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PureMathematicsnØêÆ,2022,12(8),1370-1380
PublishedOnlineAugust2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.128150
˜a‘š‚5>.^‡Œ¯K
)•35
œœœZZZJJJ
Ü“‰ŒÆ§êƆÚOÆ§[‹=²
ÂvFϵ2022c719F¶¹^Fϵ2022c819F¶uÙFϵ2022c830F
Á‡
©ïÄŒ¯K





−u
00
(t) = λh(t)f(u(t)),t∈(0,1),
αu(0)−βu
0
(0) = 0,c(u(1))u(1)+δu
0
(1) = 0
(P)
)•35§Ù¥λ•ëê§α,δ>0β≥0•~ê§c∈C([0,∞),[0,∞))§h∈C([0,1],[0,∞))§
f∈C([0,∞),R)…f>−M(M>0)§f
∞
:=lim
x→∞
f(x)
x
= ∞"ÏL$^Krasnoselskii ØÄ:½
ny²•3~êλ
0
>0§0 <λ<λ
0
ž§¯K(P)•3˜‡)"
'…c
)§Œ¯K§š‚5>.^‡§KrasnoselskiiØÄ:½n
ExistenceofPositiveSolutionsforaClass
ofSecondOrderSemi-PositoneProblems
withNonlinearBoundaryConditions
XuanrongShi
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
©ÙÚ^:œZJ.˜a‘š‚5>.^‡Œ¯K)•35[J].A^êÆ?Ð,2022,12(8):1370-1380.
DOI:10.12677/pm.2022.128150
œZJ
Received:Jul.19
th
,2022;accepted:Aug.19
th
,2022;published:Aug.30
th
,2022
Abstract
Weareconcernedwithexistenceofpositivesolutionsforthesecondordersemi-
positoneproblem





−u
00
(t) = λh(t)f(u(t)),t∈(0,1),
αu(0)−βu
0
(0) = 0,c(u(1))u(1)+δu
0
(1) = 0,
(P)
where λisapositiveparameter, α,δ>0,β≥0, c∈C([0,∞),[0,∞)), h∈C([0,1],[0,∞)),
f∈C([0,∞),R)and f>−M(M>0),f
∞
:=lim
u→∞
f(x)
x
= ∞. Byusingfixedpointtheorem
ofKrasnoselskii,weprovethatthereexistsλ
0
>0 such that (P)hasapositivesolution
for0 <λ<λ
0
.
Keywords
PositiveSolutions,Semi-PositoneProblem,NonlinearBoundaryCondition,
KrasnoselskiiFixedPointTheorem
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.Úó
‡ ©•§Sturm-Liouville >Š¯K´˜a-‡¯K, ÚåNõÆö2•'5, ¿®
²¼˜•35(J[1–14]'X,WangÚErbe[13] ïÄXeSturm-Liouville>Š¯K



u
00
(t)+a(t)g(u(t)) = 0,t∈(0,1),
αu(0)−βu
0
(0) = 0,γu(1)+δu
0
(1) = 0,
(1.1)
Ù¥α,β,γ,δ≥0…γβ+αγ+αδ>0,g∈C([0,∞),[0,∞)).a∈C([0,1],[0,∞))…3[0,1]?
¿f«mþØð•".WangÚErbe $^IþØÄ:½n,ïáXe(J:
DOI:10.12677/pm.2022.1281501371nØêÆ
œZJ
½nA([13], ½n1)eg÷ve^‡:
lim
u→0
max
t∈[0,1]
g(t,u)
u
= 0,lim
u→∞
min
t∈[0,1]
g(t,u)
u
= +∞
½
lim
u→0
min
t∈[0,1]
g(t,u)
u
= +∞,lim
u→∞
max
t∈[0,1]
g(t,u)
u
= 0,
K¯K(1.1)–•3˜‡).
3#Nf•KŠœ/e, éuŒœ/, Anuradha[14]$^IþØÄ:½nïÄ
Sturm-Liouville>Š¯K



