Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 803137, 12 pages
doi:10.1155/2011/803137
Research Article
Limit Circle/Limit Point Criteria for
Second-Order Sublinear Differential Equations
with Damping Term
Jing Shao1 and Wei Song2
1
2
Department of Mathematics, Jining University, Jining 273155, China
The Thirteen Middle School of Jining, Jining 22100, China
Correspondence should be addressed to Jing Shao, shaojing99500@163.com
Received 6 September 2011; Accepted 16 October 2011
Academic Editor: Zhenya Yan
Copyright q 2011 J. Shao and W. Song. 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.
The purpose of the present paper is to establish some new criteria for the classification of the
sublinear differential equation as of the nonlinear limit circle type or of the nonlinear limit point
type. The criteria presented here generalize some known results in the literature.
1. Introduction
In 1910, Weyl 1 published his now classical paper on eigenvalue problems for second-order
linear differential equations of the form
aty rty θy,
θ ∈ C.
1.1
He classified this equation to be of the limit circle type if each solution yt is square integrable
denoted by yt ∈ L2 , that is,
∞
y2 tdt < ∞,
1.2
0
and to be of the limit point type if at least one solution yt does not belong to L2 , that is,
∞
0
y2 tdt ∞.
1.3
2
Abstract and Applied Analysis
Weyl showed that the linear equation 1.1 always has at least one square integrable
solution if Im θ /
0. Thus, for second-order linear equations with Im θ /
0, the problem
reduces to whether 1.1 has one limit point type or two limit circle type linearly
independent square integrable solutions. This is known as the Weyl Alternative. Weyl also
proved that if 1.1 is of the limit circle type for some θ0 ∈ C, then it is of the limit circle type
for all θ ∈ C. In particular, this is true for θ 0, that is, if we can show the following equation
aty rty 0,
1.4
is of limit circle type, then 1.1 is of the limit circle type for all values of θ. There has been
considerable interest in this problem over the years see 1–10 and references cited therein.
The analogous problem for nonlinear equations is relatively new and not as extensively
studied as the linear cases. For a survey of known results on the linear and nonlinear
problems as well as their relationships to other properties of solutions such as boundedness,
oscillation, and convergence to zero, we refer the reader to the recent monograph 10. In this
paper, we will discuss the equation with damping term
aty bty rtyγ 0,
1.5
where a, r : R → R and b : R → R are continuous, a , r ACloc R , a , r L2loc R ,
at > 0, rt > 0, and 0 < γ ≤ 1, γ odd/odd, say γ 2M − 1/2N − 1, M and N are
positive integers, we can write γ 2k − 1, where k M N − 1/2N − 1. When bt ≡ 0,
then 1.5 turns into the following equation
aty rtyγ 0,
1.6
which is widely researched by many authors see 10 and references cited therein.
Definition 1.1 see 2. A nontrivial solution yt of 1.5 is said to be of the nonlinear limit
circle type if
∞
yγ1 tdt < ∞,
1.7
0
and it is of the nonlinear limit point type otherwise, that is, there exists a nontrivial solution
yt satisfying
∞
yγ1 tdt ∞.
1.8
0
Equation 1.5 is said to be of the nonlinear limit circle type if all its solutions satisfy 1.7,
and it is said to be of the nonlinear limit point type if there is at least one solution satisfying
1.8.
In this paper, we will give sufficient and necessary conditions to guarantee the
nonlinear limit circle type or nonlinear limit point type for 1.5.
Abstract and Applied Analysis
3
2. Main Results
To simplify notations, let α 1/2k 1 and β 2k 1/2k 1. We define
s
t
0
r α u
du,
aβ u
xs yt.
2.1
Then 1.5 becomes
ẍ Atẋ Btxγ 0,
2.2
where
At btr α t αatrt
,
α
aβ t
a tr α1 t
Bt atrtβ−α .
2.3
Note that in the transformation 2.1, k is no longer an integer here. In fact, 1/2 < k ≤ 1.
We begin with a boundedness result.
