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PureMathematicsnØêÆ,2022,12(7),1205-1216
PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.127132
˜a‘š‚5>.^‡‡©•§)
•359õ)5
XXXŽŽŽWWW
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2022c614F¶¹^Fϵ2022c715F¶uÙFϵ2022c722F
Á‡
©ïÄš‚5‡©•§>Š¯K





u
00
(t)−k
2
u(t)+h(t)f(u(t)) = 0,0 <t<1,
u(0) = 0,u
0
(1)−g(u(1)) = b
(P)
)•359õ)5,Ù¥b´ëê,k>0,a∈C([0,1],(0,∞)),f,g∈C([0,∞),(0,∞)).
3f,g÷v·^‡ey•3˜‡êb
∗
,¦0<b<b
∗
ž,(P)–•3ü‡);
b= b
∗
ž, (P)•3˜‡), b>b
∗
ž, (P)Ø•3). ̇(Jy²Äuÿ ÀÝnØÚ
þe)•{"
'…c
š‚5>.^‡§)§ÿÀݧþe)
ExistenceandMultiplicityofPositive
SolutionsforaClassofDifferential
EquationswithNonlinearBoundary
Conditions
XiangbingLei
©ÙÚ^:XŽW.˜a‘š‚5>.^‡‡©•§)•359õ)5[J].nØêÆ,2022,12(7):1205-1216.
DOI:10.12677/pm.2022.127132
XŽW
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Jun.14
th
,2022;accepted:Jul.15
th
,2022;published:Jul.22
nd
,2022
Abstract
Inthispaper, weareconcernedwiththeexistenceandmultiplicityofpositivesolutions
forsecondordernonlineardifferentialequationsboundaryvalueproblems





u
00
(t)−k
2
u(t)+h(t)f(u(t)) = 0,0 <t<1,
u(0) = 0,u
0
(1)−g(u(1)) = b,
(P)
wherebisapositiveparameter,k>0,a∈C([0,1],(0,∞)),f,g∈C([0,∞),(0,∞)). When
fandgsatisfytheproperconditions,weprovethatthereexistsapositivenumber
b
∗
,suchthat(P)haszero,exactlyoneandatleasttwopositivesolutionsaccording
tob>b
∗
,b=b
∗
and0<b<b
∗
,respectively. Theproofofthemainresultsisbasedon
topologicaltheoryandthemethodofupperandlowersolutions.
Keywords
NonlinearBoundaryConditions, PositiveSolutions,TopologicalDegree,Upperand
LowerSolutions
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó9̇(J
©•‘š‚5>.^‡‡©•§>Š¯K



u
00
(t)−k
2
u(t)+h(t)f(u(t)) = 0,0 <t<1,
u(0) = 0,u
0
(1)−g(u(1)) = b
(1)
)•359õ)5,Ù¥k>0,b´ëê,h!fÚg÷v:
DOI:10.12677/pm.2022.1271321206nØêÆ
XŽW
(H1)h∈[0,1] →(0,∞)´ëY¼ê, :…3[0,1]?Ûf«mþØð•";
(H2)f,g: [0,∞) →(0,∞)´ëY¼ê;
(H3)f
0
=lim
s→0
+
f(s)
s
= 0,f
∞
=lim
s→∞
f(s)
s
= ∞,g
0
=lim
s→0
+
g(s)
s
= 0.
Cc5, ~‡©•§ü:>Š¯KïÄÉ¯õÆö'5, Ù)•35•¼
NõÐ(J[1–8].AO/, ©z[1]$^I.†Ø ØÄ:½nïÄš‚5>Š¯K



u
00
(t)+a(t)f(u(t)) = 0,0 <t<1,
u(0) = u
0
(1) = 0
(2)
)•35,¿¼Xe(J:
½nA[1]b
(A1)f∈C([0,∞),[0,∞)),
(A2)a∈C([0,1],[0,∞))…3[0,1]?Ûf«mþØð•",
(A3)f
0
=lim
u→0
f(u)
u
= ∞,f
∞
=lim
u→∞
f(u)
u
= 0
¤á,K¯K(2)–•3˜‡).
©z[2]$^ØÄ:•ênØïÄ>Š¯K



