The Existence and Uniqueness of Positive Solutions for a Singular Nonlinear Three Point Boundary Value Problems

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ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2018.612217 Dec. 26, 2018 2600 Journal of Applied Mathematics and Physics

The Existence and Uniqueness of Positive

Solutions for a Singular Nonlinear Three-Point

Boundary Value Problems

Yao Dong, Baoqiang Yan

*

School of Mathematical Sciences, Shandong Normal University, Jinan, China

Abstract

Using the method of lower and upper solutions, we study the following sin-gular nonlinear three-point boundary value problems:

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

(0) 0, (1) ( ),

q p

x t K t x t x t t

x x ax

λ

η

−

 ′′− + = ∈



= =

 , where K C∈ [0,1], 0< <a 1,

0< <η 1 and λ_{is a positive parameter and present the existence,}

unique-ness, and the dependency on parameters of the positive solutions under vari-ous assumptions. Our result improves those in the previvari-ous literatures.

Keywords

Three-Point Boundary Value Problem, Positive Solution, Lower and Upper Solutions, Eigenvalue and Eigenfunction

1. Introduction and Main Results

In this paper, we consider the three-point boundary value problem

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

(0) 0, (1) ( ),

q p

x x ax

λ

η

−

 ′′− + = ∈

 ₌ ₌

 (1.1)

where K C∈ [0,1], 0< <a 1_, 0< <η 1, and λ_{is a positive parameter.}

The m-point boundary value problem for linear second-order ordinary diffe-rential equations was initiated by Ilin and Moiseev [1] [2]. Since then, there are many results on the existence of general nonlinear multi-point boundary value problems, see [3] [4] [5] [6] and their references. For examples, in [6], Rynne studied the $m$-point boundary value problem

How to cite this paper: Dong, Y. and Yan, B.Q. (2018) The Existence and Uniqueness of Positive Solutions for a Singular Nonli-near Three-Point Boundary Value Prob-lems. Journal of Applied Mathematics and Physics, 6, 2600-2620.

https://doi.org/10.4236/jamp.2018.612217

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DOI: 10.4236/jamp.2018.612217 2601 Journal of Applied Mathematics and Physics 2

1

( ), (0,1), ,

(0) 0, (1) m i ( ),i

i

u f u on u R X

u u − α ηu

= ′′

− = ∈ ×

 

 ₌ ₌



∑

where m≥3_,η ∈i (0,1), α >i 0 with

2

1 1

m i i

α

− =

<

∑

and presented the existence

of the sign changing solutions by Rabinowitz bifurcation theorem. Especially, Rynne ([7]) discussed the three-point boundary value problem

( ) , (0,1),

(0) 0, (1) ( ),

u f u h on u u α ηu

′′

− = +



 ₌ ₌



and showed the solvability and non-solvability results from either the half-eigenvalue or the Fucik spectrum approach. As we known, the method of upper and lower solutions is very important for the study of the boundary val-ue problems, see [8]-[18]. Therefore, establishing the method of upper and lower solutions for three-point boundary value problems is necessary and im-portant.

In [19], when f is nondecreasing on x, Du and Zhao got the methods of upper and lower solutions of

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

(0) ( ), (1) 0,

x t f t x t t x axη x

′′

− = ∈



 ₌ ₌



and used iterative techniques to study the existence of positive solutions. And in [3] when f is decreasing on u, Du and Zhao considered the existence and uni-queness of positive solutions of the problem

2

1

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

(0) m i ( ),i (1) 0

i

u t f t u t t

u − α ηu u

= ′′

− = ∈

 

 ₌ ₌



∑

by constructing lower and upper solutions. Wei ([15]) constructed the method of upper and lower solutions for three-point boundary value problems and gave the sufficient and necessary conditions for the existence of positive solutions of the problem

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

(0) ( ), (1) 0.

x t f t x t t x axη x

′′

− = ∈



 ₌ ₌



On the other hand, singular boundary problems arise in the contexts of chemical heterogeneous catalysts, non-Newtonian fluids and also the theory of heat conduction in electrically conducting materials, see [20]-[25] for a detailed discussion. An interesting result comes from [25], in which, using method of upper and lower solutions, Shi and Yao discussed the following problem

( ) , ,

( ) 0, ,

| 0,

q p

u K x u u x u x x

u

λ

−

∂Ω

−∆ + = ∈ Ω

 _> _{∀ ∈ Ω}



 ₌



where _{K C}_∈ 2,β_{( )}_Ω _, _{p q}_, _∈_(0,1)_and _λ_{is a positive parameter. Under}

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DOI: 10.4236/jamp.2018.612217 2602 Journal of Applied Mathematics and Physics

uniqueness of classical solutions.

Motivated by above works, under various appropriate assumptions on p, q

and K t( ), we will obtain the existence and uniqueness of positive solution of problem (1.1) for λ_{in different circumstances. In our proof, the upper and}

lower solutions theorem (see [16]) plays an important role in the paper. Define

[0,1]

*

* _[0,1]

max ( ), min ( ).

t t

K K t K K t

∈ ∈

= =

The main results of this paper are stated in the following theorems.

Theorem 1.1. When K*>0,

1) If 0< p q, <1,thereexists λ >0 _such_that_the_problem_(1.1)_has_at_least oneC[0,1]positivesolution x tλ( ) for λ λ> .

2) For λ λ> , (1.1) has amaximal solution x tλ( ) and x tλ( ) is increasing

withrespectto λ.

Theorem1.2. When _K*_<₀_,

1) If 0< p<1,0<q, (1.1) has at least one C[0,1] positive solution for all

0

λ> .

2) If 0< p q, <1, (1.1) hasanunique _C1_[0,1] _positive_solution _{x t}_{( )} λ forall

0

λ> _.

3) x tλ( ) in (2) isincreasingwithrespectto λ.

Theorem1.3. When *

* 0

K < <K ,

1) If 0< p q, <1, there exists a λ >_* 0 _such _that _the _problem_(1.1)_has _at

leastoneC[0,1] posit-ivesolution x tλ( ) for λ λ> *.

2) For λ λ> *, x tλ( ) in (1) isincreasingwithrespectto λ.

Remark 1.1: Note K t( ) 0> _{in Theorem (1.1). This is different from the}

conditions in [3] [15] [19] because K t( ) 0< _{in these references.}

Remark 1.2: The unique result in Theorem 1.2 is different from that in [3] because we remove the monotonicity of nonlinearity f in x.

Remark 1.3: Note K t( ) is sigh-changing in Theorem 1.3. This is different from the conditions in [3] [15] [19] because K t( ) 0< _{in these references and is}

different from conditions in [1] [2] [4] [5] [6] [7] [26] because f is continuous at 0

x= in these references.

This paper is organised as follows. Some preliminary lemmas are stated and proved in Section 2. And Section 3 is devoted to prove the results.