Theorem 2.1. If the condition
∞
0
atrt−
dt < ∞
atrt
2.4
holds, then each solution of 1.5 is bounded.
Proof. We rewrite 1.5 as the system
y w,
w −a tw − rtyγ − btw
,
at
2.5
and we define
V t atw2 yγ1 t
,
2rt
γ 1
then we have
at 2
atw w w yγ tw
rt
2rt
a t 2 bt 2
at 2
w −
w −
w
rt
rt
2rt
V t −
a t 2 bt 2 a t 2 atr t 2
w −
w w −
w
rt
rt
2rt
2r 2 t
2.6
4
Abstract and Applied Analysis
−
a t atr t 2 bt 2
w
w −
2rt
rt
2r 2 t
−
atrt 2 bt 2
w
w −
rt
2r 2 t
≤−
atrt 2 atrt− 2
w
w
≤
2r 2 t
2r 2 t
≤
atrt− 2 atrt− yγ1 t
w atrt γ 1
2r 2 t
atrt−
V t.
atrt
2.7
Gronwall’ inequality and condition 2.4 imply that V t is bounded, so yt is bounded.
To prove our main limit-circle result, we write 2.2 as the system
ẋ z − Atx,
ż Ȧtx − Btxγ .
2.8
Theorem 2.2. Assume that condition 2.4 holds and
∞ atrt 2 btr α t/aβ t α
αatrt
2
−α
1/2
dt < ∞,
3/2 t
3/2 tr 5/2 t
k/2k1 2
a
a
tr
atrt
0
2.9
and condition
B t ≤ γ 1 AtBt,
2.10
is satisfied. If
∞
0
1
dt < ∞,
Bt
2.11
then 1.5 is of the nonlinear limit circle type, that is, each solution yt of 1.5 satisfies
∞
yγ1 tdt < ∞.
2.12
0
Proof. Define
V x, z, s xγ1
z2
Bt
2
γ 1
t
0
AξBξ −
Ḃξ r α ξ γ1
y ξdξ.
γ 1 aβ ξ
2.13
Abstract and Applied Analysis
5
By direct calculation, we have
xγ1
Ḃt γ1
γ
V̇ zż Ḃt
Btx ẋ AtBt −
y t
γ 1
γ 1
xγ1
Btxγ z − Atx
z Ȧtx − Btxγ Ḃt
γ 1
Ḃt γ1
y t Ȧtxz.
AtBt −
γ 1
2.14
Since condition 2.4 and Theorem 2.1 are satisfied, the solution of 1.5 yt xs is
bounded, that is, there exists a constant K1 ≥ 0 such that |xt| ≤ K1 . So
Ȧtxz Ȧt|x|1−γ/2
atrtβ−α/2
1−γ/2
≤ K1
× atrtβ−α/2 |x|1γ/2 |z|
Ȧt
atrtβ−α/2
Ȧt
1−γ/2
≤ K1
β−α/2
≤
1−γ/2
K1
atrt
Ȧt
atrtβ−α/2
×
atrtβ−α x1γ z2
2
×
1γ
2
Bx
z
1γ
2
2.15
× V t.
Now
dt
aβ t
A t α ,
ds
r t
btr α t/aβ t αatrt /aα tr α1 t
Ȧt A t
A t
atrtβ−α/2
atrtβ−α/2
αatrt
a1/2 tr 3/2 t
α2 a tatrt /atrt1α 1α αr tatrt /aα tr α2 t
−
atrtβ−α/2
btr α t/aβ t
atrtβ−α/2
6
Abstract and Applied Analysis
αatrt
r tatrt
α2 a tatrt
−
αα
−
1
a1/2 tr 3/2 t
a1/2 tr 5/2 t
atrt3/2
btr α t/aβ t
atrtβ−α/2
2
αatrt
α atrt a trt atr t α2 atrt
− 3/2
2
a1/2 tr 3/2 t
a3/2 tr 5/2 t
a tr 5/2 t
btr α t/aβ t
atrtβ−α/2
2 2
α2 atrt
btr α t/aβ t
αatrt
α atrt
−
2 a3/2 tr 5/2 t
a1/2 tr 3/2 t
a3/2 tr 5/2 t
atrtβ−α/2
atrt 2 btr α t/aβ t
α
αatrt
2
−α
.
