Existence of Positive Solution for Fourth Order Superlinear Singular Semipositone Differential System

Published online December 10, 2013 (http://www.sciencepublishinggroup.com/j/pamj) doi: 10.11648/j.pamj.20130206.12

Existence of positive solution for fourth order superlinear

singular semipositone differential system

Wang Feng

1

, Junmin Zhang

2

1_{Jiangsu Union Technical Institute, Yancheng College of Mechatronic Technology, Yancheng, Jiangsu 224005, China} 2_{Department of Mathmatics and Applied Mathmatics, Hanshan Normal University, Chaozhou, Guangdong 521041, China}

Email address:

[email protected] (W. Feng), [email protected] (J.M. Zhang)

To cite this article:

Wang Feng, Junmin Zhang. Existence of Positive Solution for Fourth Order Superlinear Singular Semipositone Differential System.Pure and Applied Mathematics Journal. Vol. 2, No. 6, 2013, pp. 179-183. doi: 10.11648/j.pamj.20130206.12

Abstract:

By transforming the boundary value problem into the corresponding fixed-point problem of a completely continuous operator, the existence is obtained in the paper for two-point boundary value problem of fourth order superlinear singular semipositone differential system via the fixed point theorem concerning cone compression and expansion in norm type.

Keywords:

Superlinear Singular Semipositone Differential System, Positive Solution, Fixed Point Theorem, Cone

1. Introduction

In this paper, we study the following superlinear fourth order singular semipositone differential system

  

∈ +

=

+ =

), 1 , 0 ( ), ( )) ( ), ( , ( ) (

), ( )) ( ), ( , ( ) (

) 4 (

) 4 (

t t q t y t x t g t y

t p t y t x t f t x

λ λ

(1.1)

subject to the boundary conditions

  

= ′′ = ′′ = =

= ′′ = ′′ = =

, 0 ) 1 ( ) 0 ( ) 1 ( ) 0 (

, 0 ) 1 ( ) 0 ( ) 1 ( ) 0 (

y y y y

x x x x

(1.2)

where , 2

1≤λ< f andg:(0 ,1)×[0,+∞)×[0 ,+∞)→[0 ,+∞)are continuous and may be singular at t=0 ,1. pand q:(0,1)

) , (−∞ +∞

→ are Lebesgue integrable.

The method is fixed point theorem concerning cone compression and expansion. For the concepts and property of fixed point index theory we refer to [1].

Recently, the existence of positive solutions for nonlinear singular semipositone boundary value problems has been studied extensively, see [2-6]. However, the results are seldom for fourth order superlinear singular semipositone differential system boundary value problem. In the previous literature, a great deal of work was devoted to the case that f andgare nonnegative and have no singularities orp(t)≡0,q(t)≡0, see [7,8]. When f andgare allowed to

change sign, singular andp(t)≡0,q(t)≡0,[9]obtained the existence of positive solutions for nonlinear three-point boundary value problems. Under the given conditions that

f andgare nonnegative, singular andp(t),q(t)change

sign, [10] obtained the existence of positive solutions for two-point boundary value problems. Motivated by [6,10], we study the existence of positive solutions for the two- point boundary value problem of fourth order superlinear singular semipositone differential system by using the fixed point theorem concerning cone compression and expansion in norm type in the paper.

2. Preliminaries

LetX =C([0,1];R)×C([0,1];R)be a real Banach space equipped with for any(u,v)∈X ,

v u v

u, ) = +

( , max ()

] 1 , 0

[ u t

u

t∈

= ， max ()

] 1 , 0

[ vt

v

t∈

= ,

whereRis a real number set. Let us define a cone PofXby

{

( , )∈ ()≥ (1− ) , ( )≥ (1− ) ,∀ ∈[0 ,1]

}

.

= x y X xt t t x yt t t y t

P

Next we introduce Green’s function tou′′(0)=0 subject to boundary condition u(0)=u(1)=0,

  

≤ ≤ ≤ −

≤ ≤ ≤ − =

. 1 0

), 1 (

, 1 0

), 1 ( ) , (

s t s t

t s t s s t

For anyy∈C[0 ,1],let us define a funtion by

  

< ≥ =

∗

. 0 ) ( , 0

, 0 ) ( ), ( )] ( [

t y

t y t y t

y (2.2)

For the convenience, in this paper we make the following assumptions:

)

(H₁ There exist constant λ_i>µ_i >1,(i=1 ,2 ,3,4) such that,∀(t ,x ,y)∈(0 ,1)×[0 ,+∞)×[0 ,+∞),and c∈(0,1),

