设为首页 加入收藏 期刊导航 网站地图
  • 首页
  • 期刊
    • 数学与物理
    • 地球与环境
    • 信息通讯
    • 经济与管理
    • 生命科学
    • 工程技术
    • 医药卫生
    • 人文社科
    • 化学与材料
  • 会议
  • 合作
  • 新闻
  • 我们
  • 招聘
  • 千人智库
  • 我要投稿
  • 办刊

期刊菜单

  • ●领域
  • ●编委
  • ●投稿须知
  • ●最新文章
  • ●检索
  • ●投稿

文章导航

  • ●Abstract
  • ●Full-Text PDF
  • ●Full-Text HTML
  • ●Full-Text ePUB
  • ●Linked References
  • ●How to Cite this Article
PureMathematicsnØêÆ,2021,11(5),720-730
PublishedOnlineMay2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.115087
˜š‚5~‡©•§>Š¯K)•35
ÉÉÉeeeœœœ
Ü“‰ŒÆ§êƆÚOÆ§[‹=²
ÂvFϵ2021c46F¶¹^Fϵ2021c56F¶uÙFϵ2021c513F
Á‡
©•˜š‚5~‡©>Š¯K





u
0
(t)+a(t)u(t) = λf(t,u(t)),0 ≤t≤1,
u(0) =lu(1)
)•35,Ù¥f: [0,1]×[0,∞) →[0,∞),a: [0,1] →[0,∞) þ•ëY¼ê,…
R
1
0
a(θ)dθ>0,
λ•ëê, l•~ê…0<l<e
R
1
0
a(θ)dθ
. 3š‚5‘f÷v‡‚5, g‚5Ú ìC‚5^‡e,
©$^ØÄ:•ênؼT¯K)•35"
'…c
)§õ)§ØÄ:§I
ExistenceofPositiveSolutionsfor
NonlinearFirst-OrderOrdinary
BoudaryValueProblems
RuofeiWu
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
©ÙÚ^:Éeœ.˜š‚5~‡©•§>Š¯K)•35[J].nØêÆ,2021,11(5):720-730.
DOI:10.12677/pm.2021.115087
Éeœ
Received:Apr.6
th
,2021;accepted:May6
th
,2021;published:May13
th
,2021
Abstract
Inthispaper, weconsidertheexistenceofpositivesolutionsforthenonlinearfirst-
orderordinaryboundaryvalueproblems





u
0
(t)+a(t)u(t) = λf(t,u(t)),0 <t<1,
u(0) =lu(1)
wheref: [0,1]×[0,∞) →[0,∞),a: [0,1] →[0,∞)arecontinuous functionsand
R
1
0
a(θ)dθ>
0,λisapositiveparameter,lisaconstant,and0<l≤e
R
1
0
a(θ)dθ
.Undertheassump-
tionthatthenonlineartermfsatisfiessuperlinear, sublinearandasymptoticgrowth
conditon,theexistenceofpositivesolutionsoftheproblemisobtainedbyusingthe
fixed-pointindextheory.
Keywords
PositiveSolutions,MultipleSolutions,Fixed-Point,Cone
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
Cc5, ˜±Ï>Š¯K3²L7K+•, )Ô«+êþ(©Û, ;¾››†“£Ã
õ•¡ÑkXš~2•A^. Ïd, Ù)•35ÚåNõISÆö'5, 8c®²˜
¤J[1–12].~X, 2004c,Peng[2]ïÄ˜±Ï>Š¯K



x
0
(t) = f(t,x(t)),0 <t<ω,
x(0) = x(ω)
(1.1)
)•35, Ù¥f:[0,ω] ×R→R´ëY¼ê. T©$^I.†Ø ØÄ:½n¼Xe
(J:
DOI:10.12677/pm.2021.115087721nØêÆ
Éeœ
½nAe
(A1)•3˜‡êM>0,¦Mx−f(t,x) ≥0,x≥0,t∈[0,ω],
(A2)f
∞
>0,f
0
<0
¤á,K¯K(1.1) –•3˜‡).
2016c,Wang[3]<$^ÿÀÝnØïÄ˜±Ï>Š¯K