2
a1/2 tr 3/2 t
a3/2 tr 5/2 t
atrtβ−α/2
2.16
Let τs denote the inverse function of st, we obtain that
s Ȧτv
0
B1/2 τv
dv s A τvaβ τv/r α τv
dv
B1/2 τv
t auru 2 bur α u/aβ u α
αauru
2
−α
1/2
du
2
a3/2 ur 5/2 u auruk/2k1 ur 3/2 u
0 a
2.17
0
is convergent by condition 2.9. Hence, integrating V̇ s, applying Gronwall’s inequality,
and using condition 2.9, we obtain that V s is bounded, so
Bt
yγ1
yγ1
atrtβ−α
≤ K2
γ 1
γ 1
2.18
for some constant K2 > 0. Condition 2.11 then implies that yt is of the nonlinear limit
circle type.
When at ≡ 1, the 1.5 becomes
y bty rtyγ 0.
2.19
In this case, At btr α tαr t/r α1 t, Bt r β−α t. We get the following corollary.
Abstract and Applied Analysis
7
Corollary 2.3. Assume condition 2.4 and
∞ rt 2
α
α
t
btr
αrt
− α2 5/2
3/2
dt < ∞,
k/2k1 2
r
r
t
t
rt
0
2.20
Ḃt ≤ γ 1 AtBt
2.21
and condition
is satisfied. If
∞
0
1
dt < ∞,
Bt
2.22
then 2.19 is of the nonlinear limit circle type, that is, each solution yt of 2.19 satisfies
∞
yγ1 tdt < ∞.
2.23
0
Example 2.4. Consider the following second-order nonlinear differential equation
1/4
5/2
y 1 t2
y1/3 0,
y 1 t2
1/4
2.24
5/2
here bt 1 t2 , rt 1 t2 , γ 1/3. We can easily verify that all the conditions
in Corollary 2.3 are fulfilled, so each solution of 2.24 is of the nonlinear limit circle type. We
note further that the type of 2.24 cannot be determined since bt /
0.
Next, we give a necessary condition for the sublinear 1.5 to be of the nonlinear limit
circle type.
Theorem 2.5. Suppose condition 2.4,
∞
0
∞
0
b2 t
dt < ∞,
atrt
atr t2
dt < ∞
r 3 t
2.25
hold. If yt is a nonlinear limit circle type solution of 1.5, then
∞
0
2
at y t
dt < ∞.
rt
2.26
8
Abstract and Applied Analysis
Proof. Let yt be a nonlinear limit circle type solution of 1.5, then yt is bounded by
Theorem 2.1. Multiplying 1.5 by yt/rt, noting that
aty
2
aty y − at y ,
2.27
and integrating by parts, we obtain
aty tyt at1 y t1 yt1 −
rt
rt1 t
auy uyur u
du
r 2 u
t1
2
t t
au y u
auy u yu
−
du −
du 0.
ru
ru
t1
t1
2.28
Using 1.5, we have
aty tyt at1 y t1 yt1 −
rt
rt1 t
auy uyur u
du
r 2 u
t1
2
t
t
au y u
γ1
du 0.
y u −
ru
t1
t1
t
t1
buy uyu
du
ru
2.29
Denote
Ht t
t1
2
au y u
du,
ru
2.30
by Schwartz inequality, the boundedness of yt, and condition 2.14, we obtain
t
t1
auy uyur u
du ≤ H 1/2 t
r 2 u
t
t1
aur u2 y2 u
du
r 3 u
1/2
< M1 H 1/2 t
2.31
for some constant M1 > 0. By Schwartz inequality, the boundedness of yt, and condition
2.25,
t
t1
buy uyu
du ≤
ru
t
t1
1/2
b2 uy2 u
du
auru
H 1/2 t ≤ M2 H 1/2 t,
2.32
for some constant M2 > 0.