−u
00
(t) = λf(t,u(t)),t∈(r,R),
au(r)−bu
0
(r) = 0,cu
0
(R)+du
0
(R) = 0,
(1.2)
Ù¥λ•ëê, ¿…:
(A1)a,b,c,d≥0 …ac+ad+bc>0;
(A2)f: [r,R]×[0,∞) →RëY…f(t,u) >−M(M>0),t∈[r,R];
(A3)f
∞
=lim
x→∞
f(x)
x
= ∞.
¦‚ïáXe(J:
½nB ([14], ½n2.1)e(A1)–(A3) ¤á, λ>0 …¿©ž, ¯K(1.2) –•3˜‡
).Š5¿´, ©z[13]¤ïįK¥š‚5‘fšK, ©z[14]•,ïÄŒ¯ K•Î´
‚5>.^‡. g,‡¯: š‚5‘f•Œœ/…>.^‡•š‚5ž¯K(P) ´Ä•3)?
Ïd,©•Ä



−u
00
(t) = λh(t)f(u(t)),t∈(0,1),
αu(0)−βu
0
(0) = 0,c(u(1))u(1)+δu
0
(1) = 0
(1.3)
)•35.·‚ob½:(H1)λ>0•ëê,α,δ>0,β≥0•~ê;(H2)c: [0,∞) →[0,∞)
ëY, h: [0,1]→[0,∞) ëY, …3[0,1] ?¿f«mØð•";(H3) f:[0,∞)→RëY…f÷
vf(s) >−M(M>0);©̇(JXe:½n1.1b(H1)–(H3) …(A3) ¤á,K•3~ê
λ
0
>0 ¦0 <λ<λ
0
ž,¯K(1.3) –•3˜‡)u
λ
,…λ→0
+
ž,u
λ
→∞.
2.ý•£
-˜mX :=C[0,1],Ù3‰êkuk
∞
=max
t∈[0,1]
|u(t)|e¤Banach˜m,L
1
(0,1)3‰ê
kyk
1
=
R
1
0
|y(t)|dte¤Banach˜m.
Ún2.1[15]-E•Banach˜m,P•E¥I,T: P→P•ëYŽf.bh∈E,h6= 0
DOI:10.12677/pm.2022.1281501372nØêÆ
œZJ
…•3êr,R,r6= R¦
(a)ey∈P÷vy= θTy,θ∈[0,1],Kkyk
∞
6= r;
(b)ey∈P÷vy= Ty+ξh,ξ≥0, Kkyk
∞
6= R.
KT•3ØÄ:y, …min{r,R}<kyk
∞
<max{r,R}.Ún2.2-ω÷v



−ω
00
(t) = m(t),t∈(0,1),
αω(0)−βω
0
(0) = 0,γω(1)+δω
0
(1) = 0,
Ù¥γ>0•~ê, m∈L
1
[0,1],m(t) ≥0,t∈(0,1), …ρ:= γβ+γα+αδ>0.K
ω(t) ≥kωk
∞
σ,t∈[0,1],
Ù¥
σ= min{
δ
δ+γ
,
β
α+β
}.
y²N´y²
ω(t) =
Z
1
0
G(t,s)ω(s)ds:= Lω(t),
Ù¥
G(t,s) =







(γ+δ−γt)(β+αs)
ρ
,0 ≤s≤t≤1,
(β+αt)(γ+δ−γs)
ρ
,0 ≤t≤s≤1.
w,,G(t,s) ≥0,t∈[0,1].-kωk
∞
= ω(τ),τ∈[0,1].¯¢þ
G(t,s)
G(τ,s)
≥
G(t,s)
G(s,s)
≥







