### Abstract

In this article, the author investigates the existence and multiplicity of positive
solutions for boundary value problem of fractional differential equation with *p*-Laplacian operator

where
*α *≤ 2, 0 < *β *≤ 1, 0 < *γ *≤ 1, 0 ≤ *α *- *γ *- 1, the constant *σ *is a positive number and *p*-Laplacian operator is defined as *φ*_{p}(*s*) = |*s*|^{p-2}*s*, *p *> 1. By means of the fixed point theorem on cones, some existence and multiplicity
results of positive solutions are obtained.

**2010 Mathematical Subject Classification**: 34A08; 34B18.

##### Keywords:

fractional differential equations; fixed point index;*p*-Laplacian operator; positive solution; multiplicity of solutions

### 1 Introduction

Differential equations of fractional order have been recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, etc. (see [1-5]). There has been a significant development in the study of fractional differential equations in recent years, see the monographs of Kilbas et al. [6], Lakshmikantham et al. [7], Podlubny [4], Samko et al. [8], and the survey by Agarwal et al. [9].

For some recent contributions on fractional differential equations, see for example, [10-28] and the references therein. Especially, in [15], by means of Guo-Krasnosel'skiĭ's fixed point theorem, Zhao et al. investigated the existence of positive solutions for the nonlinear fractional boundary value problem (BVP for short)

where 1 < *α *≤ 2, *f *: [0, +∞) → (0, +∞).

In [16], relying on the Krasnosel'skiĭ's fixed point theorem as well as on the Leggett-Williams fixed point theorem, Kaufmann and Mboumi discussed the existence of positive solutions for the following fractional BVP

In [17], by applying Altman's fixed point theorem and Leray-Schauder' fixed point theorem,
Wang obtained the existence and uniqueness of solutions for the following BVP of nonlinear
impulsive differential equations of fractional order *q*

In [18], relying on the contraction mapping principle and the Krasnosel'skiĭ's fixed point
theorem, Zhou and Chu discussed the existence of solutions for a nonlinear multi-point
BVP of integro-differential equations of fractional order *q *∈ (1, 2]

On the other hand, integer-order *p*-Laplacian boundary value problems have been widely studied owing to its importance
in theory and application of mathematics and physics, see for example, [29-33] and the references therein. Especially, in [29], by using the fixed point index method, Yang and Yan investigated the existence of
positive solution for the third-order Sturm-Liouville boundary value problems with
*p*-Laplacian operator

where *φ*_{p}(*s*) = |*s*|^{p-2}*s*.

However, there are few articles dealing with the existence of solutions to boundary
value problems for fractional differential equation with *p*-Laplacian operator. In [24], the authors investigated the nonlinear nonlocal problem

where 0 < *β *≤ 1, 1 < *α *≤ 2, 0 ≤ *a *≤ 1, 0 < *ξ *< 1. By using Krasnosel'skiĭ's fixed point theorem and Leggett-Williams theorem, some
sufficient conditions for the existence of positive solutions to the above BVP are
obtained.

In [25], by using upper and lower solutions method, under suitable monotone conditions, the authors investigated the existence of positive solutions to the following nonlocal problem

where 1 < *α*, *β *≤ 2, 0 ≤ *a*, *b *≤ 1, 0 < *ξ*, *η *< 1.

No contribution exists, as far as we know, concerning the existence of solutions for
the fractional differential equation with *p*-Laplacian operator

where
*α *≤ 2, 0 < *β *≤ 1, 0 < *γ *≤ 1, 0 ≤ *α *- *γ *- 1, the constant *σ *is a positive number, the *p*-Laplacian operator is defined as *φ*_{p}(*s*) = |*s*|^{p-2}*s*, *p *> 1, and function *f *is assumed to satisfy certain conditions, which will be specified later. To obtain
the existence and multiplicity of positive solutions to BVP (1.5), the fixed point
theorem on cones will be applied.