If yt is not eventually monotonic, let {tj } → ∞ be an increasing sequence of zeros of
y t. Then, by 2.28, there exists some constant M3 > 0, such that
M1 M2 H 1/2 tj M3 ≥ H tj .
This implies Htj ≤ M4 < ∞ for all j and some M4 > 0, so 2.26 holds.
2.33
Abstract and Applied Analysis
9
If yt is eventually monotonic, then yty t ≤ 0 for all t ≥ t1 t1 ≥ 0 large enough.
Using 2.28, we can repeat the type of argument used above to obtain that 2.26 holds. This
completes the proof of Theorem 2.5.
The following theorem gives sufficient conditions to ensure that 1.5 being of the
nonlinear limit point type.
Theorem 2.6. Suppose condition 2.4, 2.9, 2.10, 2.25 hold. If
∞ 0
2
atrt
dt < ∞,
atr 3 t
2.34
∞
b2 tr 3α−β t
dt < ∞,
a3β−α t
0
∞ t AξBξ − Ȧξ/γ 1 r α ξ/aβ ξ
dξdt < ∞,
Bt
0
0
∞
0
1
du ∞,
Bu
2.35
2.36
then 1.5 is of the nonlinear limit point type.
Proof. As in the proof of Theorem 2.2, we define
V x, z, s xγ1
z2
Bt
2
γ 1
t
0
AξBξ −
Ȧξ r α ξ γ1
y ξdξ,
γ 1 aβ ξ
2.37
we differentiate it to obtain
Ḃt γ1
xγ1
γ
Btx ẋ AtBt −
V̇ zż Ḃt
y t
γ 1
γ 1
xγ1
Btxγ z − Atx
z Ȧtx − Btxγ Ḃt
γ 1
Ḃt γ1
AtBt −
y t Ȧtxz.
γ 1
2.38
10
Abstract and Applied Analysis
0.
Let yt xs be any nontrivial solution of 1.5 with yt1 xst1 xs1 /
Theorem 2.1 implies that yt xs is bounded, so |xs|1−γ/2 ≤ K1 for some constant
K1 > 0. Hence
V̇ s ≥ −Ȧt|xs|1−γ/2 |xs|1γ/2 |zs|
Ȧt
z2 Btxγ1 t
≥ −K1
×
2
2
atrtβ−α/2
Ȧt
z2 Btxγ1 t
×
≥ −K1
2
γ 1
atrtβ−α/2
Ȧt
≥ −K1
× V t.
atrtβ−α/2
2.39
Let Ht K1 |Ȧt|/B1/2 t, then we can write V̇ HtV ≥ 0. So
d
ds
V s exp
s
Hτξdξ
≥ 0.
2.40
s1
Integrating the above inequality, we obtain
V s exp
s
Hτξ ≥ V s1 .
2.41
s1
Condition 2.9 implies
s
Hτξdξ < ∞,
2.42
s1
and since V s1 > 0, we have
V s ≥ K2 > 0
for s ≥ s1 .
2.43
Rewriting V x, z, s in terms of y and t and dividing 2.42 by Bt, we obtain
2
y t at yγ1 t
bt
atrt
yty t
α
2rt
γ 1
r 2 t
atrtβ−α
2
b2 tr 3α−β t
btatrt 2
2 atrt
y t
α
α
2atr 3 t
2a3β−α t
atrt2β
α
t r ξ/aβ ξ yγ1 ξ
AξBξ − Ḃξ/ γ 1
K2
dξ ≥
.
Bt
Bt
0
2.44
Abstract and Applied Analysis
11
If yt is a limit circle type solution, then yt is bounded,
2.4 and 2.25, we get
∞
t1
yγ1 tdt < ∞, and by condition
∞ 2
y t at
dt < ∞,
2rt
t1
2.45
according to Theorem 2.5. By condition 2.34, we get
∞
t1
2
atrt
α
y2 tdt < ∞.