δ
δ+γ
,0 ≤s≤t≤1,
β
α+β
,0 ≤t≤s≤1.
Ïd
ω(t) =
Z
1
0
G(t,s)
G(τ,s)
G(τ,s)ω(s)ds≥kωk
∞
σ.
Ún2.3-k∈L
1
(0,1),…k≥0,-u∈C
1
[0,1]∩C
2
(0,1)÷v



−u
00
(t) ≥−k,t∈(0,1),
αu(0)−βu
0
(0) ≥0,γu(1)+δu
0
(1) ≥0.
-ϕ:=
β+α
α
,bkuk
∞
>2ϕ(1+
1
σ
)kkk
1
,Ku(t) ≥0…
u(t) ≥
σ
2
kuk
∞
,t∈[0,1].
DOI:10.12677/pm.2022.1281501373nØêÆ
œZJ
y²-ν
0
(t)•¯K



−ν
00
(t) = −k,t∈(0,1),
αν(0)−βν
0
(0) = 0,γν(1)+δν
0
(1) = 0
•˜),K
−ν
0
(t) =
Z
1
0
G(t,s)k(s)ds
=
Z
t
0
(γ+δ−γt)(β+αs)
ρ
k(s)ds+
Z
1
t
(β+αt)(γ+δ−γs)
ρ
k(s)ds
≤
Z
1
0
(γ+δ−γt)(β+αt)
ρ
dskkk
1
≤
(γ+δ)(β+α)
γβ+γα+αδ
kkk
1
≤
(β+α)
α
kkk
1
.
Ïd
−ν
0
(t) ≤ϕkkk
1
.
-y(t) = u(t)−ν
0
(t),k



−y
00
(t) ≥0,t∈(0,1),
αy(0)−βy
0
(0) ≥0,γy(1)+δy
0
(1) ≥0.
dÚn2.2Œ•
y(t) ≥kyk
∞
σ,t∈[0,1],
¤±
u(t) = y(t)+ν
0
(t)
≥kyk
∞
σ−ϕkkk
1
= ku−ν
0
k
∞
σ−ϕkkk
1
≥(kuk
∞
−k−ν
0
k
∞
)σ−ϕkkk
1
≥(kuk
∞
−ϕ(1+
1
σ
)kkk
1
)σ
≥
σ
2
kuk
∞
,t∈(0,1).
DOI:10.12677/pm.2022.1281501374nØêÆ
œZJ
3.̇(Jy²
½n1.1y²-λ>0, év∈XkL
λ
v= u, Ù¥u•



−u
00
(t) = λh(t)f(˜v(t)),t∈(0,1),
αu(0)−βu
0
(0) = 0,γ
v
u(1)+δu
0
(1) = 0
).…-γ
v
= c(|v(1)|), σ(t) = min{
δ
δ+γ
v
β
α+β
}
˜v(t) = max{v(t),σ(t)}.
dÚn2.2Œ•
u(t) = λ
Z
1
0
G
v
(t,s)h(s)f(˜v(s))ds,
Ù¥
G
v
(t,s) =