u
00
(t)−Mu(t)+f(t,u(t)) = 0,0 <t<1,
u(0) = u(1) = 0
(3)
)•35, Ù¥f∈C([0,1]×[0,∞),[0,∞)),M>0´~ê. 3š‚5‘f÷v·O•^‡
ž,¯K(3)–•3˜‡).
Š5¿´, ©z[1,2]¤ïįK>.^‡Ñ´‚5…´àg , g,‡¯:3š‚5…š
àg>.^‡e, ´ÄU¼)•35(J, )‡ê´ÄÉëêbK•? â·‚¤•, „
™„éda¯KïÄ. ·‚[3šàg>.^‡eïá¯K(1))•35(J, äN/, 
©̇(J´:
½n1b½(H1)-(H3)¤á,K•3˜‡êb
∗
, ¦0<b<b
∗
ž, ¯K(1)–• 3ü‡
),b= b
∗
ž,¯K(1)–•3˜‡),b>b
∗
ž,¯K(1)Ø•3).
51XJ¯K(1)¥g≡0,b= 0,k=0, K¯K(1)òz•¯K(2), g,ù´é©z[1]óŠ
ò†uÐ.
2.ý•£
-X= C[0,1],3Ù‰êkuk=sup
t∈[0,1]
|u(t)|e¤Banach˜m.
DOI:10.12677/pm.2022.1271321207nØêÆ
XŽW
5¿,¯K(1)duÈ©•§
u(t) = ϕ(t)(b+g(u(1)))+
Z
1
0
G(t,s)h(s)f(u(s))ds,t∈[0,1],(4)
Ù¥ϕ(t) =
sinhkt
kcoshk
,G(t,s)´ƒAGreen¼ê
G(t,s) =