It is worth emphasizing that our work presented in this article has the following
features which are different from those in [24,25]. Firstly, BVP (1.5) is an important two point BVP. When *γ *= 1, the boundary value conditions in (1.5) reduce to *u*(0) = 0, *u*(1) + *σu'*(1) = 0, which are the well-known Sturm-Liouville boundary value conditions *u*(0) + *bu'*(0) = 0, *u*(1) + *σu'*(1) = 0 (such as BVP (1.1)) with *b *= 0. It is a well-known fact that the boundary value problems with Sturm-Liouville
boundary value conditions for integral order differential equations have important
physical and applied background and have been studied in many literatures, while BVPs
(1.3) and (1.4) are the nonlocal boundary value problems, which are not able to substitute
BVP (1.5). Secondly, when *α *= 2, *β *= 1, *γ *= 1, then BVP (1.5) reduces to BVP (1.2) with *b *= 0. So, BVP (1.5) is an important generalization of BVP (1.2) from integral order
to fractional order. Thirdly, in BVPs (1.3) or (1.4), the boundary value conditions
*u*(1) = au(*ξ*),
*u*(1) and
*u *at *t *= 1), which brings about more difficulties in deducing the property of green's function
than the former. Finally, order *α *+ *β *satisfies that 2 < *α *+ *β *≤ 4 in BVP (1.4), while order *α *+ *β *satisfies that 1 < *α *+ *β *≤ 3 in BVP (1.5). In the case for *α*, *β *taking integral numbers, the BVPs (1.5) and (1.4) are the third-order BVP and the
fourth-order BVP, respectively. So, BVP (1.5) differs essentially from BVP (1.4).
In addition, the conditions imposed in present paper are easily verified.

The organization of this article is as follows. In Section 2, we present some necessary definitions and preliminary results that will be used to prove our main results. In Section 3, we put forward and prove our main results. Finally, we will give two examples to demonstrate our main results.

### 2 Preliminaries

In this section, we introduce some preliminary facts which are used throughout this article.

Let ℕ be the set of positive integers, **ℝ **be the set of real numbers and **ℝ**_{+ }be the set of nonnegative real numbers. Let *I *= [0, 1]. Denote by *C*(*I*, **ℝ**) the Banach space of all continuous functions from *I *into **ℝ **with the norm

Define the cone *P *in *C*(*I*, **ℝ**) as *P *= {*u *∈ *C*(*I*, **ℝ**): *u*(*t*) ≥ 0, *t *∈ *I*}. Let *q *> 1 satisfy the relation
*p *is given by (1. 5).

**Definition 2.1**. [6] The Riemann-Liouville fractional integral of order *α *> 0 of a function *y *: (*a*, *b*] → **ℝ **is given by

**Definition 2.2**. [6] The Riemann-Liouville fractional derivative of order *α *> 0 of function *y *: (*a*, *b*] → **ℝ **is given by

where *n *= [*α*] + 1 and [*α*] denotes the integer part of *α*.

**Lemma 2.1**. [34] Let *α *> 0. If *u *∈ *C*(0, 1) ⋂ *L*(0, 1) possesses a fractional derivative of order *α *that belongs to *C*(0, 1) ⋂ *L*(0, 1), then

for some *c*_{i }∈ **ℝ**, *i *= 1, 2,..., *n*, where *n *= [*α*] + 1.

A function *u *∈ *C*(*I*, **ℝ**) is called a nonnegative solution of BVP (1.5), if *u *≥ 0 on [0, 1] and satisfies (1.5). Moreover, if *u*(*t*) > 0, *t *∈ (0, 1), then *u *is said to be a positive solution of BVP (1.5).

For forthcoming analysis, we first consider the following fractional differential equation

where *α*, *γ*, *σ *are given by (1.5) and *ϕ *∈ *C*(*I*, **ℝ**).

By Lemma 2.1, we have

From the boundary condition *u*(0) = 0, we have *c*_{2 }= 0, and so

Thus,

and

From the boundary condition

Let

Substituting (2.3) into (2.2), we have

where

and

So, we obtain the following lemma.

**Lemma 2.2**. The solution of Equation (2.1) is given by

Also, we have the following lemma.

