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# Positive solution for boundary value problems with p-Laplacian in Banach spaces

- Dehong Ji
^{1}Email author and - Weigao Ge
^{2}

**2012**:51

https://doi.org/10.1186/1687-2770-2012-51

© Ji and Ge; licensee Springer. 2012

**Received:**28 December 2011**Accepted:**30 April 2012**Published:**30 April 2012

## Abstract

In this article, by using the fixed point theorem of strict-set-contractions operator, we discuss the existence of positive solution for boundary value problems with *p*-Laplacian

in Banach spaces *E*, where^{:} *θ* is the zero element of *E*. Although the fixed point theorem of strict-set-contractions operator is used extensively in yielding positive solutions for boundary value problems in Banach spaces, this method has not been used to study those boundary value problems with *p*-Laplacian in Banach spaces. So this article may be regarded as an illustration of fixed point theorem of strict-set-contractions operator in a new area.

**MSC:** 34B18.

## Keywords

- boundary value problems
*p*-Laplacian- positive solution
- strict-set-contractions

## 1 Introduction

In the last ten years, the theory of ordinary differential equations in Banach spaces has become an important new branch, so boundary value problems in Banach Space has been studied by some researchers, we refer the readers to [1–9] and the references therein.

*n*th-order differential equations in ordered Banach spaces

On the other hand, boundary value problems with *p*-Laplacian have received a lot of attention in recent years. They often occur in the study of the *n*-dimensional *p*-Laplacian equation, non-Newtonian fluid theory, and the turbulent flow of gas in porous medium [12–19]. Many studies have been carried out to discuss the existence of solutions or positive solutions and multiple solutions for the local or nonlocal boundary value problems.

However, to the authors' knowledge, this is the first article can be found in the literature on the existence of positive solutions for boundary value problems with *p*-Laplacian in Banach spaces. As is well known, the main difficulty that appears when passing from *p* = 2 to *p* ≠ 2 is that, when *p* = 2, we can change the differential equation into a equivalent integral equation easily and therefore a Green's function exists, so we can easily prove the equivalent integral operator is a strict-set-contractions operator, which is a very important result for discussing positive solution for boundary value problems in Banach space. However, for *p* ≠ 2, it is impossible for us to find a Green's function in the equivalent integral operator since the differential operator (*ϕ*_{
p
}(*u*'))' is nonlinear. To authors' knowledge, this is the first article to use the fixed point theorem of strict-set-contractions to deal with boundary value problems with *p*-Laplacian in Banach spaces. Such investigations will provide an important platform for gaining a deeper understanding of our environment.

Basic facts about an ordered Banach space *E* can be found in [1, 4]. Here we just recall a few of them. Let the real Banach spaces *E* with norm || ·|| be partially ordered by a cone *P* of *E*, i.e., *x ≤ y* if and only if *y - x* ∈ *P* , and *P** denotes the dual cone of *P. P* is said to be normal if there exists a positive constant *N* such that *θ* ≤ *x* ≤ *y* implies ||*x*|| *≤ N*||*y*||, where *θ* denotes the zero element of *E*, and the smallest *N* is called the normal constant of *P* (it is clear, *N* ≥ 1). Set *I* = 0 [1], (*C*[*I, E*], ||·|| _{
C
}) is a Banach space with ||*x*||_{
C
} = max_{t∈I}||*x*(*t*)||. Clearly, *Q* = {*x* ∈ *C*[*I, E*]*|x*(*t*) ≥ *θ* for *t* ∈ *I*} is a cone of the Banach space *C*[*I, E*].

For a bounded set *S* in a Banach space, we denote by *α*(*S*) the Kuratowski measure of noncompactness. In this article, we denote by *α*(·) the Kuratowski measure of noncompactness of a bounded set in *E* and in *C*[*I, E*].

The operator *T* : *D* → *E*(*D* ⊂ *E*) is said to be a *k*-set contraction if *T* : *D* → *E* is continuous and bounded and there is a constant *k* ≥ 0 such that *α*(*T* (*S*)) ≤ *kα*(*S*) for any bounded *S* ⊂ *D*; a *k*-set contraction with *k <* 1 is called a strict set contraction.

*p*-Laplacian

in Banach spaces *E*, where *ϕ*_{
p
}(*s*) = *s*^{p-1}, *p* > 1, (*ϕ*_{
p
})^{-1} = *ϕ*_{
q
}, $\frac{1}{p}\text{+}\frac{1}{q}=1$, *θ* is the zero element of *E, f* ∈ *C*(*P, P*).

A function *u* is called a positive solution of BVP (1) and (2) if it satisfies (1) and (2) and *u* ∈ *Q, u*(*t*) ≢ *Q*.

The main tool of this article is the following fixed point Theorems.

*K*be a cone in a Banach space

*E*and

*K*

_{ r, R }= {

*x*∈

*K, r ≤*||

*x*||

*≤ R*},

*R > r >*0. Suppose that

*A*:

*K*

_{ r, R }→

*K*is a strict-set contraction such that one of the following two conditions is satisfied:

Then, *A* has a fixed point *x* ∈ *K*_{
r, R
} such that *r* ≤ ||*x*|| ≤ *R*.

## 2 Preliminaries

**Lemma 2.1**.

*If y*∈

*C*[

*I, E*],

*then the unique solution of*

*is*

**Lemma 2.2**. *If y* ∈ *Q, then the unique solution u of the problem* (3) *and* (4) *satisfies u*(*t*) *≥ θ, t* ∈ *I, that is u* ∈ *Q*.

**Lemma 2.3**. *Let* $\delta \in \left(0,\frac{1}{2}\right)$, *J*_{
δ
} = [*δ*, 1-*δ*], *then for any y* ∈ *Q, the unique solution u of the problem* (3) *and* (4) *satisfies u*(*t*) *≥ δu*(*s*), *t* ∈ *J*_{
δ
}*, s* ∈ *I*.