sinhktcoshk(1−s)
kcoshk
,0 ≤t≤s≤1,
sinhkscoshk(1−t)
kcoshk
,0 ≤s≤t≤1,
ùpsinhx=
e
x
−e
−x
2
,coshx=
e
x
+e
−x
2
,P
m:=min
(t,s)∈[
1
4
,
3
4
]×[
1
4
,
3
4
]
G(t,s),M:=max
(t,s)∈[0,1]×[0,1]
G(t,s),
ζ:=min
t∈[
1
4
,
3
4
]
ϕ(t) =
sinh
k
4
kcoshk
,ξ:=max
t∈[0,1]
ϕ(t) =
tanhk
k
.
½ÂŽfT: X→X
Tu(t) = ϕ(t)(b+g(u(1)))+
Z
1
0
G(t,s)h(s)f(u(s))ds,t∈[0,1].(5)
P
K= {u∈X|u(t) ≥0…min
t∈[
1
4
,
3
4
]
u(t) ≥σkuk},
Ù¥σ= min{
m
M
,
ζ
ξ
},KK´X¥˜‡I.
Ún1b½(H1)-(H3)¤á, TXª(5)¤½Â, KT´ëYŽf, ¿…T(K) ⊂K.
y²é?¿u∈K, Kk
min
t∈[
1
4
,
3
4
]
Tu(t) ≥min
t∈[
1
4
,
3
4
]
ϕ(t)(b+g(u(1)))+min
t∈[
1
4
,
3
4
]
Z
1
0
G(t,s)h(s)f(u(s))ds
≥ζ(b+g(u(1)))+m
Z
1
0
h(s)f(u(s))ds
≥
ζ
ξ
ξ(b+g(u(1)))+
m
M
Z
1
0
Mh(s)f(u(s))ds
≥σ(ξ(b+g(u(1)))+
Z
1
0
Mh(s)f(u(s))ds)
= σkTuk,
ÏdT(K) ⊂K.2ŠâArzela-Ascoli½nN´T´ëYŽf.
Ún2[9]X´Banach˜m, K´X¥˜‡I,é?¿r>0, ½ÂK
r
={x∈K:
kxk<r}.bT: K
r
→K´ëYŽf,¦éx∈∂K
r
,kTx6= x, K±e(ؤá:
DOI:10.12677/pm.2022.1271321208nØêÆ
XŽW
(1)XJkTxk≥kxk,x∈∂K
r
,@oi(T,K
r
,K) = 0;
(2)XJkTxk≤kxk,x∈∂K
r
,@oi(T,K
r
,K) = 1.
3.)•359Ø•35
½n2b½(H1)-(H3)¤á,b¿©ž,¯K(1) –•3˜‡),b¿©Œž,¯
K(1)Ø•3).
y²Äky²b¿©ž, ¯K(1)–•3˜‡). Šâ(H1)Œ, L:=max
t∈[0,1]
h(t),l:=
min
t∈[
1
4
,
3
4
]
h(t).é?¿r
1
>0, -
K
r
1
= {u∈K: kuk<r
1
},
d(H3)Œ•, •3êρ
1
<r
1
, ¦0≤u≤ρ
1
ž, kf(u)≤
1
3ML
u,Ó/, •3êρ
2
<r
1
, ¦
0≤u≤ρ
2
ž, kg(u)≤
k
3
u,ρ=min{ρ
1
,ρ
2
},0≤u≤ρ…b<
k
3
ρž, é?¿u∈∂K
r
1
,
Kk
Tu(t) = ϕ(t)(b+g(u(1)))+
Z
1
0
G(t,s)h(s)f(u(s))ds
≤
1
k
b+
1
k
g((u(1))+ML
Z
1
0
f(u(s))ds
≤
1
3
ρ+
1
3
ρ+
1
3
ρ
≤r
1
,
ŠâÚn2Œ,
i(T,K
r
1
,K) = 1.
Ï•f
∞
= ∞,•3p>0, ¦u≥pž,f(u) ≥ηu, ¿…η÷v
σmηl≥1.
Àr
2
≥max{
p
σ
,r
1
},-
K
r
2
= {u∈K: kuk<r
2
},
XJu∈∂K
r
2
,K
min
t∈[
1
4
,
3
4
]
u(t) ≥σkuk≥p,
é?¿u∈∂K
r
2
,Kk
Tu(t) = ϕ(t)(b+g(u(1)))+
Z
1
0
G(t,s)h(s)f(u(s))ds
≥
Z
3
4
1
4
G(t,s)h(s)f(u(s))ds
≥mησlkuk
≥kuk,
DOI:10.12677/pm.2022.1271321209nØêÆ
XŽW
ù¿›XkTuk≥kuk,u∈∂K
r
2
,dÚn2
i(T,K
r
2
,K) = 0.
ŠâØÄ:•êŒ\5•
i(T,K
r
2
\K
r
1
,K) = −1.
ùL²T3K
r
2
\K
r
1
¥k˜‡ØÄ:,=¯K(1)–k˜‡).
5¿,ϕ´¯K



u
00
(t)−k
2
u(t) = 0,0 <t<1,
u(0) = 0,u
0
(1) = 1
).u´¯K(1))…=v= u−(b+g(u(1)))ϕ´¯K