**Lemma 2.3**. The Green's function *G*(*t*, *s*) has the following properties

(i) *G*(*t*, *s*) is continuous on [0, 1] × [0, 1],

(ii) *G*(*t*, *s*) > 0, *s*, *t *∈ (0, 1).

**Proof**. (i) Owing to the fact 1 < *α *≤ 2, 0 < *γ *≤ 1, 0 ≤ *α *- *γ *- 1, from the expression of *G*, it is easy to see that conclusion (i) of Lemma 2.3 is true.

(ii) There are two cases to consider.

(1) If 0 < *s *≤ *t *< 1, then

(2) If 0 < *t *≤ *s *< 1, then conclusion (ii) of Lemma 2.3 is obviously true from the expression of *G*.

We need to introduce some notations for the forthcoming discussion.

Let
*s *∈ [0, 1]. Set *g*(*s*) = *G*(*s*, *s*), *s *∈ [0, 1]. From 0 < *γ *≤ 1, *σ *> 0, 1 < *α *≤ 2 and
*η*_{0 }∈ (0, 1).

The following lemma is fundamental in this article.

**Lemma 2.4**. The Green's function *G *has the properties

(i) *G*(*t*, *s*) ≤ *G*(*s*, *s*),*s*, *t *∈ [0, 1].

(ii) *G*(*t*, *s*) ≥ *η*(*s*)*G*(*s*, *s*), *t *∈ [*η*_{0}, 1], *s *∈ [0, 1].

**Proof**. (i) There are two cases to consider.

**Case 1**. 0 ≤ *s *≤ *t *≤ 1. In this case, since the following relation

holds for 0 < *s *< *t *≤ 1, we have

**Case 2**. 0 ≤ *t *≤ *s *≤ 1. In this case, from the expression of *g*_{2}(*t*, *s*), it is easy to see that

(ii) We will consider the following two cases.

**Case 1**. When 0 < *s *≤ *η*_{0}, *η*_{0 }≤ *t *≤ 1, then from the above argument in (i) of proof, we know that *g*_{1}(*t*, *s*) is decreasing with respect to *t *on [*η*_{0}, 1]. Thus

and so

**Case 2**. *η*_{0 }< *s *< 1, *η*_{0 }≤ *t *≤ 1.

(a) If *s *≤ *t*, then by similar arguments to (2.5), we also have

(b) If *η*_{0 }≤ *t *≤ *s*, then the following relation

holds in view of the expression of *g*_{2}(*t*, *s*).

To summarize,

Now, we shall show that

In fact, for *s *∈ (0, 1), we have

and so

On the other hand, for *s *∈ (*η*_{0}, 1), we have

Since

holds for *s *= 1. Thus,

Since 1 < *α *≤ 2, it follows from (2.9) that

Hence, from (2.8) and (2.11), we immediately have

Thus, from (2.6) and (2.12 ), it follows that

Also, by (2.8), the following inequality

holds, and therefore

from the proof in Case 1.

Summing up the above relations (2.13)-(2.14), we have

and so

The proof of Lemma 2.4 is complete.

To study BVP (1. 5), we first consider the associated linear BVP

where *h *∈ *P*.

Let

is given by

From the relations *v*(0) = 0, 0 < *β *≤ 1, it follows that *C*_{1 }= 0, and so

Noting that

By Lemma 2.2, the solution of Equation (2.17) can be written as

Since *h*(*s*) ≥ 0, *s *∈ [0, 1], we have
*s *∈ [0, 1], and so

from (2.18). Thus, by Lemma 2.3, we have obtained the following lemma.

**Lemma 2.5**. Let *h *∈ *P*. Then the solution of Equation (2.15) in *P *is given by

We also need the following lemmas to obtain our results.

**Lemma 2.6**. If *a*, *b *≥ 0, *γ *> 0, then

**Proof**. Obviously, without loss of generality, we can assume that 0 < *a *< *b*, *γ *≠ 1.

Let *ϕ*(*t*) = *t*^{γ}, *t *∈ [0, +∞).