), , , ( ) , , ( ) , ,

( 1

1_{f t x y f t cx y c f t x y}

cλ ≤ ≤ µ

), , , ( )

, , ( ) , ,

( 2

2_{f t x y f t x cy c f t x y}

cλ ≤ ≤ µ

), , , ( ) , , ( ) , ,

( 3

3_{gt x y g t cx y c g t x y}

cλ ≤ ≤ µ

). , , ( ) , , ( ) , ,

( 4

4_{g t x y gt x cy c gt x y}

cλ ≤ ≤ µ

) (H₂

, 0 d )

( ₁

1

0 = >

∫

p− s s r ( )d 3 0, 1

0 = >

∫

q− s s r

, 1 d

)] ( ) 1 , 1 , ( )[ ,

( ₂

1

0 + = <λ

∫

G s s f s p+ s s r

, 1 d

)] ( ) 1 , 1 , ( )[ ,

( ₄

1

0 + = < λ

∫

G s s g s q+ s s r

, 0 ) 1 , 0 , (t >

f f(t ,1 ,0)>0, g(t,0,1)>0, ,

0 ) 0 , 1 , (t >

g ∀t∈(0,1).

∫

₀1G(s,s)f(s,1,1)dsand

∫

1

0G(s,s)g(s,1,1)dsare convergent.

, 1 , }

,

max{ 1 1 3

1 r < − m>

r m

n

λ where

}, , max{r₂ r₄

n= m=max{λ₁+λ₂ ,λ₃+λ₄}, },

0 ), ( max{ )

(s p s

p₊ = p₋(s)=max{−p(s),0}, },

0 ), ( max{ )

(s q s

q₊ = q₋(s)=max{−q(s),0}. If (x ,y)∈(C2[0 ,1]∩C4(0 ,1))×(C2[0 ,1]∩C4(0 ,1)) satisfies (1.1), (1.2) and x(t)>0,y(t)>0,∀t∈(0 ,1),then we call that(x,y)is a positive solution of BVP (1.1), (1.2).

Let

, d ) ( ) , ( d ) , ( ) ( 1

0

1

0

1 t =

∫

Gt

∫

G s p− s s

w ξ ξ ξ

, d ) ( ) , ( d ) , ( ) ( 1

0

1

0

2 t =

∫

Gt

∫

G sq− s s

w ξ ξ ξ

By(H₂)we have

, d ) ( ) , ( d ) , ( ) ( 1

0

1

0

1 t =

∫

G t

∫

G s p− s s<+∞

w ξ ξ ξ

() ( , )d ( , ) ( )d .

1

0

1

0

2 t =

∫

Gt

∫

G sq− s s<+∞

w ξ ξ ξ

By direct computation, we know that w₁(t) and w₂(t) are positive solutions of the following BVP:

   

= ′′ = ′′ = =

< <

= −

, 0 ) 1 ( ) 0 ( ) 1 ( ) 0 (

, 1 0 ), ( ) (

) 4 (

u u u u

t t p t u

and

   

= ′′ = ′′ = =

< < =

− ₋

, 0 ) 1 ( ) 0 ( ) 1 ( ) 0 (

, 1 0 ), ( ) (

) 4 (

v v v v

t t q t v

respectively.

We consider the following ordinary differential system

   

    

 

= ′′ = ′′ = =

= ′′ = ′′ =

= − +

− =

+ −

− =

+ ∗

∗+ ∗

∗

0 ) 1 ( ) 0 ( ) 1 (

) 0 ( ) 1 ( ) 0 ( ) 1 ( ) 0 (

), ( ) )] ( ) ( [

, )] ( ) ( [ , ( ) (

), ( ) )] ( ) ( [

, )] ( ) ( [ , ( ) (

2 1 )

4

( 2

1 )

4 (

y y y

y x x x x

t q t w t y

t w t x t g t y

t p t w t y

t w t x t f t x

λ λ

(2.3)

It is known that(x,y)∈(C2[0 ,1]∩C4(0 ,1))×(C2[0 ,1] ))

1 , 0 (

4

C

∩ is a solution of system (2.3) if and only if ]

1 , 0 [ ] 1 , 0 [ ) ,

(x y ∈C ×C is a solution of the following nonlinear integral equations system

  

   

 

+ −

− =

+ −

− =

+ ∗ ∗

+ ∗ ∗

∫

∫

∫

∫

, d )] ( ) )] ( ) ( [ , )] (

) ( [ , ( [ s) , ( d ) , ( ) (

, d )] ( ) )] ( ) ( [ , )] (

) ( [ , ( [ ) , ( d ) , ( ) (

2 1

1 0

1 0

2 1

1

0

1

0

s s q s w s y s w

s x s g G t

G t y

s s p s w s y s w

s x s f s G t

G t x

λ ξ ξ ξ

λ ξ ξ ξ

(2.4)