x
0
(t)+a(t)x(t) = f(t,x),0 <t<T,
x(0) = x(T)
(1.2)
)•35,Ù¥a: [0,T] →R´ëY¼ê,f: [0,T]×R→R´ëY¼ê,T©äN(JXe
½nBe•3ü‡ê0 <a<b¦
(B1)f(t,b<0),t∈[0,T],
(B2)f(t,x) >0,(t,x) ∈[0,T]×[0,a],
(B3)f(t,x) ≥−κx,(t,x) ∈[0,T]×[0,b].
¤á,K¯K(1.2) –•3˜‡)x
∗
,a≤kx
∗
k≤b.
Š5¿´, 3¯K(1.2) ¥,a(t)≡0,t∈[0,1] ž, T¯Kòz•(1.1). d, ±þü‡¯
KÑ´>.^‡Xê•1œ/. ¤±,˜‡g,¯K´: 镘„/ª˜>Š¯K



u
0
(t)+a(t)u(t) = λf(t,u(t)),0 ≤t≤1,
u(0) = lu(1)
(1.3)
´Ä•3). e•3), ëêλI‡÷vŸo^‡? eÃ), ëêλqI‡÷vŸo^‡? Ä
u±þóŠ,©ò$^ØÄ:•ênØïĘš‚5~‡©•§>Š¯K(1.3) )•35.
©ob½:
(H1)f: [0,1]×[0,∞) →[0,∞)•ëY¼ê;
(H2)a: [0,1] →[0,∞)•ëY¼ê, …
R
1
0
a(θ)dθ>0;
(H3)0 <l<e
R
1
0
a(θ)dθ
•~ê.λ>0 •ëê;
P
f
0
=lim
s→0
f(t,s)
s
,f
∞
=lim
s→∞
f(t,s)
s
.
©Ì‡(JXe:
½n1.1b(H1)-(H3)¤á,
DOI:10.12677/pm.2021.115087722nØêÆ
Éeœ
(a)ef
0
= 0½f
∞
= 0,K•3˜‡λ
0
>0,λ>λ
0
ž,¯K(1.1) –•3˜‡).
(b)ef
0
= ∞½f
∞
= ∞,K•3˜‡λ
0
>0,0 <λ<λ
0
ž,¯K(1.1) –•3˜‡).
(c)ef
0
= 0…f
∞
= 0,K•3˜‡λ
0
>0,λ>λ
0
ž,¯K(1.1) –•3ü‡).
(d)ef
0
= ∞…f
∞
= ∞,K•3˜‡λ
0
>0,0 <λ<λ
0
ž,¯K(1.1) –•3ü‡).
(e)ef
0
<∞…f
∞
<∞, K•3˜‡λ
0
>0,0 <λ<λ
0
ž,¯K(1.1) Ã).
(f)ef
0
>0 …f
∞
>0, K•3˜‡λ
0
>0,λ>λ
0
ž,¯K(1.1) Ã).
51~êlŠ‰Œ´•yŽf5.
©{eSNSüXe: 312 Ü©, ·‚‰Ñƒ'Ún9Ùy²; 313 Ü©, ‰Ñ̇(J
y²9~f.
2.ý•£
Ún2.1[4]X´Banach˜m,K⊂X´˜‡I.éup>0,½ÂK
p
= {u∈K: kuk= p},
bT: K
p
→K´˜‡;Žf,u∈∂K
p
= {u∈K: kxk= p}ž,Tx6= x.
(i)kTuk≥kuk, u∈∂K
p
,K
i(T,K
p
,K) = 0,
(ii)kTuk≤kuk,u∈∂K
p
,K
i(T,K
p
,K) = 1.
Ún2.2b(H1)-(H3) ¤á,>Š¯K