v
00
−k
2
v+hf(v+(b+g(u(1)))ϕ) = 0,0 <t<1,
v(0) = 0,v
0
(1) = 0
(6)
šK).
u´¯K(1)),Kv= u−(b+g(u(1)))ϕ´¯K(6)),dÚn1Œ
inf
t∈[
1
4
,
3
4
]
v(t) ≥σkvk,
inf
t∈[
1
4
,
3
4
]
ϕ(t) ≥
ζ
ξ
kϕk≥σkϕk,u´
inf
t∈[
1
4
,
3
4
]
v+(b+g(u(1)))ϕ≥σ(kvk+(b+g(u(1)))kϕk)
≥σkv+(b+g(u(1)))ϕk.
-
e
f(t) = inf
t≤s
f(s),Kk
v(t) =
Z
1
0
G(t,s)h(s)f(v+(b+g(u(1)))ϕ(s))ds
≥
Z
3
4
1
4
G(t,s)h(s)f(v+(b+g(u(1)))ϕ(s))ds
≥ml
Z
3
4
1
4
f(v+(b+g(u(1)))ϕ(s))ds
≥ml
e
f(σkv+(b+g(u(1)))ϕk),
?
e
f(δkv+(b+g(u(1)))ϕk)
kv+(b+g(u(1)))ϕk
≤
e
f(δkv+(b+g(u(1)))ϕk)
kvk
≤
1
ml
,
DOI:10.12677/pm.2022.1271321210nØêÆ
XŽW
d(H2)9
e
f½Â•lim
s→∞
e
f(s)
s
= ∞,Ïd,•3κ>0,¦kv+(b+g(u(1)))ϕk≤κ,b´k..
4.þ)Úe)
!ÏL½Â¯K(1)þe),‘31o!¥¯K(1)õ‡).
½Â1α∈C
2
[0,1]´¯K(1)þ),XJα÷v



α
00
(t)−k
2
α(t)+h(t)f(α(t)) ≤0,0 <t<1,
α(0) ≥0,α
0
(1)−g(α(1)) ≥b.
β∈C
2
[0,1]´¯K(1)e),XJβ÷v



β
00
(t)−k
2
β(t)+h(t)f(β(t)) ≥0,0 <t<1,
β(0) ≤0,β
0
(1)−g(β(1)) ≤b.
Ún3b½(H1)-(H3)¤á,αÚβ©O´¯K(1)þ)Úe), …kβ(t)≤α(t),K¯
K(1)–•3˜‡)u÷v
β(t) ≤u(t) ≤α(t),t∈[0,1].
y²•Ä9ϯK



u
00
(t)−k
2
u+h(t)f
∗
(u(t)) = 0,0 <t<1,
u(0) = 0,u
0
(1)−g
∗
(u(1)) = b
(7)
Ù¥
f
∗
(u(t)) =









