一类四阶非线性边值问题多个正解的存在性 Existence of Multiple Positive Solutions for a Class of Nonlinear Fourth-Order Boundary Value Problems

doi:10.12677/PM.2022.127131

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PureMathematicsnØêÆ,2022,12(7),1196-1204

PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.127131

˜aoš‚5>Š¯Kõ‡)•35

xxxhhhhhh

Ü“‰ŒÆêÆ†ÚOÆ§[‹=²

ÂvFÏµ2022c614F¶¹^FÏµ2022c715F¶uÙFÏµ2022c722F

Á‡

©$^IþØÄ:½nïÄoš‚5>Š¯K











(ρ

)

= λa(t)f(u(t)),t∈(0,1),

u(0) = u(1) = 0,u

(0) = u

(1) = 0

õ‡)•35,Ù¥λ>0´ëê,ρ

∈C

4−i

([0,1],(0,∞)),i= 0,1,2,3,4,f∈C([0,∞),[0,∞)),

a∈C([0,1],[0,∞)) …3[0,1]?Ûf«mþØð•""

'…c

o§ØÄ:½n§šÝ§[ê

ExistenceofMultiplePositive

SolutionsforaClassofNonlinear

Fourth-OrderBoundaryValueProblems

CongcongKang

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Jun.14

,2022;accepted:Jul.15

,2022;published:Jul.22

,2022

©ÙÚ^:xhh.˜aoš‚5>Š¯Kõ‡)•35[J].nØêÆ,2022,12(7):1196-1204.

DOI:10.12677/pm.2022.127131

xhh

Abstract

Thispaperconsiderswiththeexistenceofmultiplepositivesolutionsfornonlinear

fourth-orderboundaryvalueproblems











(ρ

)

= λa(t)f(u(t)),t∈(0,1),

u(0) = u(1) = 0,u

(0) = u

(1) = 0

byusingﬁxedpointtheoremincones.Whereλ>0isaparameter,ρ

∈C

4−i

([0,1],(0,∞)),

i=0,1,2,3,4,f∈C([0,∞),[0,∞)),a∈C([0,1],[0,∞))anddoesnotvanishidenticallyon

anysubintervalof[0,1].

Keywords

Fourth-Order,FixedPointTheorem,Disconjugacy,Quasi-Derivatives

2022byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

oü:>Š¯KŒ±£ãØÓÉå^‡e5ù/C,duÙ-‡A^µ,Æö‚é

ÙïÄl™ÊŽ,¿…¼˜-‡´L(J[1–10].Š5¿´, üà{ü| Ú½|

5ù•§

0000

(t) = λa(t)f(u(t))

)•35®éõÆöïÄ[1,5–10], 'u˜„/ª•§ék<ïÄ[2–4].

©Äk0[êŽf½Â.béz‡i=0,1,2,3,4, ¼êρ

∈C

4−i

([0,1],(0,∞)), é

?¿u∈C

[0,1],-

u= ρ

i−1

,i= 1,2,3,4,

DOI:10.12677/pm.2022.1271311197nØêÆ

xhh

K¼êL

u,i= 1,2,3,4 ¡•u[ê.e•Ä>.^‡

)u(0) = 0, t∈{i

)u(1) = 0, t∈{j

Ù¥{i

},{j

}´{0,1,2,3}f8.AO/, -i

=r−1,r=1,2,Kþã>.^‡Œ

±=z•½ù|

u(0) = u(1) = u

(0) = u

(1) = 0.(1.1)

Ïd©•Äoš‚5>Š¯K

(

(ρ

)

= λa(t)f(u(t)),t∈(0,1),

u(0) = u(1) = 0,u

(0) = u

(1) = 0

(1.2)

õ‡)•35,Ù¥λ>0,ρ

∈C

4−i

([0,1],(0,∞)),a∈C([0,1],[0,∞))…3[0,1]?Ûf«m

þØð•",f∈C([0,∞),[0,∞)). AO/, ρ

= 1,i=1,2,3,4 ž, ρ

(ρ

)

= u

0000

aq¯K3Bai[10]¥?Ø, Šö$^IþØÄ:½nïÄo>Š¯K







(4)