u
0
(t)+a(t)u(t) = λh(t),0 ≤t≤1,
u(0) =lu(1)
(1.4)
duÈ©•§
u(t) = λ
Z
t
0
h(s)e
R
s
t
a(θ)dθ
ds+λ
l
e
R
1
0
a(θ)dθ
−l
Z
1
0
h(s)e
R
s
t
a(θ)dθ
ds.
y²d(1.4) ª
e
R
t
0
a(θ)dθ
(u
0
(t)+a(t)u(t)) = λe
R
t
0
a(θ)dθ
h(t),
(e
R
t
0
a(θ)dθ
u(t))|
t
0
= λ
Z
t
0
e
R
s
0
a(θ)dθ
h(s)ds,
Kk
e
R
t
0
a(θ)dθ
u(t)−u(0) = λ
Z
t
0
e
R
s
0
a(θ)dθ
h(s)ds,
DOI:10.12677/pm.2021.115087723nØêÆ
Éeœ
‘\>.^‡Œ
u(1) =
λ
R
1
0
e
R
s
0
a(θ)dθ
h(s)ds
e
R
1
0
a(θ)dθ
−l
,u(0) =
λl
R
1
0
e
R
s
0
a(θ)dθ
h(s)ds
e
R
1
0
a(θ)dθ
−l
,
¤±
u(t) = λ
Z
t
0
h(s)e
R
s
t
a(θ)dθ
ds+λ
l
e
R
1
0
a(θ)dθ
−l
Z
1
0
h(s)e
R
s
t
a(θ)dθ
ds.
X= C[0,1],Ù3‰êkuk=max
t∈[0,1]
|u(t)|e¤Banach˜m. 
K= {u∈C[0,1] :u(t) ≥0,min
06t61
u(t) >σkuk}
´X¥I, Ù¥σ= le
−
R
1
0
a(θ)dθ
.½ÂŽfT
λ
:K→X
T
λ
u(t) = λ
Z
t
0
h(s)e
R
s
t
a(θ)dθ
ds+λ
l
e
R
1
0
a(θ)dθ
−l
Z
1
0
h(s)e
R
s
t
a(θ)dθ
ds.(1.5)
´„
λl
e
R
1
0
a(θ)dθ−l
Z
1
0
e
R
s
0
a(θ)dθ
h(s)ds≤T
λ
u(t) ≤
λe
R
1
0
a(θ)dθ
e
R
1
0
a(θ)dθ
−l
Z
1
0
h(s)e
R
s
0
a(θ)dθ
ds.
Ún2.3b½(H1)-(H3) ¤á,KT
λ
(K) ⊂K,…T
λ
: K→K´˜‡;Žf.
y²bu∈K, KT
λ
u(t) >0, t∈[0,1], …
min
06t61
T
λ
u(t) >
λl
e
R
1
0
a(θ)dθ−l
Z
1
0
h(s)e
R
s
0
a(θ)dθ
ds≥σkTu(t)k,
=T
λ
(K) ⊂K.w,T
λ
: K→K´˜‡;Žf,K¯K(1.1))duŽf•§T
λ
u= uØÄ:.
3.̇(Jy²
½n1.1y²
(a).bH(1)-H(3) ¤á,p
1
>0, u∈∂K
p
1
ž,Òk
kT
λ
uk≥
ˆm
p
1
λl
e
R
1
0
a(θ)dθ
−l
Z
1
0
e
R
s
0
a(θ)dθ
ds.
Ù¥ˆm
p
1
=min
σp
1
≤u≤p
1
{f(t,u)},
λ
0
≥
p
1
(e
R
1
0
a(θ)dθ
−l)
ˆm
p
1
l
R
1
0
e
R
s
0
a(θ)dθ
ds
,
λ>λ
0
ž,
kT
λ
uk≥kuk.