(γ
v
+δ−γ
v
t)(β+αs)
αδ+γ
v
β+γ
v
α
,0 ≤s≤t≤1,
(β+αt)(γ
v
+δ−γ
v
s)
αδ+γ
v
β+γ
v
α
,0 ≤t≤s≤1.
½ÂX¥I
P= {u∈X|u(t) ≥
σ
2
||u||
∞
,t∈[0,1]}.
eu∈P, (ÜÚn2.3 Œ•
L
λ
u(t) = λ
Z
1
0
G(t,s)h(s)f(˜v(s))ds
≥λ
Z
1
0
G(t,s)
G(s,s)
G(s,s)h(s)f(˜v(s))ds
≥σ||L
λ
u||
∞
≥
σ
2
||L
λ
u||
∞
,t∈[0,1].
ÏdL
λ
(P) ⊂P.eyL
λ
ëY.
Äk, S⊂C[0,1] •k.8, K•3êB, ¦é?¿v∈Sk||v||
∞
≤B. dfëY
5•,•3D>0,k
f(v) ≤D,v∈S.
-v
n
∈S…v
n
→v,K
u
n
= L
λ
v
n
,u= L
λ
v.
d.‚KF¥Š½nŒ•,•3~êN>0, k
|G
v
n
(t,s)−G
v
(t,s)|≤N|γ
v
n
−γ
v
|.
DOI:10.12677/pm.2022.1281501375nØêÆ
œZJ
Ï•
|f(˜v
n
(s))−f(˜v(s))|→0,n→∞,s∈[0,1],
|L
λ
v
n
−L
λ
v|= |λ
Z
1
0
G
v
n
(t,s)h(s)f(˜v
n
(s))ds−λ
Z
1
0
G
v
(t,s)h(s)f(˜v(s))ds|
≤λ(
Z
1
0
|G
v
n
(t,s)−G
v
(t,s)|h(s)|f(˜v
n
(s))|ds+
Z
1
0
G
v
(t,s)h(s)|f(˜v
n
(s))−f(˜v(s))|ds)
<λ(
Z
1
0
N|γ
v
n
−γ
v
|h(s)|f(˜v
n
(s))|ds+
Z
1
0
G
v
(t,s)h(s)|f(˜v
n
(s))−f(˜v(s))|ds),
¤±
|L
λ
v
n
−L
λ
v|→0,n→∞,
L
λ
v
n
→L
λ
v. ¤±L
λ
ëY5y.eyL
λ
´;Žf.
éuv∈Sk
||L
λ
v||≤D
Z
1
0
G
v
(s,s)h(s)ds,
Ïd,L
λ
: C[0,1] →C[0,1]˜—k..é?¿t
1
,t
2
∈[0,1] (t
1
<t
2
)k
|L
λ
v(t
1
)−L
λ
v(t
2
)|= |λ
Z
1
0
G
v
(t
1
,s)h(s)f(˜v(s))ds−λ
Z
1
0
G
v
(t
2
,s)h(s)f(˜v(s))ds|
≤λD
Z
1
0
|G
v
(t
1
,s)−G
v
(t
2
,s)|h(s)ds.
dG(t,s) ëY5Œ•,L
λ
ÝëY.dArz`ela-Ascoli½n•,L
λ
: C[0,1] →C[0,1]•ëYŽf.
-a>1 k
f(z) >0,z≥a.
dfëY5Œ•
|f(z)|≤M,z∈(0,a).(3.1)
Ïd
|f(z)|≤(M+
ˆ
f(max{z,a}),z>0,(3.2)
Ù¥
ˆ
f(x) =sup
a≤z≤x
f(z),x≥a.
bλ<
a
2(c
1
+c
2
f(a))
,Ù¥
c
1
=
(α+β)M
α
Z
1
0
h(s)ds,c
2
=
(α+β)
α
Z
1
0
h(s)ds.
DOI:10.12677/pm.2022.1281501376nØêÆ
œZJ
eyÚn2.2béL
λ
¤á.
(a)•3r
λ
>0, eu∈P÷v
u= θL
λ
u,θ∈[0,1],
K||u||
∞
6= r
λ
.¯¢þ, -u∈P÷vu= θL
λ
u,θ∈[0,1], K
u(t) = θλ
Z
1
0
G
u
(t,s)h(s)f(˜u(s))ds,t∈[0,1].
Ï•a>1, σ<1,d(3.2) Œ•
|f(u)|≤M+
ˆ
f(max{u,a}),
…G
u
(t,s) ≤
α+β
α
,t∈(0,1).Ïd
u(t) ≤λ
α+β
α
(M+
ˆ
f(max{u,a})
Z
1
0
h(s)ds
≤λ(c
1
+c
2
ˆ
f(max{||u||
∞
,a})),t∈[0,1].
=
||u||
∞
c
1
+c
2
ˆ
f(max{||u||
∞
,a}
≤λ.(3.3)
Ï•
a
(c
1
+c
2
f(a))
>2λ,
(Ü(A4),K•3r
λ
>a¦
r
λ
c
1
+c
2
ˆ
f(r
λ
)
= 2λ.(3.4)
d(3.3),(3.4) ||u||
∞
6= r
λ
…
r
λ
→∞,λ→0.
(b)•3R
λ
>0, éuu∈Pk
u= L
λ
u+ξ,ξ≥0.
K||u||
∞
6= R
λ
.¯¢þ,-u∈P÷vu= L
λ
u+ξ,ξ≥0, K
u(t)−ξ= λ
Z
1
0
G
u
(t,s)h(s)f(˜u(s))ds.
Ï•u(t)−ξ÷v