f(α(t)),u(t) >α(t),
f(u(t)),α(t) ≤u(t) ≤β(t),
f(β(t)),u(t) <β(t),
g
∗
(u(t)) =









g(α(t)),u(t) >α(t),
g(u(t)),α(t) ≤u(t) ≤β(t),
g(β(t)),u(t) <β(t).
¯¢þ,‡y²¯K(1)•3˜‡)u,…kβ(t) ≤u(t) ≤α(t),t∈[0,1],•Iy²¯K(7)•3
)u…÷vT^‡.
dª(5)Œ,¯K(7)duÈ©•§
u(t) = ϕ(t)(b+g
∗
(u(1)))+
Z
1
0
G(t,s)h(s)f
∗
(u(s))ds,t∈[0,1],
DOI:10.12677/pm.2022.1271321211nØêÆ
XŽW
½ÂŽfT
∗
: X→X
T
∗
u(t) = ϕ(t)(b+g
∗
(u(1)))+
Z
1
0
G(t,s)h(s)f
∗
(u(s))ds,t∈[0,1],
duf
∗
,g
∗
´ëY¼ê,N´y,T
∗
´ëYŽf,
B= {u∈X: β(t) ≤u(t) ≤α(t), t∈[0,1]},
w,B´X¥k.48, K|f
∗
(u(t)) |≤max
u(t)∈B
|f(u(t))|, Ïdf
∗
k., Ón,g
∗
k., lT
∗
´
k.Žf,ŠâSchauderØÄ:½n,T
∗
kØÄ:u, =u´¯K(7)).
eyu(t)≤α(t). ‡é,t
0
∈[0,1], ku(t
0
)>α(t
0
), -ω(t)=α(t)−u(t), e¡©o«œ
/?Ø.
(i)é?¿t∈[0,1],bÑkω(t) <0, dž,
f
∗
(u(t)) = f(α(t)),g
∗
(u(t)) = g(α(t)),
Ïd
ω
00
(t) = α
00
(t)−u
00
(t)
≤k
2
α(t)−h(t)f(α(t))−k
2
u(t)−h(t)f
∗
(u(t))
≤0,
,˜•¡
ω(0) = α(0)−u(0) ≥0,
ω
0
(1) = α
0
(1)−u
0
(1) ≥b+g(α(1))−b−g
∗
(u(1)) = 0,
Šâ4ŒŠn,ω(t
0
) ≥0,t
0
∈[0,1],ù†bgñ.
(ii)0 <a<1÷vω(a) = 0, bω(t) <0,t∈[a,1],Kk
ω
00
(t) ≤0,ω(a) = 0, ω
0
(1) ≥0,
ù†(i)aqŒgñ,lkω(t) ≥0.
(iii)0 <a<1÷vω(a) = 0, bω(t) <0,t∈[0,a),Kk
ω
00
(t) ≤0,ω(0) ≥0,ω
0
(a) ≥0,
Ón,Œgñ.
(iv)0 <a,b<1÷vω(a) = 0,ω(b) = 0,ω(t) <0,t∈[a,b],Kk
ω
00
(t) ≤0,ω(a) = 0, ω
0
(b) ≥0,
DOI:10.12677/pm.2022.1271321212nØêÆ
XŽW
aq/,d«œ/e½kω(t) ≥0.
éuu(t) ≥β(t),t∈[0,1],^Ó•{Œy,ùpØ3Kã. u´¯K(7))u÷v
β(t) ≤u(t) ≤α(t),t∈[0,1],
Šâf
∗
,g
∗
½Â´•u´¯K(1)).
5.õ)59̇(Jy²
•¯K(1)),•Bå„,!ob
(H4)f(u) = 0,g(u) = 0,u<0.
Ún4b½(H1)-(H3)¤á, I⊂(0,∞)•;f«m, eb∈I,K•3~êem>0,¦¯
K(1)¤k)u÷vkuk≤em.
y²:{u
n
}´¯K(1)Ã.)S, †ÙéA{b
n
}∈I. dÚn1•, u
n
∈K, Ï•f
∞
=
∞,K•3~êq>0,u≥qž,f(u) ≥ηu, Ù¥η>0÷v
ησ
Z
3
4
1
4
G(
1
2
,s)h(s)ds≥2,
qÏn→∞ž,ku
n
k→∞,K•3N,n>N,žk
min
t∈[
1
4
,
3
4
]
u
n
(t) ≥σku
n
k≥q,
l
u
n
(
1
2
) ≥
Z
3
4
1
4
G(
1
2
,s)h(s)f(u
n
(s))ds
≥η
Z
3
4
1
4
G(
1
2
,s)h(s)u
n
(s)ds
≥ησ
Z
3
4
1
4
G(
1
2
,s)h(s)ku
n
k
≥2ku
n
k,
w,ù´˜‡gñ,·Ky.
Ún5PΓ = {b>0 |¯K(1)–k˜‡)}, supΓ = b
∗
,KΓk.,¿…b
∗
∈Γ.
y²d½n2Œ•,Γ´k..{b
n
}∈Γ…÷v
b
n
→b
∗
,n→∞,
DOI:10.12677/pm.2022.1271321213nØêÆ
XŽW
w,{b
n
}´k., dÚn4•b
n
éAu¯K(1))u
n
k.,(ÜŽfT;5´•b
∗
∈Γ.
éε>0, u
∗
´éAub
∗
¯K(1)).-
e
f(u(t)) =











f(u
∗
(t)+ε),u(t) >u
∗
(t)+ε,
f(u(t)),−ε≤u(t) ≤u
∗
(t)+ε,
f(−ε),u(t) <−ε,
eg(u(t)) =











g(u
∗
(t)+ε),u(t) >u
∗
(t)+ε,
g(u(t)),−ε≤u(t) ≤u
∗
(t)+ε,
g(−ε),u(t) <−ε,
½Â
e
Tu(t) = ϕ(t)(b+eg(u(1)))+
Z
1
0
G(t,s)h(s)
e
f(u(s))ds,t∈[0,1],(8)