(t) = λf(u(t)),t∈(0,1),

u(0) = u(1) = 0, u

(0) = u

(1) = 0

õ‡ )•35,Ù¥,λ>0´~ê, f∈C([0,∞),[0,∞)) …f

:=lim

u→0

f(u)

∞

:=lim

u→∞

f(u)

6∈

{0,∞}.

g,,/, ·‚•ÄUÄ3ρ

6= 1,i= 1,2,3,4,a∈C([0,1],[0,∞))…3[0,1] ?Ûf«m

þØð•",f

∞

∈{0,∞}œ/e¯K(1.2) õ‡)•35?3ùp, ·‚ob:

(A1)ρ

∈C

4−i

([0,1],(0,∞)),i= 0,1,2,3,4, f∈C([0,∞),[0,∞));

(A2)a∈C([0,1],[0,∞)) …3[0,1] ?Ûf«mþØð•";

(A3)f

= ∞,f

∞

= +∞(f

= 0,f

∞

= 0);

(A4)•3h>0¦

f(u) ≤γh,0 ≤u≤h(f(u) ≥ηh,

h≤u≤h),

γ= (λM

a(s)ds)

−1

(η= (λm

a(s)ds)

−1

),(1.3)

Ù¥G(t,s) ´¯K(1.2) éAàg¯K‚¼ê, m= minG(t,s),M= maxG(t,s). ¿Xe

(J:

½n1.1b(A1)-(A4) ¤á.K¯K(1.2) –kü‡)u

¦

0 <ku

k<h<ku

DOI:10.12677/pm.2022.1271311198nØêÆ

xhh

2.ý•£

½Â2.1[11]Ω´R

¥,k.m8,f: Ω →R

,f´C

N”(=f= (f

,···,f

),f

,··

·,x

)3ΩþäkëYˆ ê, i= 1,2,···,n), p∈R

\f(∂Ω),u´τ=inf

x∈∂Ω

kf(x)−pk>0.

ŠëY¼êΦ : [0,+∞) →R

,¦§÷ve¡ü‡^‡:

(1)•3σ,τ

∗

÷v0 <σ<τ

∗

≤τ, …r6∈(σ,τ

∗

)žðkΦ(r) = 0;

(2)

Φ(kzk)dz= 1.

½ÂÿÀÝ

deg(f,Ω,p) =

Ω

Φ(kf(x)−pk)J

(x)dx,

Ù¥,J

(x)L«f3:xJacobi 1ª

(x) =

D(f

,···,f

)

D(x

,···,x

)

= |

∂f

∂x

½Â2.2[11]X•Banach˜m,K⊂X•4à8,D⊂Kk.…•ƒém8,

F: D→K•ëYN.k.m8Ω ⊂X÷vD= Ω∩K,∂D= ∂Ω∩K, FëYòÿ

∗

: Ω →K, -f

∗

= I−F

∗

,½Âi(F,D,K) = deg(f

∗

,Ω,θ)•F3Dþ'uKØÄ:•ê.

Ún2.1[10]X´˜‡Banach ˜m,K⊆X´˜‡I,ép>0,½ÂK

={u∈K:

kuk≤p}. bT: K

→K´˜‡ë YŽf,é?¿u∈∂K

= {u∈K:kuk=p}, kTu6=u,

(1)ekTuk≥kuk,u∈∂K

i(T,K

,K) = 0;

(2)ekTuk≤kuk,u∈∂K

i(T,K

,K) = 1.

½Â2.3[2]p

∈C[a,b],k= 1,···,n.e‚5n‡©•§

Ly≡y

(x)y

n−1

+···+p

(x)y= 0(2.1)

z‡š²…)":3«m[a,b] þÑun‡, K¡•§(2.1) 3«m[a,b] þšÝ, Ù¥-ŠU

-êOŽ.

½Â2.4[2]e¼êy

,···,y

∈C

[a,b]÷vnKdÄ1ª

:= W[y

,···,y

] =



···y

·········

(k−1)

···y

(k−1)



,(k= 1,···,n)

3«m[a,b] þð•,K¡¼êy

,···,y

¤˜‡êŒÅXÚ.