2atr 3 t
2.46
2
By the Schwartz inequality,
∞
t1
atrt
yty tdt ≤
α
r 2 t
∞
t1
1/2
2
atrt y2 t
dt
α
atr 3 t
2
1/2
∞ 2
y t at
dt
×
< ∞.
rt
t1
2.47
Noticing condition 2.35 and yt being bounded, we get
∞
t1
b2 tr 3α−β t 2
y tdt < ∞,
a3β−α t
2.48
so, by the Schwartz inequality, we have
∞
t1
btatrt
atrt2β
y tdt ≤
2
∞
t1
1/2
b2 tr 3α−β t 2
y tdt
a3β−α t
∞ 1/2
2
atrt
2
y tdt
×
< ∞,
atr 3 t
t1
∞
t1
bt
atrt
β−α
yty tdt ≤
∞
t1
b tr
ty t
dt
3β−α
a
t
2
3α−β
2
1/2
2.49
1/2
∞ 2
y t at
dt
×
< ∞,
rt
t1
Consequently, integrating both sides of 2.44, we see that the integrand of the left side of
2.44 is bounded, but the integrand of the right side of 2.44 tends to infinity according to
condition 2.36. This leads to a contradiction, so yt is a limit point case solution of 1.5,
and 1.5 is of the nonlinear limit point type.
12
Abstract and Applied Analysis
Similarly, when at ≡ 1, 1.5 becomes 2.19 and we get the following corollary.
Corollary 2.7. Suppose condition 2.4, 2.10, 2.20 hold. If
2
∞ ∞
rt
b2 t
dt < ∞,
dt < ∞,
b2 tr 3α−β tdt < ∞,
r 3 t
0 rt
0
0
∞
∞ t AξBξ − Ȧξ/ γ 1 r αξ
1
dξdt < ∞,
du ∞,
β−α u
Bt
r
0
0
0
∞
2.50
then 2.19 is of the nonlinear limit point type.
Acknowledgments
The authors thank the referee for his helpful suggestions on this paper which improved some
of our results. This research was partially supported by the NSF of China Grants 10801089
and 11171178.
References
1 H. Weyl, “Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen
Entwicklungen willkürlicher Funktionen,” Mathematische Annalen, vol. 68, no. 2, pp. 220–269, 1910.
2 M. Bartusek, Z. Dosla, and J. R. Graef, “On the definitions of the nonlinear limitpoint/limit-circle
properties,” Differential Equations and Dynamical Systems, vol. 9, pp. 49–61, 2001.
3 M. Bartušek and J. R. Graef, “Some limit-point and limit-circle results for second order Emden-Fowler
equations,” Applicable Analysis, vol. 83, no. 5, pp. 461–476, 2004.
4 J. R. Graef, “Limit circle type criteria for nonlinear differential equations,” Proceedings of the Japan
Academy, Series A, vol. 55, no. 2, pp. 49–52, 1979.
5 J. R. Graef, “Limit circle criteria and related properties for nonlinear equations,” Journal of Differential
Equations, vol. 35, no. 3, pp. 319–338, 1980.
6 J. R. Graef and P. W. Spikes, “On the nonlinear limit-point/limit-circle problem,” Nonlinear Analysis,
vol. 7, no. 8, pp. 851–871, 1983.
7 P. W. Spikes, “Criteria of limit circle type for nonlinear differential equations,” SIAM Journal on
Mathematical Analysis, vol. 10, no. 3, pp. 456–462, 1979.
8 M. V. Fedoryuk, Asymptotic Analysis, Springer, New York, NY, USA, 1993.
9 P. Hartman and A. Wintner, “Criteria of non-degeneracy for the wave equation,” American Journal of
Mathematics, vol. 70, pp. 295–308, 1948.
10 M. Bartušek, Z. Došlá, and J. R. Graef, The Nonlinear Limit-Point/Limit-Circle Problem, Birkhäuser,
Boston, Mass, USA, 2005.
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