Suppose that (H1 )–(H4 ) hold. Assume that δ − Σni1 ai expλti − ati − a > 0, then
there exist positive numbers k2 , r2 , and L2 with expλtn /tn − ar2 < L2 such that
B1 ft, w ≥ k2 , t, w ∈ a, b × r2 , ∞,
B2 ft, w ≥ r2 /m2 , t, w ∈ tn , b × u3 , ∞,
B3 ft, w ≤ L2 /M2 , t, w ∈ a, tn × u4 , ∞.
Then the operator T2 has at least one fixed-point v such that r2 ≤ αv and βv ≤ L2 , and the
problem 1.4, 1.6 has at least one positive solution u such that
⎛
⎞
t
ut expλt⎝c2 v vω exp−λωdω⎠.
2.39
a
3. Examples
Consider the problems
u t − u t htft, ut 0,
u 0 − u0 0,
0 < t < 1,
3.1
1
,
2
3.2
u 1 − u1 0.
3.3
u1 2u 1 2u
1
,
2
u0 2u 0 2u
Let
1
,
ht t1 − t
⎧
2
⎪
⎪
,
⎨10−3 t √
w2
ft, w ⎪
⎪
⎩10−3 t 2 − √2,
w < 2,
w ≥ 2.
3.4
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International Journal of Mathematics and Mathematical Sciences
It is easy to check that hypotheses H1 –H4 hold. For the problem 3.1, 3.2, by some
calculations, we have d1 ≈ 4.858, m1 ≈ 0.348, and M1 ≈ 3.142. Taking k1 0.001, r1 0.2, and
L1 2, satisfying the following conditions: ft, w ≥ 0.001, t, w ∈ 0, 1 × 0.2, ∞, ft, w ≥
r1 /m1 ≈ 0.575, t, w ∈ 0, 1/2 × 0.141, ∞, ft, w ≤ L1 /M1 ≈ 0.637, and t, w ∈ 1/2, 1 ×
2.242, ∞. Thus, the hypotheses of Lemma 1.1 are fulfilled, and so the operator T1 has at
least one fixed point, that is to say, the problem 3.1, 3.2 has at least one positive solution.
For the problem 3.1, 3.3, by some calculations, we obtain d2 ≈ 0.297, m2 ≈ 0.348, and
M2 ≈ 3.142. Taking k2 0.001, r2 0.1, and L2 2, combining with the following conditions:
ft, w ≥ 0.001, t, w ∈ 0, 1 × 0.1, ∞, ft, w ≥ r2 /m2 ≈ 0.287, t, w ∈ 1/2, 1 × 0.272, ∞,
ft, w ≤ L2 /M2 ≈ 0.637, and t, w ∈ 0, 1/2 × 2.364, ∞. So the problem 3.1, 3.3 has at
least one positive solution.
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