2. Preliminaries

In this section, we first consider the following problem ( ) ( , ( ), ( )), (0,1),

(0) 0, ( ) (1),

x t f t x t x t t x xη ax

′′ ′

− = ∈



 ₌ ₌

 (2.1)

where η ∈(0,1)_, 0< <a 1_and f ∈[0,1]× ×_{ }_.

Let _C1_{[0,1] { :[0,1]}₌ _x _→__{| ( )}_{x t} _{is differential continuous on} _[0,1]}_with

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|| || max{| | ,| | }x = x ∞ x′∞ ,

where | | max | ( ) |_[0,1] t

x′ =∞ _∈ x t . Obviously, C1[0,1] is a Banach space. Now we give the definitions of lower and upper solutions for problem (2.1).

Definition 2.1. A function α( )t is called a lower solution to the problem (2.1), if _α_{( )}_t _∈_C_[0,1]_∩_C2_(0,1) _{and satisfies}

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

(0) 0, (1) ( ).

t f t t t t a

α α α

α α α η

′′ ′

− ≤ ∈



 _≤ _≤

 (2.2)

Upper solution is defined by reversing the above inequality signs in problem (2.2).

If there exists a lower solution α( )t and an upper solution β( )t to problem (2.1) such that α( )t ≤β( )t _{, then}( ( ), ( ))α t β t _{is called a couple of upper and} lower solutions of problem (2.1).

Set Dβ {( , ) (0,1)t x , ( )t x ( ),t t (0,1)}.

α = ∈ ×+ α ≤ ≤β ∈

We list a lemma for the eigenvalues and eigenfunctions for the following li-near problem

( ) ( ), (0,1),

(0) 0, (1) ( ).

x t x t t x x ax

λ

η

′′

− = ∈



 ₌ ₌

 (2.3)

Lemma 2.1. (see [6]) The spectrum σ( )L _of _problem₍₂_.₃₎_consists _of _a

strictly increasing sequence ofeigenvalues λ >k 0, k=1, 2,, with

eigenfuc-tions _sin( 12 ₎

k kt

φ = λ _._In_addition_,

1) lim _k

k→+∞λ = +∞;

2) φk( )t has exact k−1 simple zeros in (0,1), k=2,3, and φ1 is

strictlypositiveon (0,1).

Lemma 2.2. Supposethat _{h L}_∈ 1_(0,1)_._Then_,_for_each _λ_>₀_,_the_problem

( ) ( ), (0,1),

(0) 0, ( ) (1)

x t x h t t x x x

λ

η α

′′

− + = ∈



 ₌ ₌

 (2.4)

hasanuniquesolutioninC[0,1].

Proof. Assume that v t1( ) and v t2( ) satisfies that

( ) ( ), (0,1),

(0) 0, (0) 1

x t x h t t x x

λ

′′

− + = ∈



 ₌ _′ ₌

 and

( ) ( ), (0,1),

(1) 0, (1) 1

x t x h t t x x

λ

′′

− + = ∈



 ₌ _′ _{= −}

 respectively. Define

2 1

1 1

( ) ( ), 0 1,

1 ( , )

( ) ( ), 0 1,

v t v s s t G t s

v t v s t s

ω

≤ ≤ ≤ 

=  _{≤ ≤ ≤}

 and

1 ₁ 1

0 0

1 1

( )

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

(1) ( )

e t

x t G t s h s ds G s h s ds s

e α ηe α η

= + ∈

−

∫

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( ) ( )

1 0

2 1 1 2

1

2 1 1 2

1 1

0

1 1

2 1 1 2

1

1 1

( ) ( ) ( ) ( ) ( ) ( ) ]''

''( ) _{( , ) ( )}

(1) ( )

( ) _{( ,}

1

''( ) ( ) [

( )

1 [ '( ) ( ) '( ) ( )] ( )

1 [ ( ) (

) (1) ( ) ] ) 0 t t t

x t x t

x t

v t v t v t v t h t t

v t v s h s ds v t v s

v t v s h s ds v t v

h s d s h s ds

e t _G _{s h s ds}

e e

e t _G

e s s h e λ α ω λ ω λ λ ω η α η

λ _α _η

α η − + = − = − − − + − + − + − −

∫

1 0

2 1 1 2

1 1 0 1 0 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] ( ) _{( , ) ( )} ( ) 1 ( ) [ ( ) ( ),

(1) ( )

(0,1)

t t

s ds

v t v s h s ds v t v s h s ds

x t

h t

x t

h t

e t _G _{s h s ds}

e e

t

λ

λ ω

λ α η λ

α η + + − = − − = ∈ +

∫

and

1 ₁ 1

0 0

1 1

1 ₁ 1

0 0 1 1 1 1 0 1 1

1 ₁ 1

0 0

1 1

(1)

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

(1) ( )

( )

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

(1) ( )

(1) _{( , ) ( )}

(1) ( )

( )

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

(1) ( )

e

x x G s h s ds G s h s ds

e e

e

G s h s ds G s h s ds

e e

e _G _{s h s ds}

e e

e

G s h s ds G s h s ds

e e

α η α η

α η η

α η α η

α η

η

α η α η

α η − = + − − + − = − − + −

∫

0. =

Hence, x t( ) is a C[0,1] solution to problem(2.4). Since λ>0, Lemma 2.1 guarantees that problem (2.4) has an unique C[0,1] solution. The proof is com-plete. 

Theorem 2.1. Let

α

_and _β_∈_C_([0,1])_∩_C1_(0,1) _be _lower _and_upper

solu-tions of (2.1) such that α β≤ . Let ψ∈L1[0,1] _and φ : + _→ +₀

  be a conti-nuousfunctionthatsatisfies

0

1 _.

( )s ds

φ ∞

= +∞

∫

(2.5)

Suppose f D: β

α× →  isan L1-Carathéodory-functionsuchthat

| ( , , ) |f t x v ( ) (| |), ( , )t v t x Dβ, v .

α ψ φ

≤ ∀ ∈ ∈_ (2.6)

Thentheproblem (2.1) hasatleastonesolution _{x C}_∈ 1_[0,1] _such_that_for_all [0,1]

t∈ ,

( )t x t( ) ( ).t

α ≤ ≤β

Proof. The proof proceeds in five steps.

Step 1. We consider a new modified problem. From (2.5), there is an R>0 be large enough so that

1 0

1 _{|| || .}

( ) R

ds

s ψ

φ >

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And (2.6) guarantees that there is an N t( ) with _{N L}_∈ 1_[0,1]_{such that}

| ( , , ) |f t x v N t( ), ( , )t x Dβ, | |v R.

α

≤ ∀ ∈ ≤ (2.8)

Define then

( ), ( ),

( , ) , ( ) ( ),

( ), ( )

t if x t t x x if t x t

t if x t

α α

χ α β

β β

< 



=_ ≤ ≤

 _>



(2.9)

and

( , , ) max{min{ ( , ( , ), ), ( )}, ( )}.

g t x v = f t χ t x v N t −N t (2.10) Choose a λ >0 and consider the new boundary value problem

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

(0) 0, (1) ( ),

x t x g t x t x t t x t t x x ax

λ λχ

η

′′ ′

− + = + ∈



 ₌ ₌

 (2.11)

where 0< <a 1_, 0< <η 1.