DOI:10.12677/pm.2021.115087724nØêÆ
Éeœ
dÚn2.1Œ
i(T
λ
,K
p
1
,K) = 0.
ef
0
= 0,K•3˜‡0 <r
1
<σp
1
,0 ≤u≤r
1
ž,f(t,u) ≤ε
1
u,Ù¥
ε
1
λe
R
1
0
a(θ)dθ
R
1
0
e
R
s
0
a(θ)dθ
ds
e
R
1
0
a(θ)dθ
−l
≤1,
u∈∂K
r
1
ž,Kk
kT
λ
uk≤
λe
R
1
0
a(θ)dθ
e
R
1
0
a(θ)dθ
−l
Z
1
0
h(s)e
R
s
0
a(θ)dθ
ds≤kuk.
dÚn2.1Œ
i(T
λ
,K
r
1
,K) = 1,
¤±
i(T
λ
,K
p
1
/
˚
K
r
1
,K) = −1.
ef
∞
= 0,K•3˜‡R
1
>p
1
,u≥R
1
ž,f(t,u) ≤ε
1
u,u∈∂K
R
1
,Kk
kT
λ
uk≤
λe
R
1
0
a(θ)dθ
e
R
1
0
a(θ)dθ
−l
Z
1
0
h(s)e
R
s
0
a(θ)dθ
ds≤kuk.
Ù¥
R
1
λe
R
1
0
a(θ)dθ
R
1
0
e
R
s
0
a(θ)dθ
ds
e
R
1
0
a(θ)dθ
−l
≤1,
dÚn2.1Œ
i(T
λ
,K
R
1
,K) = 0,
¤±
i(T
λ
,K
R
1
/
˚
K
p
1
,K) = −1,
KT
λ
3K
R
1
/
˚
K
p
1
½K
p
1
/
˚
K
r
1
pk˜‡ØÄ:,=T
λ
u= u. ¯K(1.1) –•3˜‡).
(b).b(H1)-(H3) ¤á,p
2
>0, u∈∂K
p
2
ž,Òk
kT
λ
uk≤
λe
R
1
0
a(θ)dθ
ˆ
M
p
2
e
R
1
0
a(θ)dθ
−l
Z
1
0
e
R
s
0
a(θ)dθ
ds,
Ù¥
ˆ
M
p
2
= 1+max
σp
2
≤u≤p
2
{f(t,u)}.-
0 <λ
0
≤
p
2
(e
R
1
0
a(θ)dθ
−l)
e
R
1
0
a(θ)dθ
ˆ
M
p
2
R
1
0
e
R
s
0
a(θ)dθ
ds
,
λ<λ
0
ž,
kT
λ
uk≤kuk,
DOI:10.12677/pm.2021.115087725nØêÆ
Éeœ
dÚn2.1Œ
i(T
λ
,K
p
2
,K) = 1.
ef
0
= ∞,K•30 <r
2
<σp
2
,¦f(t,u) ≥M
1
u,0 ≤u≤r
2
,Ù¥
M
1
λl
R
1
0
e
R
s
0
a(θ)dθ
ds
e
R
1
0
a(θ)dθ
−l
≥1,
u∈∂K
r
2
ž,Kk
kT
λ
uk≥
λl
e
R
1
0
a(θ)dθ
−l
Z
1
0
h(s)e
R
s
0
a(θ)dθ
ds≥kuk.
dÚn2.1Œ
i(T
λ
,K
r
2
,K) = 0,
¤±
i(T
λ
,K
p
2
/
˚
K
r
2
,K) = −1.
ef
∞
= ∞, K•3
ˆ
R
2
>p
2
¦f(t,u) ≥M
1
u, u≥
ˆ
R
2
,R
2
= max{2p
2
,
ˆ
R
2
/σ}, u∈∂K
R
2
ž
min
0≤t≤1
u(t) ≥σkuk≥
ˆ
R
2
,
¤±
kT
λ