DOI:10.12677/pm.2022.1271311199nØêÆ

xhh

Ún2.2[2]•§(2.1)3«m[a,b] þ•3˜‡êŒÅ)X…=Ù3[a,b]þšÝ.

Ún2.3[2]•§(2.1)3«m[a,b] þšÝ…=LŒ±L«•

Ly≡v

···v

D···D

Ù¥D= d/dt,…

1 = W

, v

= W

, v

= W

k−2

k−1

, (k= 2,···,n).

Ún2.4[2]b•§(2.1)3[a,b] þ´šÝ,K

(−1)

n−k

G(t,s) >0,a<s<b,a<t<b.

5©•Äoü:>Š¯K, Ïdn= 4,k= 2, w,¯K(1.2)éAàg¯K‚¼ê

G(t,s) >0.

X= {u∈C

[0,1],u÷vª(1.1)},Y= C[0,1],

KX3‰êk·k

e¤Banach˜m, Y3‰ê|·|

∞

e¤Banach˜m, ••Bå„,k·k

|·|

∞

þ{P•k·k.

½ÂXþIK:

K= {u∈X:u≥0,min

0≤t≤1

u(t) ≥

kuk}.

½ÂŽfT: X→Y, äN/ª•:

Tu=

G(t,s)λa(s)f(u(s))ds.

3.½n1.1y²

Ún3.1é?¿u∈K, KŽfT: K→K, …TëY.

y²dumaxG(t,s) = M,minG(t,s) = m,0 ≤t≤1,0 ≤s≤1,é?¿u∈K,K

min

0≤t≤1

(Tu)(t) =min

0≤t≤1

G(t,s)λa(s)f(u(s))ds

≥λm

a(s)f(u(s))ds

= λ·

Ma(s)f(u(s))ds

≥λ

max

0≤t≤1

G(t,s)a(s)f(u(s))ds

≥

kTuk.

DOI:10.12677/pm.2022.1271311200nØêÆ

xhh

Tu∈K. d,dAscoli-Arzeia½n´ŽfT: K→K´ëY.

½n1.1y²ky²f

= ∞,f

∞

= ∞œ/.

Q>0 ¦

λQ

,s)a(s)ds>1.(3.1)

= ∞, •30 <R

<h¦f(u) ≥Qu,0 ≤u≤R

.eykTuk>kuk,u∈∂K

¯¢þ,u∈∂K

ž,

Tu(

) =

,s)λa(s)f(u(s))ds

≥λQ

,s)a(s)kukds

>kuk.(3.2)

Ïd,dÚn2.1

i(T,K

,K) = 0.(3.3)

2df

∞

=∞,•3R>0¦f(u)≥Qu,u≥R.R

>max{h,

},t∈[0,1],K

min

0≤t≤1

u(t) ≥

kuk>R, u∈∂K

,Óª(3.2),dÚn2.1 Œ

i(T,K

,K) = 0.(3.4)

,˜•¡,d(A4)Œ•,éu∈∂K

kTuk=max

0≤t≤1

G(t,s)λa(s)f(u(s))ds

≤λM

a(s)f(u(s))ds

≤λM

a(s)γkukds

= kuk(dª(1.3)).

Ïd,kTuk≤kuk,u∈∂K

.w,Tu6= u,u∈∂K

,KdÚn2.1

i(T,K

,K) = 1.(3.5)

dØÄ:•êŒ\59ª(3.3), (3.4),(3.5) Œ

i(T,K

◦

,K) = −1,i(T,K

◦

,K) = 1.