Step 2. We discuss the existence of a _C1_[0,1]_{solution of (2.11).}

Now Lemma 2.2 guarantees that for each _{h L}_∈ 1_[0,1]_{, the linear problem}

( ) , (0,1),

(0) 0, (1) ( )

x t x h t x x ax

λ

η

′′

− + = ∈



 ₌ ₌

 has an unique C[0,1] solution

1 ₁ 1

0 0

1 1

( )

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

(1) ( )

e t

v t G t s h s ds a G s h s ds s

e ae η η

= + ∈

−

∫

For _{x C}_∈ 1_[0,1]_{, define}

( )( )Fx t =g t x t x t( , ( ), ( ))′ +λχ( , ( )),t x t t∈[0,1] and

1 ₁ 1

0 ₁ ₁ 0

( )

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

(1) ( )

η

= + ∈

−

∫

e t

∫

Tx t G t s Fx s ds a G s Fx s ds s

e ae

From (2.9) and (2.10), we have

[0,1] [0,1]

| ( , ( ), ( )) ( , ( )) | ( ) max{sup | ( ) |, sup | ( ) |},

t t

g t x t x t

λχ

t x t N t

λ

α

t

β

t

∈ ∈

′ + ≤ + _which

implies that the functions belonging to _{{( )( ) :}_{Tx t x C}_∈ 1_[0,1]}_and 1

{( ) ( ) :Tx t x C′ ∈ [0,1]} are bounded and equicontinuous. The Arzela-Ascoli Theorem guarantees that _TC1_[0,1]_{is relatively compact. The proof of the}

con-tinuity of T is standard. Using the Schauder’s fixed point theorem, we assert that

T has at least one fixed point _{x C}_∈ 1_[0,1]_.

Step 3. The solution x of (2.11) is such that α( )t ≤x t( )≤β( )t .

We prove that x t( )≤β( )t _for t∈[0,1] only. In fact, suppose that there ex-ist a t0∈[0,1) such that x t( )0 >β( )t0 . Since x(0) 0= ≤β(0), t0>0. Let

( ) ( ) ( )

w t =x t −β t _, t∈[0,1]. Then w(0) 0≤ _and w t( ) 0₀ > _.

Let t∗=sup{∣t w s( ) 0,> s t t∈[ , ]}0 , t∗=inf{∣t w s( ) 0,> s t t∈[ , ]}.0

It is obvious that w t( ) 0> _{for all} * *

( , )

t∈ t t , w t( ) 0* = and w t( ) 0* ≥ . If *

( ) 0

w t = , then there exists a *

*

( , )

t′∈ t t such that _*

*

[ , ]

( ) max ( )

t t t

w t w t

∈

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DOI: 10.4236/jamp.2018.612217 2606 Journal of Applied Mathematics and Physics *

( ) 0

w t > _{, obviously} _t*₌₁_and _w₍₁₎₌_x₍₁₎₋_β_{(1 0}₎_> _{. Since}

1 1

( ) ( ) ( ) ( (1) (1)) (1) (1)

w x x w w

a a

η = η −β η = −β = > _{, there exists} *

*

( , )

t′∈ t t

such that _*

*

[ , ]

( ) max ( )

t t t

w t w t

∈

′ = also. Hence, w t′ ′ =( ) 0₍_i.e._, β ′ ′( )t =x t′ ′( )_{) and} ( ) 0

w t′′ _′

− ≥ . On the other hand, since

( ) ( ) ( )

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

( , ( ), ( )) max{min{ ( , ( ), ( )), ( )}, ( )}

( ) ( )

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

w t t x t

f t t t g t x t x t t x t x t f t t t f t t t N t N t

t x t

f t t t f t t t t

β

β β λχ λ

β β β β

λβ λ

β β β β λβ

′′ _′ ′′ _′ ′′ _′

− = −

′ ′ ′ ′ ′ ′ ′ ′ ′

≤ − + + −

′ ′ ′ ′ ′ ′ ′

= − + −

′ ′

+ −

′ ′ ′ ′ ′ ′ ′ ′ ′

= − + + ( )

( ( ) ( )) 0.

x t t x t

λ λ β

′ −

′ ′

= − <

This is a contradiction.

A similar argument holds to prove x t( )≤β( )t _{for all} t∈[0,1]. Hence, from (2.10), one know that x satisfies that

( ) ( , ( ), ( )) max{min{ ( , ( ), ( )), ( )}, (0,1),

(0) 0, (1) ( ).

x t g t x t x t f t x t x t N t t

x x ax

η

′′ ′ ′

− = = ∈

 ₌ ₌

 (2.12)

Step 4. The solution x of (2.11) is such that | |x′ ≤∞ R.

On the contrary, suppose that there is a t′∈(0,1)_{such that}| ( ) |x t′ ′ >R_. Without loss of generality, we assume that x t′ ′ >( ) R. Since x(0) 0= _and

(1) ( )

x =axη _with 0< <a 1_{, there is a} t₀∈(0,1)_{such that} x t′( ) 0₀ = _{. Without}

loss of generality, we assume that x t′( ) 0> _{for all}( , )t t′ ₀ _{. Observe that, for all}

( , )t x Dβ α

∈ , v∈__,

max{min{ ( , , ), ( )},f t x v N t −N t( )}≤ψ φ( ) (| |).t v

Then, from (2.12), one has

0

0 0

( )

0 ( )

1

1 _| 1 _{| |} 1 _{( ) |}

( ) ( ) ( ( ))

( ) ( , ( ), ( ))

| | | |

( ( )) ( ( ))

( ) ( ( )) _{( )} _{|| || .}

( ( ))

R x t t

x t t

t t

ds ds dx t

s s x t

x t _dt g t x t x t _dt x t x t t _{x t dt} _{t dt}

x t

φ φ φ

φ φ

ψ φ _ψ _ψ

φ

′ ′

′

= =

′

′′ ′

= =

′ ′

′

= = =

′

∫

This contradicts to (2.7).

Hence | ( , ( ), ( )) |f t x t x t′ ≤N t( )_{, which together with}u∈[ , ]α β guarantees that

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

g t x t x t′ = f t x t x t′ ∀ ∈t

Step 5. We claim that x t( ) satisfies (2.1).

Since | |x′ ≤∞ R and α( )t ≤x t( )≤β( )t , by (2.8), (2.10) and (2.12), we have

( ) max{min{ ( , ( ), ( )), ( )} ( , ( ), ( )), (0,1),

(0) 0, (1) ( ),

x t f t x t x t N t f t x t x t t

x x ax

η

′′ ′ ′

− = = ∈

 ₌ ₌



that is, x t( ) is a _C1_[0,1]_{solution of (2.1). The proof is complete.}_

Now we consider the following problem

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

(0) 0, ( ) (1),

x t f t x t t x xη ax

′′

− = ∈



 ₌ ₌

(8)

where η ∈(0,1), 0< <a 1 and f ∈[0,1]× ×_{ }_.