uk≥kuk,u∈∂K
R
2
.
dÚn2.1Œ
i(T
λ
,K
R
2
,K) = 0,
¤±
i(T
λ
,K
R
2
/
˚
K
p
2
,K) = −1,
KT
λ
3K
R
2
/
˚
K
p
2
½K
p
2
/
˚
K
r
2
pk˜‡ØÄ:,=T
λ
u= u. ¯K(1.1) –•3˜‡).
(c).b(H1)-(H3) ¤á,p
3
>0, u∈∂K
p
3
ž,Òk
kT
λ
uk≥
ˆm
p
3
λl
e
R
1
0
a(θ)dθ
−l
Z
1
0
e
R
s
0
a(θ)dθ
ds,
Ù¥ˆm
p
3
=min
σp
3
≤u≤p
3
{f(u)},
λ
0
≥
p
3
(e
R
1
0
a(θ)dθ
−l)
ˆm
p
3
l
R
1
0
e
R
s
0
a(θ)dθ
ds
.
λ>λ
0
ž,
kT
λ
uk≥kuk,
DOI:10.12677/pm.2021.115087726nØêÆ
Éeœ
dÚn2.1Œ
i(T
λ
,K
p
3
,K) = 0.
ef
0
= 0,K•30 <r
3
<σp
3
,¦f(t,u) ≤ε
2
u,0 ≤u≤r
3
,Ù¥
ε
2
λe
R
1
0
a(θ)dθ
R
1
0
e
R
s
0
a(θ)dθ
ds
e
R
1
0
a(θ)dθ
−l
≤1,
u∈∂K
r
3
ž,Kk
kT
λ
uk≤
λe
R
1
0
a(θ)dθ
e
R
1
0
a(θ)dθ
−l
Z
1
0
h(s)e
R
s
0
a(θ)dθ
ds≤kuk.
dÚn2.1Œ
i(T
λ
,K
r
3
,K) = 1,
¤±
i(T
λ
,K
p
3
/
˚
K
r
3
,K) = −1.
ef
∞
= 0,K•3R
3
¦f(t,u) ≤ε
2
u,u≥R
3
,u∈∂K
R
3
,Kk
kT
λ
uk≤kuk.
dÚn2.1Œ
i(T
λ
,K
R
3
,K) = 1,
¤±
i(T
λ
,K
R
3
/
˚
K
p
3
,K) = 1,
KT
λ
3K
p
3
/
˚
K
r
3
,K
R
3
/
˚
K
p
3
pˆk˜‡ØÄ:,=T
λ
u= u. ¯K(1.1) –•3ü‡).
(d).b(H1)-(H3) ¤á,p
4
>0, u∈∂K
p
4
ž,Òk
kT
λ
uk≤
λe
R
1
0
a(θ)dθ
ˆ
M
p
4
e
R
1
0
a(θ)dθ
−l
Z
1
0
e
R
s
0
a(θ)dθ
ds,
Ù¥
ˆ
M
p
4
= 1+max
σp
4
≤u≤p
4
{f(u)},-
0 <λ
0
≤
p
4
(e
R
1
0
a(θ)dθ
−l)
e
R
1
0
a(θ)dθ
ˆ
M
p
4
R
1
0
e
R
s
0
a(θ)dθ
ds
.
λ<λ
0
ž,
kT
λ
uk≤kuk,
dÚn2.1Œ
i(T
λ
,K
p
4
,K) = 1.
DOI:10.12677/pm.2021.115087727nØêÆ
Éeœ
ef
0
= ∞,K•30 <r
4
<σp
4
,¦f(t,u) ≥M
2
u,0 ≤u≤r
4
,Ù¥
M
2
λl
R
1
0
e
R
s
0
a(θ)dθ
ds
e
R
1
0
a(θ)dθ
−l
≥1,
u∈∂K
r
4
ž,Kk
kT
λ
uk≥
λl
e
R
1
0
a(θ)dθ