−u
00
(t) = λh(t)f(u(t)),t∈(0,1),
u(0)−βu
0
(0) = ξ≥0,γ
v
u(1)+δu
0
(1) = γ
v
ξ≥0.
DOI:10.12677/pm.2022.1281501377nØêÆ
œZJ
-k(t) = Mh(t),t∈[0,1].d(3.1)Œ•
h(t)f(˜u(t)) ≥−Mh(t) = −k(t),t∈[0,1].
dÚn2.3Œ•
u(t) ≥(||u||
∞
−ϕ(1+
1
σ
)||k||
1
)σ,t∈[0,1].
b||u||
∞
>max{2ϕ(1+
1
σ
),
2
σ
},K
u(t) ≥(||u||
∞
−
||u||
∞
2
)σ
=
σ||u||
∞
2
,t∈[0,1].
s∈Λ = [
1
4
,
3
4
]k
G
u
(t,s) ≥
(δ+
1
4
γ)(
1
4
α+β)
(δ+γ)(α+β)
≥
(4+γ)(1+4β)
16(1+γ)(1+β)
≥
1
16
.
Ïd
u(t) = λ
Z
1
0
G
u
(t,s)h(s)f(˜u(s))ds
≥λ
Z
Λ
G
u
(t,s)h(s)f(˜u(s))ds+λ
Z
Λ
c
G
u
(t,s)h(s)f(˜u(s))ds
≥λ(
1
16
ˇ
f(
||u||
∞
2
σ)
Z
Λ
h(s)ds−
(α+β)
α
M||k||
1
),
Ù¥
ˇ
f(x) =inf
z≥x
f(z). ¯¢þ
1
16
ˇ
f(
||u||
∞
2
σ)
R
Λ
h(s)ds−
(α+β)
α
K||k||
1
||u||
∞
≤
1
λ
.(3.5)
(Ü(A3),(3.5) Œ•,•3R
λ
>1 …||u||
∞
<R
λ
,Ïd(b) ¤á.
Ïd,dÚn2.1 Œ•,¯K(1.3)•3˜‡)u
λ
(t),λ→0 ž
u
λ
(t) →∞,t∈[0,1].
4.A^
~•Ä¯K



−u
00
(t) = λt
2
(u
2
(t)−1),t∈(0,1),
u(0)−u
0
(0) = 0,u
3
(1)+u
0
(1) = 0
(4.1)
DOI:10.12677/pm.2022.1281501378nØêÆ
œZJ
)•35,Ù¥λ>0.
) ùpf(u) = u
2
(t)−1,α= β= δ= 1,h(t) = t
2
,c(u(1)) = u
2
(1).
éu¯K(4.1)ó, (H1), (H2) w,¤á, Ï•lim
u→∞
f(u)
u
=∞…f(u)>−1,u∈[0,∞), Kf
÷v(H3),(A3).
Šâ½n1.1,•3~êλ
∗
>0, ¦0 <λ
∗
<λ
0
ž,¯K(4.1) •3˜‡)u
λ
,λ→0
+
ž,u
λ
→∞.
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(1OÒ:12061064).
ë•©z
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