Now we give the definitions of lower and upper solutions for problem (2.13). Definition 2.2. (see [16]) A function α( )t is called a lower solution to the problem (2.13), if _α_{( )}_t _∈_C_[0,1]_∩_C2_(0,1)_{and satisfies}

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

(0) 0, (1) ( ).

t f t t t a

α α

α α α η

′′

− ≤ ∈



 _≤ _≤

 (2.14)

Upper solution is defined by reversing the above inequality signs in problem (2.14).

By Theorem 2.1, we have following result.

Corollary 2.1. Suppose that there exists a lower solution α( )t _{and an upper} solution β( )t of problem (2.1) such that α( )t ≤β( )t _, t∈[0,1] and there ex-ists _{F L}_∈ 1_[0,1]_{such that} _{| ( , ) |}_{f t x} _≤_{F t}_{( )}_{for all} _{( , )}_{t x} _Dβ

α

∈ _{. Then the problem}

(2.13) has at least one C[0,1] solution x t( ) satisfies α( )t ≤x t( )≤β( )t _,

[0,1]

t∈ .

Remark 2.1: This result can be found in [15]. So our theorem improves the works in the previous literature.

Lemma 2.3. Suppose that f : (0,1) [0,× +∞ →) __{is a continuous functions} such that _{s f t s}−1 _{( , )}_{is strictly decreasing for}_s_>₀_{at each}_t_∈_(0,1)_{. Let}

2

, [0,1] (0,1)

w v C∈ ∩C satisfies:

1) w′′+ f t w( , ) 0≤ ≤ +v′′ f t v( , )_, t∈(0,1)_;

2) w v, >0, t∈(0,1)_and w(0)≥v(0)_, w(1)≥aw( )η _, v(1)≤av( )η _;

3) _v_′′∈_L1_[0,1]_.

Then w t( )≥v t( )_, t∈[0,1].

Proof. By _v_′′∈_L1_(0,1)_{, we know that}_v_{′ +}_{(0 )}_and _v_{′ −}_{(1 )}_{exist and then} 1_[0,1]

v C∈ .

Suppose conversely v t( )≤/w t( ) on [0,1]. We may assume without loss of ge-nerality that there exists t0∈(0,1) such that v t( )0 −w t( ) max( ( )0 = ₀_{≤ ≤}_t ₁ v t −w t( )) 0> .

Let

* inf{ | 01 1 0, ( ) ( ), ( , )},1 0

t = t ≤ <t t v t >w t t∈ t t

*

2 0 2 0 2

sup{ | 1, ( ) ( ), ( , )}.

t = t t ≤ <t v t >w t t∈ t t

It’s obvious that *

*

0≤ < ≤t t 1 and v t( )* =w t( )* , v t′ + ≥( )* D w t+ ( )*+ , where

D+_{denote Dini derivatives.}

For _t*_≤₁_{, there are three cases.}

1) _t*_<₁_{. Then} _{v t}_{( )}* ₌_{w t}_{( )}* _, _{v t}_′_{( )}* _≤_{w t}_′_{( )}* _, _{v t}_{( )}_>_{w t}_{( )}_{for all} * *

( , )

t∈ t t _.

2) _t*₌₁ _and _{v t}_{( )}* ₌_{w t}_{( )}* _, _{v t}₍* ₎ _{D w t}₍* ₎

−

′ − ≤ − , v t( )>w t( ) _{for all}

* *

( , )

t∈ t t _{, where} D− denotes Dini derivatives.

3) _t*₌₁_and _{v t}_{( )}* _>_{w t}_{( )}* _, _{v t}_{( )}_>_{w t}_{( )}_{for all} * *

( , ]

t∈ t t . Since

(1) (1) ( ( ) ( )) ( ) ( )

v −w ≤a vη −wη <vη −wη _{, then there is} t′∈[ ,1]η _{such that}

( ) ( ) 0, ( ( ) ( )) 0.

v t_′ ₋w t_′ _> v t_′ ₋w t_′ ′_<

Combining above (1), (2) and (3), there is a t′ >t* such that

* * * *

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

v t w t v t D w t+ v t w t v t D w t

−

′ ′ ′ ′ ′ ′

(9)

and

*

( ) ( ), ( , ).

v t >w t ∀ ∈t t t′

Let y t( )=v t w t w t v t′( ) ( )− ′( ) ( )_, t∈( , )t t_* ′ _{. Then we have}

*

lim inf ( ) 0 lim sup ( ).

t t→ + y t ≥ ≥t t→ −′ y t (2.15)

On the other hand,

( ) ( ) ( ) ( ) ( )

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

( , ( )) ( , ( ))

( ) ( )( )

( ) ( )

0

y t w t v t w t v t

w t f t v t v t f t w t

f t w t f t v t

w t v t

w t v t

′ = ′′ − ′′

= − +

= −

≥

for t∈( , )t t* ′ and y t′( ) 0≡/ on ( , )α β . This implies y t( )′ > y t( )* . This

con-tradicts (2.15), so v t( )≤w t( )_{. The proof is complete.} _

By analogous methods in [19], we establish the following maximal theorem, which can be used in the proof of the uniqueness of positive solutions.

Lemma 2.4. (maximal theorem) Suppose that 0< <η 1, and

2

{ [0,1] (0,1), (1) ( ) 0, (0) 0}

F = x C∈ ∩C x −axη ≥ x ≥ , if x t( )∈F _{such that}

''( ) 0 x t

− ≥ _for t∈(0,1)_{, then} x t( ) 0≥ _for t∈[0,1].

3. Proofs of Main Theorems

In this section, we’ll always assume that _{f t x}( , )₌_λ_xp₋_{K t x}( ) −q_.

(A) The proof of Theorem 1.1. Proof.

1) We consider the problem

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

(0) 0, (1) ( ),

q p

x x ax

λ

η

−

 ′′− + = ∈

 ₌ ₌

 (3.1.1)

where 0<q p, <1, K C∈ [0,1], K*>0, 0< <a 1, 0< <η 1 and λ is a

positive parameter.