−l
Z
1
0
h(s)e
R
s
0
a(θ)dθ
ds≥kuk,
dÚn2.1Œ
i(T
λ
,K
r
4
,K) = 0.
ef
∞
= ∞,K•3
ˆ
R
4
>p
4
,¦f(t,u) ≥M
2
u,u≥R
4
.R
4
= max{2p
2
,
ˆ
R
4
/σ},u∈∂K
R
4
ž
min
0≤t≤1
u(t) ≥σkuk≥
ˆ
R
4
,
¤±
kT
λ
uk≥kuk,u∈∂K
R
4
,
dÚn2.1Œ
i(T
λ
,K
R
4
,K) = 0,
¤±
i(T
λ
,K
p
4
/
˚
K
r
4
,K) = 1,i(T
λ
,K
R
4
/
˚
K
p
4
,K) = −1,
KT
λ
3K
p
4
/
˚
K
r
4
,K
R
4
/
˚
K
p
4
pˆk˜‡ØÄ:,=T
λ
u= u. ¯K(1.1) –•3ü‡).
(e)ef
0
<∞…f
∞
<∞, @o•3ü‡êε
∗
1
,ε
∗
2
,±9r
∗
1
<R
∗
1
,¦
f(t,u) ≤ε
∗
1
u,u∈[0,r
∗
1
],
f(t,u) ≤ε
∗
2
u,u∈[R
∗
1
,∞).
-
ε
∗
3
= {maxε
∗
1
,ε
∗
2
,max
r
∗
1
≤u≤R
∗
1
f(t,u)
u
},
¤±
f(t,u) ≤ε
∗
3
uu∈[0,∞).
bv(t) ´¯K(1.1) ˜‡).¤±t∈[0,1] ž,Òkv(t) = T
λ
v(t), -
λ
0
=
e
R
1
0
a(θ)dθ
−l
ε
∗
3
e
R
1
0
a(θ)dθ
R
1
0
e
R
s
0
a(θ)dθ
ds
,
λ<λ
0
ž,
kvk= kT
λ
vk≤
λe
R
0
1a(θ)dθ
ε
∗
3
e
R
0
1a(θ)dθ
−l
Z
1
0
e
R
0
sa(θ)dθ
dskvk<kvk.
DOI:10.12677/pm.2021.115087728nØêÆ
Éeœ
ù´˜‡gñ,Ïd•3˜‡λ
0
>0, ¦0 <λ<λ
0
ž,¯K(1.1) Ã).
(f)ef
0
>0 …f
∞
>0, @o•3ü‡êη
∗
1
,η
∗
2
,±9r
∗
2
<R
∗
2
,¦
f(t,u) ≥η
∗
1
uu∈[0,r
∗
2
],
f(t,u) ≥η
∗
2
uu∈[R
∗
2
,∞),
-
η
∗
3
= max{η
∗
1
,η
∗
2
,max
r
∗
2
≤u≤R
∗
2
f(t,u)
u
},
¤±
f(t,u) ≥η
∗
3
uu∈[0,∞).
bv(t) ´¯K(1.1) ˜‡),¤±t∈[0,1] ž,Òkv(t) = T
λ
v(t), -
λ
0
=
e
R
1
0
a(θ)dθ
−l
η
∗
3
l
R
1
0
e
R
s
0
a(θ)dθ
ds
,
λ>λ
0
ž,
kvk= kT
λ
vk≥
λlη
∗
3
e
R
0
1a(θ)dθ
−l
Z
1
0
e
R
0
sa(θ)dθ
dskvk>kvk.
ù´˜‡gñ,Ïd•3˜‡λ
0
>0, ¦λ>λ
0
ž,¯K(1.1) Ã),y..
~1-¯K(1.3)¥~êl=
1
2
,‚5‘a(t)≡1,‚5‘a(t)≡1,š‚5‘f(t,u(t))=
t(u
2
+u
1
2
),Ù¥λ´˜‡ëê,=