Let

, d )] (

) )] ( ) ( [ , )] ( ) ( [

, ( [ s) , ( d ) , ( ) )( , (

2 1

1

0

1

0

s s p

s w s y s w s x

s f G t

G t y x A

+

∗ ∗

+

− −

=

_∫

ξ ξ

_∫

ξ λ

, d )] (

) )] ( ) ( [ , )] ( ) ( [

, ( [ s) , ( d ) , ( ) )( , (

2 1

1

0

1

0

s s q

s w s y s w s x

s g G

t G t y x B

+

∗ ∗

+

− −

=

∫

ξ ξ

∫

ξ λ

Define a nonlinear integral operator F:X →X, by )).

, ( ), , ( ( ) ,

(x y A x y B x y

F = Thus, system (2.4) is equivalent to the fixed point equationF(x,y)=(x,y) in the Banach spaceX =C([0,1];R)×C([0,1];R).

The proof of main results will be based on the following lemmas.

Lemma 2.1. Let

Ω

₁ and

Ω

₂ be two bounded open sets in Banach space E such that θ∈Ω₁, Ω₁⊂Ω₂. A:

P

i. Au ≤ u,∀u∈P∩∂Ω₁; Au ≥ u ,∀u∈P∩∂Ω₂.

ii. Au ≥ u ,∀u∈P∩∂Ω₁; Au ≤ u,∀u∈P∩∂Ω₂.

ThenAhave a fixed point in P∩(Ω₂ \Ω₁).

Lemma 2.2. If f(t,x,y) and g(t,x,y) satisfy(H₁), ),

(H₂ thenf andgare nondecreasing in bothx,y∈[0,+∞) for any fixedt∈(0,1), we have

, ) , (

) , , ( lim

0

+∞ =

≥ +∞ → _{x y}

y x t f

y

x _{( , )} ,

) , , ( lim

0

+∞ =

≥ +∞ → _{x y}

y x t f

x y

, ) , (

) , , ( lim

0

+∞ =

≥ +∞ → _{x y}

y x t g

y

x =+∞

≥ +∞ → _{( , )}

) , , ( lim

0

y x

y x t g

x

y ,

where (x,y) = x+ y.

Proof.For any fixedt∈(0 ,1),0<x₁<x₂,it follows from

) (H₁ that

), , , ( ) , , ( ) (

) , , ( ) , , (

2 2

2 1

2 2 1 1

1_{f t x y f t x y} x

x

y x x x t f y x t f

≤ ≤

=

µ

Thus,f(t,x,y)is nondecreasing inx∈[0 ,+∞).

In the same way, we knowf(t,x,y)is nondecreasing in

). , 0 [ +∞ ∈

y g(t,x,y)are nondecreasing inx,y∈[0,+∞)for any fixedt∈(0,1).

On the other hand, for anyx>1,y≥0, it follows from

),

(H₂ when x≥y,we have

, 0 ) 0 , 1 , ( 2 1 2

) , 1 , (

) , , ( ) , (

) , , (

1

1 1

> ≥

≥ + =

− _{f t}

x x

y t f x

y x

y x t f y x

y x t f

µ µ

whenx< y, we havey>1 and

, 0 ) 1 , 0 , ( 2

1 2

) 1 , , (

) , , ( ) , (

) , , (

1

2 2

> ≥

≥ + =

− _{f t}

y y

x t f y

y x

y x t f y x

y x t f

µ µ

Therefore, we obtain =+∞

≥ +∞ → _{( , )}

) , , ( lim

0 y x

y x t f

y

x .

In the same way , we have

, )

, (

) , , ( lim

0

+∞ = ≥

+∞ → _{x y}

y x t f

x y

,

) , (

) , , ( lim

0

+∞ = ≥

+∞ → _{x y}

y x t g

y x

. )

, (

) , , ( lim

0

+∞ = ≥

+∞ → x y

y x t g

x y

Lemma 2.3.[10] If (u,v)withu(t)>w₁(t),v(t)>w₂(t) for anyt∈(0,1)is a positive solution of system (2.4), then

) ,

(u−w₁ v−w₂ is a positive solution of the semipositone singular differential system (1.1), (1.2).

Lemma 2.4. Suppose that (H₁),(H₂) hold , Then F:

P

P→ is a completely continuous operator.