Ïd,T3K

◦

þk˜‡ØÄ:P•u

(t),3K

◦

þk˜‡ØÄ:P•u

(t),K

DOI:10.12677/pm.2022.1271311201nØêÆ

xhh

(t),u

(t)Ñ´¯K(1.2)), …÷v0 <ku

k<h<ku

2yf

= 0,f

∞

= 0œ/. é?¿ε>0, •3R

>0 ¦

f(u) <εu,0 <u<R

.(3.6)

ε= (λM

a(s)ds)

−1

…R

<h, Kéu∈∂K

kTuk=max

0≤t≤1

G(t,s)λa(s)f(u(s))ds

≤λM

a(s)f(u(s))ds

<ελM

a(s)kukds

= kuk.(3.7)

Ïd,dÚn2.1,

i(T,K

,K) = 1.(3.8)

aq/,R

>h>0 vŒ,¦f(u) <εu,u>R

.Óª(3.7),dÚn2.1 Œ,

i(T,K

,K) = 1.(3.9)

,˜•¡,d(A4)Œ, éu∈∂K

kTuk=max

0≤t≤1

G(t,s)λa(s)f(u(s))ds

≥λ

G(t,s)a(s)ηhds

≥ληm

a(s)kukds

= kuk(dª(1.3)).

Ïd,kTuk≥kuk,u∈∂K

.w,, Tu6= u,u∈∂K

.KdÚn2.1

i(T,K

,K) = 0.(3.10)

dØÄ:•êŒ\59(3.8), (3.9),(3.10) ªŒ

i(T,K

◦

,K) = 1,i(T,K

◦

,K) = −1.

Ïd,T3K

◦

ÚK

◦

þˆk˜‡ØÄ:P•u

(t)Úu

(t),…u

(t),u

(t)Ñ´¯K

(1.2)), ¿÷v0 <ku

k<h<ku

k.½n1.1y..

DOI:10.12677/pm.2022.1271311202nØêÆ

xhh

4.~f

~1 •Ä>Š¯K







0000

(t)−l

u(t) = u

,t∈(0,1),

u(0) = u(1) = 0, u

(0) = u

(1) = 0,

(4.1)

Ù¥l∈[0,l

),l

≈4.73´•§

cos(l)cosh(l) = 1

1˜‡Š,dž¯K(4.1)kš²…).

éz‡½i∈{1,2,3,4}, -u

´ÐŠ¯K

0000

(t)−l

u(t) = 0,

(j)

(0) = 0, j6= 4−i,

(4−i)

(0) = l

4−i

•˜),K

(t) =

[sinh(lt)−sin(lt)],u

(t) =

[cosh(lt)−cos(lt)],

(t) =

[sinh(lt)+sin(lt)],u

(t) =

[cosh(lt)+cos(lt)].

-y

(t)=u

(t+σ),y

(t)=u

(t+σ),y

(t)=−u

(t+σ),y

(t)=u

(t+σ),Ù¥σ∈(0,1) v

.K,

](t) =

[sinh(l(t+σ))−sin(l(t+σ))],

](t) =

[cosh(l(t+σ))cos(l(t+σ))−1],

](t) =

[sinh(l(t+σ))−sin(l(t+σ))],

](t) = W

](0) = l

w,, W

>0,i=1,2,3,4, K¼ê{y

}´•§u

0000

(t)−l

u(t)=0 3«m[0,1] þ˜‡

êŒÅ)X,ÏL{ü/OŽ,

0000

(t)−l

u(t) =

sinh(l(t+σ))−sin(l(t+σ))

(

(sinh(l(t+σ))−sin(l(t+σ)))

8l(cosh(l(t+σ))cos(l(t+σ))−1)

(

2(cosh(l(t+σ))cos(l(t+σ))−1)

l(sinh(l(t+σ))−sin(l(t+σ)))

(

(sinh(l(t+σ))−sin(l(t+σ)))

2l(cosh(l(t+σ))cos(l(t+σ))−1)

(

sinh(l(t+σ))−sin(l(t+σ))

)

ŠâÚn2.39Ún2.4•,Žfu

0000

(t) −l

u(t)šÝ,…3>.^‡u(0)=u(1)=u

(0)=

(1) = 0eéA‚¼ê•. ,˜•¡,w,u

÷v^‡(A3)-(A4), K¯K(4.1)–•

3ü‡).

DOI:10.12677/pm.2022.1271311203nØêÆ

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Ä7‘8

I[g,‰ÆÄ7]Ï‘8(1OÒ:12061064)"

ë•©z

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DOI:10.12677/pm.2022.1271311204nØêÆ