u
0
(t)+u(t) = λt(u
2
+u
1
2
),0 ≤t≤1,
u(0) =
1
2
eu(1)
y²éuf(t,u(t))=t(u
2
+u
1
2
), f∈C([0,1]×R
+
,R
+
), ^‡(H1) ¤á.w ,, a(t)≡1
…l=
1
2
e÷v^‡(H2) †(H3), ´f
0
= ∞,f
∞
= ∞.
λ
0
=
p
2
2
ˆ
M
p
2
(e−1)
,
d½n1.1, 0 <λ≤λ
0
ž,¯K(1.3) –•3ü‡).

4.Ä7‘8
I[g,‰ÆÄ7]Ï‘8(12061064).
DOI:10.12677/pm.2021.115087729nØêÆ
Éeœ
ë•©z
[1]Lakshmikantham,V.(2008)PeriodicBoundaryValueProblemsofFirstandSecondOrder
DifferentialEquations.JournalofAppliedMathematicsandSimulation,No.3,131-138.
[2]Peng,S.G.(2004)PositiveSolutionsforFirstOrderPeriodicBoundaryValueProblem.Ap-
pliedMathematicsandComputation,158,345-351.https://doi.org/10.1016/j.amc.2003.08.
090
[3]Wang,F.,Zhang,F.,Zhu,H.L.andLi,S.J.(2016)PeriodicOrbitsofNonlinearFirst-Order
GeneralPeriodicBoundaryValueProblem.Filomat,30,3427-3434.
https://doi.org/10.2298/FIL1613427W
[4]Wu,X.R.andWang,F.(2008)ExistenceofPositiveSolutionsofSingularSecond-Order
PeriodicBoundaryValueProblems.MathematicsinPracticeandTheory,No.23,227-232.
[5]Ma,R.Y.,Chen,R.P.andHe,Z.Q.(2014)PositivePeriodicSolutionsofSecond-OrderD-
ifferentialEquationswithWeakSingularities.AppliedMathematicsandComputation,232,
97-103.https://doi.org/10.1016/j.amc.2013.12.142
[6]Torres,P.J.(2003)ExistenceofOne-SignedPeriodicSolutionsofSomeSecond-OrderDiffer-
entialEquationsviaa KrasnoselskiiFixedPointTheorem.JournalofDifferentialEquations,
190,643-662.https://doi.org/10.1016/S0022-0396(02)00152-3
[7]Ma,R.Y.,Gao,C.H.andChen,R.P.(2010)ExistenceofPositiveSolutionsofNonlinear
Second-OrderPeriodicBoundaryValueProblems.BoundaryValueProblems,2010,Article
No.626054.https://doi.org/10.1155/2010/626054
[8]Jiang,D.Q.,Chu,J.F.andZhang,M.R.(2005)MultiplicityofPositivePeriodicSolutionsto
SuperlinearRepulsiveSingularEquations.JournalofDifferentialEquations,211,282-302.
https://doi.org/10.1016/j.jde.2004.10.031
[9]Zhang,Z.X.andWang,J.Y.(2003)OnExistenceandMultiplicityofPositiveSolutionsto
PeriodicBoundaryValueProblemsforSingularNonlinearSecond-OrderDifferentialEqua-
tions.JournalofMathematicalAnalysisandApplications,281,99-107.
https://doi.org/10.1016/S0022-247X(02)00538-3
[10]Ma,R.Y.,Xie,C.J.andAhmed,A.(2013)PositiveSolutionsoftheOne-Dimensionalp-
LaplacianwithNonlinearityDefinedonaFiniteInterval.AbstractandAppliedAnalysis,
2013,ArticleID:492026.https://doi.org/10.1155/2013/492026
[11]Wang,H.Y.(2003)OntheNumberofPositiveSolutionsofNonlinearSystems.Journalof
MathematicalAnalysisandApplications,281,287-306.
https://doi.org/10.1016/S0022-247X(03)00100-8
[12]HŒ.š‚5•¼©Û[M].LH:ìÀ‰ÆEâч,1985.
DOI:10.12677/pm.2021.115087730nØêÆ

版权所有：汉斯出版社 (Hans Publishers) Copyright © 2021 Hans Publishers Inc. All rights reserved.