Proof. For any fixed(x,y)∈P,choose0<a,b<1,such that a x <1,b y <1,then

, 1 )

( )] ( ) (

[xt −w1 t ∗≤ax t ≤a x < a

. 1 )

( )] ( ) (

[yt −w2 t ∗≤byt ≤b y < b

Hence, by(H1)we have

). 1 , 1 , (

) )] ( ) ( [ , )] ( ) ( [ , (

2 1 2 2 1 1

2 1

t f y x b a

t w t y t w t x t f

µ µ λ µ λ µ− −

∗ ∗

≤

− −

Consequently, for anyt∈[0 ,1],we have

. d )] ( ) 1 , 1 , ( )[ , (

) 1 (

4 1

d )] ( ) 1 , 1 , (

)[ , ( 4 1

d )] ( ) )] ( ) ( [

, )] ( ) ( [ , ( )[ , ( 4 1 ) )( , (

1 0 1 0

2 1

0 1

2 1 2 2 1 1

2 1 2 2 1 1

+∞ < +

⋅ + ≤

+

⋅ ≤

+ −

− ≤

∫

∫

∫

+ − −

+

− −

+ ∗

∗

s s p s f s s G

y x b a

s s p s f

y x b a s s G

s s p s w s y

s w s x s f s s G t

y x A

µ µ λ µ λ µ

µ µ λ µ λ µ

λ

λ λ

In the same way, we also haveB(x,y)(t)<+∞.Thus F:

X

P→ is well defined.

There exists a t₁∈[0,1]such that

. ) , ( ) )( ,

(x y t₁ A x y

A =

SinceG(t ,s)≥t(1−t)G(t₁ ,s),∀t ,s∈[0 ,1],then, we have

. ) , ( ) 1 ( ) 1 ( ) )( , (

d )] (

) )] ( ) ( [ , )] ( ) ( [ , ( [

) , ( )d , ( ) 1 ( ) )( , (

1

2 1

1

0 1

0 1

y x A t t t t t y x A

s s p

s w s y s w s x s f

s G t

G t t t y x A

− = − ≥

+

− −

⋅ −

≥

+

∗ ∗

∫

∫

λ

ξ ξ ξ

In the same way, there exists a t₂∈[0 ,1]such that )

, ( ) )( ,

(x y t₂ B x y

B = .

Using the same way as the above proof, we also have .

) , ( ) 1 ( ) )( ,

(x y t t t B x y

B ≥ −

Then, F(P)⊂P.

By proceeding as for the proof of Lemma 2.4 in [10],F:

P

3. Main Result

Theorem 3.1. Suppose that (H₁),(H₂)are satisfied, then two-point boundary value problem (1.1), (1.2) has at least a

)) 1 , 0 ( ] 1 , 0 [ ( )) 1 , 0 ( ] 1 , 0 [

(C2 ∩C4 × C2 ∩C4 positive solution. Proof.Choose R₁ such that

. }

1 , ,

max{ 1 1 1 3

1 r <R <m− _n

r _λ

Let

{

( , ) ₁ , ₁

}

,

1 x y P x R y R

R = ∈ < <

Ω

{

,

}

,

max x y

z = then∀(x,y)∈P∩∂ΩR₁,we can obtain 1

>

z and

, ) 2 ( 4 1 ) 2 ( 4 1 ) 1 ( 4 1 ) 1 ( 4 1 ) ( 4 1 ) ( 4 1 ) 1 ( 4 1 ) 1 ( 4 1 d )] ( ) 1 , 1 , ( )[ , ( ) 1 ( 4 1 d )] ( ) 1 , 1 , ( )[ , ( ) 1 ( 4 1 d )] ( ) 1 , 1 , ( )[ , ( 4 1 d )] ( ) 1 , 1 , ( )[ , ( 4 1 d )] ( )) ( ), ( , ( [ ) , ( ) , ( d )] ( )) ( , ) ( , ( [ ) , ( ) , ( ) )( , ( ) )( , ( ) )( , ( 1 1 1 1 1 4 1 1 2 1 1 1 1 4 1 1 1 1 2 4 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 4 3 2 1 4 3 2 1 R R R R R n r R R n r R R R r R R R r r z r z s s q s g s s G z s s p s f s s G z s s q s g z s G s s p s f z s G s s q s y s x s g s G d t G s s p s y s x s f s G d t G t y x B t y x A t y x F m m m m = + ≤ + + + ≤ + + + ≤ + + + ≤ + + + + + ≤ + + + ≤ + + + ≤ + = − − + + + + + + + + + +

∫

∫

∫

∫

∫

∫

∫

∫

λ λ λ λ λ λ λ λ λ λ λ ξ λ ξ λ ξ ξ ξ λ ξ ξ ξ λ λ λ λ λ λ λ λ

As a consequence,

, ) , ( ) ,

(x y x y x y

F ≤ + = ( , ) .