In [19], when f t x( , ) is increasing in x, the problem

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

(0) ( ), (1) 0

x t f t x t x axη x

′′

− = ∈



 ₌ ₌



has an unique _C1_[0,1]_{positive solution. From that, suppose that} *( )

x t is an unique _C1_[0,1]_{positive solution of the problem}

( ) ( ), (0,1),

(0) 0, (1) ( ),

p

x t x t t

x x ax

η

 ′′− = ∈

 ₌ ₌

 (3.1.2)

where 0< <a 1_, 0< <η 1. Set

1 1

*

( )_t p_{x t}( )

β

₌

λ

− _{. Then}

1

1 1

* *

1

1 1

* * *

1 1

*

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( ),

q

q p p q

q q

p p

t K t t x t K t x t

x t K x t

x t

β β λ λ

λ λ

λ

−

− − − −

− −

− ′′

− + = +

> +

(10)

1 *

( ) ( ).

p _t p_{x t}p

λβ

₌

λ

−

Thus ₋_β_′′( )_t ₊_{K t}( )_β−q( )_t _>_λβp( )._t _{Combining it with (3.1.2) we obtain}

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

(0) 0, (1) ( ).

q p

t K t t t t

a

β

λβ

β

β η

−

− ′′ + > ∈

 ₌ ₌



Consequently, β( )t _{is a upper solution of (3.1.1).} Set

2 1 1

( )_t _M q

α

₌

ϕ

+ _{, where}_M_{is a positive constant and} 1

ϕ is the first eigen-function. Then 1 1 2 1 2 2 1 1 2 1 2 1 2 1

1 1 2

2 1 1 1

2 1

1 1 2

2 1 1 2 * 1 1 2 2 1 1 1

2(1 ) | ' |

2 ( )

''( ) ( ) ( ) ( ) ''( )

1

(1 )

2(1 ) | ' |

2 ( )

1

(1 )

2(1 ) | ' |

2 _. (1 ) q q q q q q q q q q q q q q q q q q q q q M

M K t

t K t t t t

q

q M

q M

M K t

q M q q M M K M q ϕ

α α ϕ ϕ

ϕ ϕ

ϕ

λ ϕ ϕ

ϕ ϕ λ _ϕ ϕ ϕ − + + + + + + − + + + − − + = − + − + + − = + − + + −

< + −

+

By Lemma 2.1 we have

ϕ

1( ) sin(t =

λ

1t),

ϕ

1( )t =

λ

1cos(

λ

1t). Thus there

exists δ >0 0 and b∈(0,1) such that

1' 1 1 0

| ( ) | |

ϕ

t =

λ

cos(

λ

t) |>

δ

, t∈[0, ),b

1 1 0

| ( ) | | sin(

ϕ

t =

λ

t) |>

δ

, t b∈[ ,1].

a) On [0, )b , choosing

1 2 * 1 1 2 0 (1 ) [ ] 2(1 ) q q K M M q

δ

+ + ≥ =

− , then we have

2 * 1 1 1 2 1 1 1 q q q q M K q M λ _ϕ ϕ + + ≤ + .

b) On [ ,1]b , choosing

1 2 * 1 2 2 0 (1 ) [ ] 2(1 ) q q K M M q

δ

+ + ≥ =

− , then we have

2 * 1 1 1 2 1 1 . 1 q q q q M K q M λ _ϕ ϕ + + ≤ +

Fixing M=max{ ,M M1 2}, then

2 1 1 1 3 ( ) ( ) ( ) 1

q M q

t K t t

q

λ

α

_′′

α

−

ϕ

+

− + ≤ + and 2 1 1 ( ) . q

p _t _Mp q

λα

₌

λ

ϕ

+

Set

2 2 1

1 0 3₁ | |1

p q q M q

λ

ϕ

− − + ∞ =

(11)

2 2

1 1

1

1 1 0

3 _, _.

1

p p

q q

M _M

q

λ

_ϕ

+ _<

_λ

_ϕ

+ _{∀ >}

_{λ λ}

+

Hence, ₋_α_′′( )_t ₊_{K t}( )_α−q( )_t _<_λαp( )_t _,

0 λ λ

∀ > .

It follows from Lemma (2.1) that

2 1 1

(0) _M q(0) 0

α

₌

ϕ

+ ₌

and

2 2 2 2 2

1 1 1 1 1

1 1 1 1

(1) _M q(1) _{M a}[ ( )] q _Ma q q( ) _aM q( ) _a ( ).

α

₌

ϕ

+ ₌

ϕ η

+ ₌ +

ϕ

+

η

_<

ϕ

+

η

₌

α η

Set

1 1 1 1

2 1

*

( | | | | )

q p q M

x

ϕ

λ

ϕ

− − +

∞ ∞

= _{. Then} 12 11

1 *

( )_t _M q( )_t p_{x t}( ) ( )_t

α

₌

ϕ

+ _≤

λ

− ₌

β

_for

all λ λ> 2. Thus we choose λ =max{ , }λ λ0 2 and λ λ> , then ( ( ), ( ))α t β t is

a couple of upper and lower solutions of (3.1.1).

We choose _{F t}_{( )}₌_λβp₊_K*_β−q_{, then} _{| ( , ) |}_{f t x} _≤_{F t}_{( )}_{for all} _{( , )}_{t x} _Dβ α

∈ . It’s easy to see that _{F t}_{( )}_∈_L1_[0,1]_{. From Corollary 2.1, the problem (3.1.1) has at}

least one C[0,1] positive solution x t( ) satisfying α( )t ≤x t( )≤β( )t _for λ λ> . 2) (Existence of the maximal solution) We observe the problem

( ) ( ), (0,1),

(0) 0, (1) ( ).

p

x t x t t

x x ax

λ

η

 ′′− = ∈

 ₌ ₌

 (3.1.3)

From [19], we note the unique solution of (3.1.3) is w tλ( ) for any λ>0. In (1) we obtained the solution x tλ( ) of (3.1.1) then we have

( ) ( ) 0 ( ) (

' ' )

' p ' p

w tλ +λw tλ = <x tλ +λx tλ

and _{x f t x}1 _{( , )} _xp 1_{( )}_t λ λ

− ₌ − _{is decreasing in}_x_{. Noting that} _{x t}_{( )} _L1_[0,1]

λ ∈ by (1).

From Lemma 2.3, we have x tλ( )≤w tλ( ). Let

0

1

[ ,1)

j _i _j

Ω =

+ , j=1, 2, and w tj( ) be the solution of

1 1

1

0

( ) ( ) ( ) ( ), ,

1

( ) ( ), [0, ),

(1) ( )

q p

j j j

j

x t K t w t w t t

x t w t t

i j

x ax

λ

η −

− −

−

 ′′− + = ∈ Ω



 ₌ _∈

 ₊



 ₌



(3.1.4)

for j=1, 2,, with w t0( )=w tλ( ) defined in (3.1.3). Let x tλ( ) be a solution

of (3.1.1).

In (3.1.4), letting j=1 we have

1 1

1

0

1 1

( ) ( ) ( ) ( ), ,

1

( ) ( ), [0, ),

(1) ( ).

q p

w t K t w t w t t

w t w t t

i j

w aw

λ λ

λ

η −

− ′′ + = ∈ Ω



 ₌ _∈

 ₊



 ₌



(3.1.5)

Combining (3.1.5) with (3.1.3) we have w t w t1''( )− λ''( ) 0≥ for t∈ Ω1. By

maximum principle, we have w t1( )≤w t0( )=w tλ( ). Similarly, we can obtain that

1( ) ( ) ( )

j j

(12)

Furthermore, we observe problem (3.1.1)

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

(0) 0, (1) ( ).

q p

x x ax

λ

η

−

 ′′− + = ∈

 ₌ ₌



Combining it with (3.1.5) we have

1''( ) ''( ) ( )( q( ) q( )) ( ( )p p( )) 0,

w t x tλ K t w tλ− x tλ− λ w tλ x tλ

− + + − = − ≥

thus x t w tλ''( )− 1''( )≥0 for t∈ Ω1. It’s easy to verify that x tλ( )≤w t1( ) for [0,1]

t∈ by maximum principle. By similar method we can obtain

1

( ) j ( ) j( ) ( )

x tλ ≤w+ t ≤w t ≤w tλ for t∈[0,1].