**Lemma 2.4**.

*We define an operator T by*

*Then u is a solution of problem* (1) *and* (2) *if and only if u is a fixed point of T*.

In the following, the closed balls in spaces *E* and *C*[*I, E*] are denoted by *T*_{
r
} = {*x* ∈ *E|*||*x*|| ≤ *r*} (*r >* 0) and *B*_{
r
} = {*x* ∈ *C*[*I, E*]|||*x*||_{
c
} ≤ *r*}, *M* = sup {||*f*(*u*)||: *u* ∈ *Q* ⋂ *B*_{
r
}}.

**Lemma 2.5**.

*Suppose that, for any r >*0,

*f is uniformly continuous and bounded on P*⋂

*T*

_{ r }

*and there exists a constant L*

_{ r }

*with*

*such that*

*Then, for any r >* 0, *operator T is a strict-set-contraction on D* ⊂ *P* ⋂ *T*_{
r
}.

*Proof*. Since *f* is uniformly continuous and bounded on *P* ⋂ *T*_{
r
}, we see from Lemma 2.4 that *T* is continuous and bounded on *Q* ⋂ *B*_{
r
}. Now, let *S* ⊂ *Q* ⋂ *B*_{
r
} be given arbitrary, there exists a partition $S={\cup}_{i=1}^{m}{S}_{i}.$ We set *α*{*y* : *y* ∈ *S*} = *α*(*S*)·

*Ty|y*∈

*S*} are uniformly bounded and equicontinuous, and so by [11],

where *T* (*S*(*t*)) = {*Tu*(*t*)|*u* ∈ *S, t* is fixed}⊂ *P* ⋂ *T*_{
r
} for any *t* ∈ *I*.

*u*

_{1},

*u*

_{2}∈

*S*

_{ i },

*B*= {

*y*(

*s*)|

*s*∈

*I, y*∈

*S*}⊂

*P*⋂

*T*

_{ r }. Similarly, to the proof of [10], we have

*α*(

*B*) ≤ 2

*α*(

*S*)·It follows from (6), (7), and (8), that

and consequently *T* is a strict-set-contraction on *S* ⊂ *Q* ⋂ *B*_{
r
} because of (*q*-1)*M*^{q-2}*L*_{
r
} *<* 1. □

## 3 Existence of positive solution to BVP (1) and (2)

where *β* = 0 or ∞, *ψ* ∈ *P** and ||*ψ*|| = 1.

Furthermore, we list some condition:

_{1}): For any

*r >*0,

*f*is uniformly continuous and bounded on

*P*⋂

*T*

_{ r }and there exists a constant

*L*

_{ r }with (

*q*- 1)

*M*

^{q-2}

*L*

_{ r }< 1 such that

**Theorem 3.1**. *Let* (H_{1}) *hold, cone P be normal. If* ${\varphi}_{q}\left({f}^{0}\right)<1<\frac{1}{2}\delta {\varphi}_{q}\left[\left(\frac{1}{2}-\delta \right){\left(\psi f\right)}_{\infty}\right]$, *then BVP* (1) *and* (2) *has at least one positive solution*.

*Proof*. Set

*K*is a cone of the Banach space

*C*[

*I, E*] and

*K*⊂

*Q*. By Lemma 2.4, we know

*T*(

*Q*) ⊂

*K*, and so

*ϕ*

_{ q }(

*f*

^{0}) < 1 Then, there exists a constant ${\stackrel{\u0304}{r}}_{1}>0$ such that, for any

*u*∈

*K*, $\u2225u\u2225\le {\stackrel{\u0304}{r}}_{1},$ we have ||

*f*(

*u*)|| ≤ (

*f*

^{0}+

*ε*

_{1})

*ϕ*

_{ p }(||

*u*||), where

*ε*

_{1}

*>*0 satisfies

*ϕ*

_{ q }(

*f*

^{0}+

*ε*

_{1}) ≤ 1. Let ${r}_{1}\in \left(0,{\stackrel{\u0304}{r}}_{1}\right),$ then for any

*t*∈

*I, u*∈

*K*, ||

*u*||

_{ C }=

*r*

_{1}, we have

i.e., *u* ∈ *K*, ||*u*||_{
C
} = *r*_{1} implies ||*Tu*||_{
C
} ≤ ||*u*||_{
C
}·

where *ε*_{2} *>* 0 satisfies $\frac{1}{2}\delta {\varphi}_{q}\left[\left(\frac{1}{2}-\delta \right)\left({\left(\psi f\right)}_{\infty}-{\epsilon}_{2}\right)\right]\ge 1$.

*t*∈

*J*

_{ δ }

*, u*∈

*K*, ||

*u*||

_{ C }=

*r*

_{2}, we have

*u*∈

*K*, ||

*u*||

_{ C }=

*r*

_{2}, we have

On the other hand, by Lemma 2.5, *T* is a strict set contraction from ${B}_{r}^{\left(1\right)}$ into ${B}_{r}^{\left(1\right)}$. Consequently, Theorem 1 implies that *T* has a fixed point in ${B}_{r}^{\left(1\right)}$, and the proof is complete. □

## Declarations

### Acknowledgements

This study was sponsored by the National Natural Science Foundation of China (No. (11071014))and the Tianjin City High School Science and Technology Fund Planning Project (No. (20091008)) and Tianyuan Fund of Mathematics in China (No. (11026176)) and Natural Science Foundation of Shandong Province of China (No. (ZR2010AM035)). The authors thank the referee for his/her careful reading of the article and useful suggestions.

## Authors’ Affiliations

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## Copyright

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