1

R

P y

x ∈ ∂Ω

∀ ∩

We choose constant L,M such that

, )] d ) , ( min )( ( ) 1 ( [ 3 4 1 ] , [ 2 2

∫

− ∈ − − > β α β

α ξ ξ

α β β

λα Gt

M t . 1 R L>

From Lemma 2.2, whenx≥L,y≥0,we have

, ) , ( ) , , ( M y x y x t f ≥

that is, f(t,x,y)≥M (x,y).

In the same way, whenx≥L,y≥0,we haveg(t,x,y) )

, (x y M

≥ , (where[α ,β]∈(0 ,1)).

Choose , ) 1 ( 3 4

2 _{α β}

− > L

R then R₂>L>R₁, thus 1.

2 1 _< R R

Let

{

( , ) ₂ , ₂

}

,

2 x y P x R y R

R = ∈ < <

Ω

Then∀(x ,y)∈P∩∂Ω_R2, x and y have at least a R₂. Noticing that . ) 1 ( 4 1 d ) ( ) 1 ( 4 1 d ) ( ) , ( d ) , ( ) ( 1 0 1 1 0 1 0 1

∫

∫

∫

− ≤ − ≤ = − − r t t s s p t t s s p s G t G t

w ξ ξ ξ

In the same way, ₂ (1 )₃ 4 1 )

(t t t r

w ≤ − .

1. Both

x

and

y

are

R

₂,

. ) 1 ( 4 3 ) 1 ( 4 3 ) ( 4 ) ( 4 ) ( ) ( ) 1 ( 4 1 ) ( ) 1 ( 4 1 ) ( ) ( ) ( 2 2 1 1 1 1 1 L R x t t t x R R t x R x t x t x R t t t x r t t t x t w t x ≥ − ≥ − ≥ − ≥ − ≥ − − ≥ − − ≥ − β α

In the same way,y(t)−w₂(t)≥L.

2. One of x and y isR₂,without loss of the generality, let x =R₂,we havex(t)−w₁(t)≥L,and[y(t)−w₂(t)]∗≥0.

This together with Lemma 2.2 yields

. ) , ( ], , [ , d ) , ( ) ( ) 1 ( 4 3 d ) , ( ) ( ) 1 ( 4 3 d ) ( ) ( ) , ( ) , ( d ) ( ) ( ) , ( ) , ( ds ) )] ( ) ( [ ), ( ) ( ( ) , ( ) , ( ds ) )] ( ) ( [ ), ( ) ( ( ) , ( ) , ( ) )( , ( ) )( , ( ) )( , ( 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1 R P y x t R R t G R M t G R M s s w s x M s G d t G s s w s x M s G d t G s w s y s w s x M s G d t G s w s y s w s x M s G d t G t y x B t y x A t y x F Ω ∂ ∈ ∀ ∈ ∀ + ≥ − − + − − ≥ − + − ≥ − − ⋅ + − − ⋅ ≥ + =

∫

∫

∫

∫

∫

∫

∫

∫

∫

∫

∗ ∗ ∩ β α ξ ξ α β β α λ ξ ξ α β β α λ λ ξ ξ ξ λ ξ ξ ξ λ ξ ξ ξ λ ξ ξ ξ β α β α β α β α β α β α β α β α β α β α Thus, . ) , ( , ) , ( ) , ( 2 R P y x y x y x y x

As a consequence, by lemma 2.1Fhas at least one fixed point ( , ) ( \ )

1 2

0

0 y P R R

x ∈ ∩ Ω Ω with R₁≤ x₀ ≤R₂, and

2 0

1 y R

R ≤ ≤ .

Hence, for anyt∈(0,1),we have

). 1 , 0 ( , 0 ) 1 ( 4 1 ) 1 (

d ) ( ) , ( d ) , ( ) 1 (

) ( ) (

1 0

1

0

1

0 0

1 0

∈ ∀ > − − − ≥

− − ≥

−

∫

∫

−

t r

t t t t x

s s p s G t

G t t x

t w t x

ξ ξ ξ

In the same way, we have

). 1 , 0 ( , 0 ) ( ) ( ₂

0 t −w t > ∀t∈

y

It follows from lemma 2.3 that(x₀−w₁,y₀−w₂)is one positive solution of the boundary value problem (1.1), (1.2) in(C2[0 ,1]∩C4(0 ,1))×(C2[0 ,1]∩C4(0 ,1)).

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