Furthermore, we have { ( )}w t_j _{j N}∈ is bounded from below by x tλ( ).

Because w tj( ) is a solution to (3.1.3),

1 1

1 * 1

1 *

( ) ( ) ( ) ( )

( ) ( )

[ ( ) ] ( )

[ ( )

''

] ( ).

p q

j j j

p q

j j

p q q

j j

p q q

j j

w t w t K t w t

w t K w t

w t K w t

λ λ λ λ

−

− −

−

− −

+ −

− −

+ −

−

− = −

≤ −

Suppose that t0∈(0,1), w tj( ) max ( )0 = ₀_{≤ ≤}_t ₁w tj , then w tj'( )0 =0 and w tj( ) is

increasing on ( , )t t0 . By integration of −w tj''( ) from t to t0, we have

0 0

1 *

''( ) [ ( ) ] ( )

t t _{p q} _q

j j j

t w s ds t λw s K w s ds

+ −

−

− ≤ −

∫

.

So w t w tj'( ) ( )qj ≤

λ

wjp q−+1( )t0 −K*. Similarly, by integration of −w tj''( ) from

0

t to t, we can obtain |w t w tj'( ) ( ) |j ≤

λ

wjp q−+1( )t0 −K*. For giving t t1 2, ∈[0,1],

we have

2 2 2

1 1 1

'

1 0 *

'( ) ( ) | ( ) ( ) | [ ( ) ] .

t _q t _q t _{p q}

j j j j j

t w s w s ds t w s w s ds t

λ

w t K ds

+ −

≤ ≤ −

∫

We can find K large such that |

λ

wp qj−+1 ( )t0 −K*|<K. Then

2

1 '( ) ( ) | 2 1|

t _q

j j

t w s w s ds K t t≤ −

∫

, 1 1

2 1 2 1

| q ( ) q ( ) | | | .

j j

w + t ₋w + t _≤K t t₋

(3.1.4)

We define an operator _{I w}_{( )}₌_wq+1_{, then}

1 1_{( )} q 1

I w− ₌w + _{. It follows from}

(3.1.4) that { ( ( ))}I w tj j N∈ is a uniformly bounded and equicontinuous

func-tions in [0,1]. Obviously, _I−1_{is uniformly continuous in a bounded and closed}

domain Ω, i.e., for all ε >0, there exists a δ >0_{such that when}w₁,

2

w ∈ Ω _, |w w₁− ₂|<δ _{, we have} 1 1

1 2

|I w− ( )₋I w− ( ) |_<ε _{. Since} 0

0<w tj( )<w t( ), there exists a M >0 such that w tj( ) (0, ]∈ M . From (3.1.4),

for the above δ >0_{, there exists}δ′ >0_{such that when}|t t₁− <₂| δ ′_{, we have}

1 1

2 1

| q ( ) q ( ) |

j j

w + t ₋w+ t _<

δ

_.

Therefore, for all ε >0_{, there exists}δ′ >0_{such that when}|t t₁− <₂| δ ′_{, we}

have

1 1 1 1

2 1 2 1

| ( ) ( ) | | ( q ( )) ( q ( )) |

j j j j

w t ₋w t ₌ I w− + t ₋I w− + t _<

ε

_.

Thus { ( )}w t_j _{j N}∈ is equicontinuous. Using Arzela-Ascoli theorem, there

ex-ists a subsequence {w tjk( )}jk∈{ }i such that _j_klim→+∞w tjk( )=x tλ( ). Without loss of

(13)

lim j( ) ( ), [0,1].

j→+∞w t =x tλ t∈ (3.1.5)

In the following, we shall show that x tλ( ) is a C[0,1] positive solution of (3.1.1).

Fixing (0,1)( 1)

2

t∈ t≠ , then w tj( ) can be stated

1 1 1

2

1 1 1

( ) ( ) ( )( ) ( )[ ( ) ( ) ( )] .

2 ' 2 2

t _q _p

j j j j j

w t w w t s t K s w− s λw s ds

− −

= + − +

_∫

− − (3.1.6)

Fixing j N∈ , by Lagrange mean value theorem, there exists ( ,1)1 2

n

t ∈ such

that xλ(1)−wj( )1₂ ≤wj(1)−wj( )₂1 =w tj′( )(1n −₂1)<w0(1).

So there exists M1>0 such that |w tj′( ) | 2n < M1 . Since { ( )}w tj j N∈ is

bounded in [0,1], we may assume that m w t< j( )<M2, t∈[ , ]1₂ tn ,

''

1 1 1 1

2 2

1 1 * 1

2

*

| ( ) | | [ ( ) ( ) ( )] |

| [ ( ) ( )] |

.

n n

n

t t _p _q

j j j

t _p _q

j j

p q

w s ds w s K s w s ds

w s K w s ds

M K m

λ

−

− −

−

− −

−

− = −

≤ −

∫

Thus

* 2

1 1

| '( ) | | '( ) | | '( ) '( ) |

2 2 p q

j j n j j n

w ₋ w t _≤ w ₋w t _≤λM ₋K m−

i.e.,

*

1 2

1

| ( ) | 2 .

2

' p q

j

w _≤ M ₊λM ₋K m−

Thus both { ( )}1 2 '

j j N

w ∈ and { ( )}wj 1₂ j N∈ are bounded. Then they all have a

convergence subsequence. Without loss of generality, we note the subsequences are 1

{ ( )} 2

j j N

w ∈ and{wj'( )}1₂ j N∈ . And fixing j N∈ , we assume lim_j_→∞wj' 1( )₂ =r0.

In equation (3.1.6), letting j→ ∞ we have

1 0

2

1 1

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

2 2

t _q _p

x tλ =xλ +r t− +

∫

s t K s x− λ− s −λx s dsλ

for t∈(0,1)_,_i.e._, _{x t}''( ) _{K t x t}( ) q( ) _{x t}p( )

λ λ− λ λ

− + = . Therefore x tλ( ) is a C[0,1]

positive solution of (3.1.1). Therefore x tλ( ) is the maximal solution of (3.1.1). Next we shall verify the dependence on λ_{of maximal solution} x tλ( ). Let H = {µ>0: (3.1.1) has a C[0,1] positive solution with λ µ= }.