Ω = {u∈X: −ε<u(t) <u
∗
(t)+ε,t∈[0,1]}.
Ún6b½(H1)-(H4)¤á,¿©êε,¦é?¿u∈C[0,1]…÷v
e
Tu=u,
0 <b<b
∗
ž,ku∈Ω.
y²:dª(8)Œ•, u≥0. e¡y²u≤u
∗
+ε. Šâf˜—ëY5, 0<ε<ε
0
ž, •3
êc, ¦cL≤k
2
,Kk
|f(u
∗
+ε)−f(u
∗
)|<cε,
u´
(u
∗
+ε)
00
= (u
∗
)
00
= k
2
u
∗
−hf(u
∗
)
= k
2
(u
∗
+ε)−hf(u
∗
+ε)+h(f(u
∗
+ε)−f(u
∗
))−k
2
ε
≤k
2
(u
∗
+ε)−hf(u
∗
+ε)+(cL−k
2
)ε
≤k
2
(u
∗
+ε)−hf(u
∗
+ε),
,˜•¡,
(u
∗
+ε)
0
(1) = b
∗
+g(u
∗
(1)) ≥b+g(u
∗
(1)),
l,u
∗
+ε´¯K(1)þ), dÚn3Œu≤u
∗
+ε.
½n1y²½n2¿›Xb>b
∗
ž, ¯K(1)Ø•3). duu
∗
Ú0©O´¯K(1)þ
)Úe), ŠâÚn3Œ•, •3¯K(1))u
b
… ÷v0≤u
b
≤u
∗
. Ïd, 0<b≤b
∗
ž, ¯K(1)
•3˜‡),du
b
∈Ω. e¡ïá0 <b<b
∗
ž¯K(1)1‡).
DOI:10.12677/pm.2022.1271321214nØêÆ
XŽW
B(u
b
,R
1
)´X¥±u
b
•¥%,R
1
•Œ»¥, é¿©ŒR
1
, d
e
T éb3;«mþžk .
•
deg(I−
e
T,B(u
b
,R
1
),0) = 1,
XJ•3u∈∂Ω, ¦
e
Tu= u, Kkf=
e
f,g=eg, dž,u´¯K(1)1‡).‡
e
Tu6= u,u∈
∂Ω, Kdeg(I−
e
T,Ω,0)û½. Ún6L²,
e
T3B(u
b
,R
1
)\Ω¥vkØÄ:, dÿÀÝƒØ5
deg(I−T,Ω,0) = deg(I−
e
T,Ω,0) = 1.
,˜•¡,dÚn4•,¯K(1)¤k)3b;«mþÑkk., é¿©ŒR
2
k
deg(I−T,B(0,R
2
),0) = d,(d•~ê)
Ï•b>b
∗
ž,¯K(1)Ø•3),ld= 0, u´
deg(I−T,B(0,R
2
)\Ω,0) = −1,
ùL²0 <b<b
∗
ž,¯K(1)•31‡).
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(1OÒ:12061064)"
ë•©z
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theAnnulus.JournalofDifferentialEquations,109,1-7.
https://doi.org/10.1006/jdeq.1994.1042
[2]Li,Y.X.(2003)PositiveSolutionsofSecond-OrderBoundaryValueProblemswithSign-
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[3]Benmeza¨ı,A.(2010)PositiveSolutionsforaSecondOrderTwoPointBoundaryValueProb-
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[6]Lian,W.C.,Wong,F.H.andYeh,C.C.(1996)OntheExistenceofPositiveSolutionsof
NonlinearSecondOrderDifferentialEquations.ProceedingsoftheAMS,124,1117-1126.
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[7]Yang,Z.L.(2021)PositiveSolutionsofaSecond-OrderNonlinearRobinProblemInvolving
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DOI:10.12677/pm.2022.1271321216nØêÆ

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