Obviously, by (1), H ≠ ∅_{. Let}λ ∈₁ H_{. and} x tλ( ) be the corresponding maximal solution of (3.1.1) for λ λ= 1 . Then for any λ2 >λ1>λ ,

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

p

xλ t +λx tλ ≥ t∈ . By Lemma (2.3), x tλ₁( )≤w tλ₂( ) in [0,1]. Just

re-placing x tλ( ) by x tλ₁( ) in above proof. We can easily find that

1 1 1 1

2 2 2

''

1 2

''

2

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

( ) ( ) ( ) ( ).

q p p

q p

x t K t x t x x t w t K t w t w t

λ λ λ λ

λ λ λ

λ λ

λ

− −

− + = ≤ ∈

 

− + ≥

(14)

Combining it with boundary conditions, we can obtain that ( ( ),x t w tλ₁ λ₂( )) is a couple of lower and upper solutions of (3.1.1) for λ λ= 2 >λ1. One can be

prove that there is a solution x tλ₂( ) of (3.1.1) with λ λ= 2 such that

1( ) 2( ) 2( ).

x tλ ≤x tλ ≤w tλ

Therefore λ ∈2 H. Moreover, by (ii), for any λ2 >λ1≥λ, x tλ2( )≥x tλ1( ). This completes the proof of Theorem 1.1. 

(B) The proof of Theorem 1.2. Proof. 1) We consider the problem

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

(0) 0, (1) ( ),

q p

x x ax

λ

η

−

− + = ∈

 ₌ ₌

 (3.2.1)

where q>0,0< p<1, K t( )∈C[0,1]_, _K*_<₀_, ₀_{< <}_a ₁_, ₀_{< <}_η ₁_and _λ

is a positive parameter.

Now we consider an approximate problem of (3.2.1) as follows

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

1 1

(0) , (1) ( ) ,

q p

x t K t x t x t t

x x ax

n n

λ

η −

− + = ∈

 

= = +

 (3.2.2)

where 0< <a 1, 0< <η 1, n≥1.

Let

ε

_{very small. We’ll verify that} _n( )t ₁( )t 1 n

α =εϕ + _{is a lower solution of}

(3.2.2). Indeed, when n is big enough, we can obtain that εϕ1( )t +1_n is close to

0. Since 1 (0, ₂) π

λ ∈ _{(see [6]), we can deduce}

1 1 1 1

1 1 1

1

1 1 1

( ) ( ) ( ) ( )

1 1

( ) ( )( ( ) ) ( (

'

) )

1

( ) ( ( ) )

1

( )[ ( ( ) ) ]

0,

' q p

n n n

q p

p

t K t t t

n n

t t

n

t t

n

α α λα

λ εϕ εϕ λ εϕ

λ εϕ λ εϕ

εϕ λ λ εϕ

−

− + −

= + + − +

< − +

<

1

(0) (0) 0

n _n

α − =εϕ =

and αn(1) [− aα ηn( )+_n1]=ε ϕ ηa 1( )+ −1_n aεϕ η1( )− − <a_{n n}1 0, which imply that

( ) n t

α is a lower solutions of (3.2.2).

In the following, we’ll construct an upper solution of (3.2.2). Let

2

( )t Mt (M aM t M)

β = − + + + ,

where M is big enough for

1

1 1

{(2 ) , }

(1 )

p M

n a

λ

− >

(15)

*

''( ) ( ) ( ) 2 ( )[ ( ) ]

2 ,

q q

q

t K t t M K t Mt M aM t M M K M

M

β β− −

−

− + = + − + + +

> +

>

2

( ) [ ( ) ]

(1 )

[ ]

4 (2 ) ,

p p

p

t Mt M aM t M M a _M

M

λβ λ

λ

= − + + +

+

< +

<

( )_t _{K t}( ) q( )_t p( ),_t

β_′′ β− λβ

− + ≥

2

1 1

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

1

( 1) 2

1

0

a a M a M M aM M

n n

a M aM n M aM

n

β − β η + = + − − η + + η+ −

> + − −

= − −

>

and (0) 1 M 1 0

n n

β − = − > . It’s easy to see that β( )t _{is na upper solution of}

(3.2.2).

Choosing F tn( )=λβp−K*αn−q, then | ( , ) |f t x ≤F tn( ), for all ( , )t x Dn β α

∈ . It’s

easy to verify that _{( )} 1_[0,1]

n

F t ∈L . Because that

ε

_{is small and}_n_{is big enough,}

( ) ( )

n t t

α ≤β . From Corollary 2.1, ( ( ), ( ))αn t β t is a couple of upper and lower solutions of (3.2.2). And for all n N∈ _{, (3.2.2) has at least one C[0,1] positive} solution x tn( ) such that αn( )t ≤x tn( )≤β( )t .

In the following, we shall obtain a result as follows, there exists a subsequence { ( )}x t_n_k and x t( ) such that lim _k( ) ( )

k n

n→∞x t =x t .

Since _β_{( )}_t _∈_C_[0,1]_∩_C2_(0,1)_, _β_{( )}_t _{is bounded. Therefore { ( )}}

n n N

x t ∈ is a

uniformly bounded sequence of functions in [0,1]. Because x tn( ) is a C[0,1] positive solution of (3.2.2), x tn( ) satisfies

*

( ) ( ) ( ) ( )

( ) ( )

[ ( ) ] (

''

).

p q

n n n

p q

n n

p q q

n n

x t x t K t x t

x t K x t

x t K x t

λ λ λ

−

+ −

− = −

≤ −

Suppose that t0∈(0,1),

0 1

0

( ) max ( )

t

n n

x t x t

≤ ≤

= , then x tn'( )0 =0 and x tn( ) is increasing on ( , )t t0 . By integration of −x tn''( ) from t to t0, we have

0 0

*

''( ) [ ( ) ] ( ) .

t t _{p q} _q

n n n

t x s ds t λx s K x s ds

+ −

− ≤ −

∫

So n'( ) q1_{( )}[ np q( )0 *]

n

x t x t K

x t λ

+

≤ − _{. We can find a} K>0_{such that}

) ( '( q )

n n

x t x t ≤K. And by integration of −x tn( ) from t0 to t, we have

0 0

*

''( ) [ ( ) ] ( ) .

t t _{p q} _q

n n n

t x s ds t λx s K x s ds

+ −

− ≤ −

∫

So n'( ) q1₍₎[ np q( )0 *]

n

x t x t K

x t λ

+

− ≤ − _{. For above}_K_{, we have}| '( ) ( )q |

n n

x t x t K

(16)

DOI: 10.4236/jamp.2018.612217 2615 Journal of Applied Mathematics and Physics i.e., | ' ) ( )( q |

n n

x t x t ≤K.

For giving t t1 2, ∈[0,1], we have

2 2 2

1 '( ) ( ) 1| ( ) ( ) |' 1 .

t _q t _q t

n n n n

t x s x s ds≤ t x s x s ds≤ t Kds

∫

Then 2

1 '( ) ( ) | 2 1|

t _q

n n

t x s x s ds K t t≤ −

∫

. The above inequality can be rewritten as

2

1

( ) ₁ ₁

2 1 2 1 2 1

( )

| n ( ) ( ) | | |, | ( ) ( ) | | | .

n

x t _q _q _q

n n n n

x t x s dx s K t t x t x t K t t

+ +

≤ − − ≤ −

∫

(3.2.3)

We now define an operator _{I x}_{( )}₌_xq+1_{, then}

1 1_{( )} q1

I x− ₌x + _{. It follows from}

(3.2.3) that { ( ( ))}I x tn n N∈ is a uniformly bounded and equicontinuous

func-tions in [0,1]. Obviously, _I−1_{is uniformly continuous in a bounded and closed}

domain Ω , i.e., for all ε >0 _{, there exists a}δ >0 _{such that}

1 1

1 2

|I x−( )₋I x− ( ) |_<ε _for

1 2

|x x− |<δ _, x x₁, ₂∈ Ω _{. Since}0<x tn( )<β( )t , there exists a M>0_{such that}x tn( ) (0, ]∈ M . From (3.2.3), for the above

0

δ > _{, there exists} δ′ >0_{such that} 1 1

2 1

| q ( ) q ( ) |

n n

x + t ₋x+ t _<δ_for 1 2

|t t− <| δ ′_.

Therefore, for all ε >0_{, there exists}δ′ >0 such that

1 1 1 1

2 1 2 1

| ( ) ( ) | | ( q ( )) ( q ( )) |

n n n n

x t ₋x t ₌ I x− + t ₋I x− + t _<ε

for |t t1− <2| δ ′. Consequently, { ( )}x tn n N∈ is equicontinuous. Using

Arze-la-Ascoli theorem, there exists a subsequence { ( )}x tn_k such that _n_klim→+∞x tnk( )=x t( ).

Without loss of generality, we assume that

lim ( )n ( ), [0,1].

n→+∞x t =x t t∈ (3.2.4)

In the following, we shall show that x t( ) is a C[0,1] positive solution of (3.2.1). Fixing (0,1)( 1)

2

t∈ t≠ , x tn( ) can be stated

1 2

1 1 1

( ) ( ) '( )( ) ( )[ ( ) ( ) ( )] .

2 2 2

t _q _p

n n n n n

x t ₌x ₊x t₋ ₊

_∫

s t K s x s₋ − ₋λx s ds _(3.2.5)

Fixing n N∈ _{, by Lagrange mean value theorem, there exists} ( ,1)1 2

n

t ∈ such

that (1) ( )1 (1) ( )1 '( )(1 1) (1)

2 2 2

n xn xn xn x tn n

α − ≤ − = − ≤β .

So there exists M1>0 such that |x tn'(n) |≤2M1. Since { ( )}x tn n N∈ is

bounded in [0,1], we may assume that m x t≤ n( )≤M2, t∈[ , ]1₂ tn .

''

1 1

2 2

| tn ( ) | | [tn p( ) ( ) q( )] | .

n n n

x s ds λx s K s x s ds−

− = −

∫

We can obtain

2 * 2

1

| ( ) '( ) |

2

' p q

n n n

x t x λM K M−

− + ≤ − and |xn'( ) | 21₂ ≤ M1+λM2p−K M* 2−q.

Therefore both { ( )}1 2

n n N

x ∈ and { ( )}xn' 1₂ n N∈ are bounded. They all have a

convergence subsequence. Without loss of generality, we note the subsequences are { ( )}1

2

n n N

(17)

From (3.2.5), letting n→ ∞_{, we obtain}

1 0

2

1 1

( ) ( ) ( ) ( )[ ( ) ( ) ( )] .

2 2

t _q _p

x t ₌x ₊r t₋ ₊

_∫

s t K s x s₋ − ₋λx s ds

By derivation twice of x t( ), we have

( ) ( ) q( ) p( ).

x t_′′ K t x t− λx t

− + =

Combining it with (3.2.4), we can obtain that x t( ) is a C[0,1] positive solu-tion of (3.2.1).

2) We study the uniqueness of _C1_[0,1]_{positive solution of problem (3.2.1).}

Let F t( )=λβp−K*(εϕ1)−q. Obviously, when 0< <q 1, F t( ) is integrable

over (0,1). Since | ( ) |x t′′ ≤F t( )_, x t( ) is absolutely integrable over (0,1). Then both x′ +(0 )_and x′ −(1 )_exist,_i.e._, _{x t}_{( )}_∈_C1_[0,1]_.

Suppose conversely that x t1( ), x t2( ) are two C1[0,1] positive solutions of

the problem (3.2.1), x t1( )≡/x t2( ) on [0,1]. We may assume without loss of

ge-nerality that there exists _t*_∈_(0,1)_{such that}

* *

2( ) 1( ) max( ( )₀_t₁ 2 1( )) 0

x t x t x t x t

≤ ≤

− = − > . Let

* *

1 1 2 1 1

inf{ | 0t t t x t, ( ) x t t( ), ( , )},t t

α= ≤ < > ∈

* *

2 2 2 1 2

sup{ |t t t 1, ( )x t x t t( ), ( , )}.t t

β = ≤ < > ∈

It’s obvious that 0≤α β< ≤1 and

1 2 1 2 1 2

1 2 1 2

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

( ) ( ), ( ) ( ),

' '

' ' ( , ).

x x x x x x x x x t x t t

α α α α β β

β β α β

= ≤ ≤

+ ≥ + < ∈

Let y t( )=x t x t1( ) ( )2' −x t x t t2( ) ( ),1' ∈( , )α β . Then we have

lim inf ( ) 0 lim sup ( ).

t→ +α y t ≥ ≥t→ +β y t (3.2.6)

On the other hand,

'' '' 1 2 2 1

1 2 2 2 1 1

1 2 1 2 1 2 1 2

1 1 1 1

1 2 2 1 1 2 1 2

'( )

( ) ( )

0

q p p q

q q p p

y t x x x x

x Kx x x x Kx

Kx x x x x x Kx x

Kx x x x x x x x

λ λ

λ

− −

− − − − − −

= −

= − + −

≥

for t∈( , )α β _and y t′( ) 0_≡/ _on _{( , )}α β _{. This implies} y( )β− > y( )α+ _,

contra-dicts (3.2.6), so x t1( )≡x t2( ). Thus the C1[0,1] positive solution of (3.2.6) is

unique.

3) We assume that 0<λ1<λ2 and x tλ1( ), x tλ2( ) are the corresponding unique _C1_[0,1]_{positive solutions to (3.2.1). Obviously,}

1

( ) [

'' 0, ]1

xλ t ∈L . In

(3.2.1), _{f t x}( , )₌_λ_{x t}p( )₋_{K t x t}( ) −q( )_{is continuous.} Since p q, ∈(0,1)_, K_*<0_{, it’s easy to see that}

1 _{( , )} p1_{( )} _{( )} q1_{( )}

x f t x− ₌λx − t ₋K t x− − t _{is decreasing for} _x_>₀_{at each} _t_∈_[0,1]_.

2''( ) ( ) 2( ) 2 2( ) 0 1''( ) ( ) 1( ) 2 1()

q p q p

xλ t −K t x tλ− +λ x tλ = <xλ t −K t x tλ− +λ x tλ

for t∈(0,1)_,

2(0) 1(0)