(i) If *γ *> 1, then *ϕ*(*t*) is convex on (0, +∞), and so

i.e.,

(ii) If 0 < *γ *< 1, then *ϕ*(*t*) is concave on [0, +∞), and so

Thus, *ϕ*(*a*) + *ϕ*(*b*) ≥ *ϕ*(*a *+ *b*), namely,

By (i), (ii) above, we know that the conclusion of Lemma 2.6 is true.

**Lemma 2.7**. Let *c *> 0, *γ *> 0. For any *x*, *y *∈ [0, *c*], we have that

(i) If *γ *> 1, then |*x*^{γ }- *y*^{γ}| ≤ *γc*^{γ-1 }|*x *- *y*|,

(ii) If 0 < *γ *≤ 1, then |*x*^{γ }- *y*^{γ}| ≤ |*x *- *y*|^{γ}.

**Proof**. Obviously, without loss of generality, we can assume that 0 < *y *< *x *since the variables *x *and *y *are symmetrical in the above inequality.

(i) If *γ *> 1, then we set *ϕ*(*t*) = *t*^{γ}, *t *∈ [0, *c*]. by virtue of mean value theorem, there exists a *ξ *∈ (0, *c*) such that

i.e.,

(ii) If 0 < *γ *< 1, then by Lemma 2.6, it is easy to see that

and so

Now we introduce some notations, which will be used in the sequel.

Let

By simple calculation, we know that

In this article, the following hypotheses will be used.

(*H*_{1}) *f *∈ *C*(*I *× **ℝ**_{+}, **ℝ**_{+}).

(*H*_{2})

(*H*_{3}) There exists a *r*_{0 }> 0 such that *f*(*t*, *x*) is nonincreasing relative to *x *on [0, *r*_{0}] for any fixed *t *∈ *I*.

By Lemma 2.5, it is easy to know that the following lemma is true.

**Lemma 2.8**. If (*H*_{1}) holds, then BVP (1.5) has a nonnegative solution if and only if the integral equation

has a solution in *P*. Let *c *be a positive number, *P *be a cone and *P*_{c }= {*y *∈ *P *: ∥*y*∥ ≤ *c*}. Let *α *be a nonnegative continuous concave function on *P *and

We will use the following lemma to obtain the multiplicity results of positive solutions.

**Lemma 2.9**. [35] Let
*α *be a nonnegative continuous concave function on *P *such that *α*(*y*) ≤ ∥*y*∥ for all
*a*, *b *and *d *with 0 < *a *< *b *< *d *≤ *c *such that

(C1)
*α*(*Ay*) > *b*, for all *y *∈ *P*(*α*, *b*, *d*);

(C2) ∥*Ay*∥ < *a*, for ∥*y*∥ ≤ *a*;

(C3) *α*(*Ay*) > *b*, for *y *∈ *P*(*α*, *b*, *c*) with ∥*Ay*∥ > *d*.

Then *A *has at least three fixed points *y*_{1}, *y*_{2}, *y*_{3 }satisfying

### 3 Main results

In this section, our objective is to establish existence and multiplicity of positive
solution to the BVP (1.5). To this end, we first define the operator on *P *as

The properties of the operator *A *are given in the following lemma.

**Lemma 3.1**. Let (*H*_{1}) hold. Then *A *: *P *→ *P *is completely continuous.

**Proof**. First, under assumption (*H*_{1}), it is obvious that *AP *⊂ *P *from Lemma 2.3. Next, we shall show that operator *A *is completely continuous on *P*. Let

Step 1. We shall show that the operator *A *is compact on *P*.

Let *B *be an arbitrary bounded set in *P*. Then exists an *M *> 0 such that ∥*u*∥ ≤ *M *for all *u *∈ *B*. According to the continuity of *f*, we have

Thus,

That is, the set *AB *is uniformly bounded.

On the other hand, the uniform continuity of *G*(*t*, *s*) on *I *× *I *implies that for arbitrary *ε *> 0, there exists a *δ *> 0 such that whenever *t*_{1}, *t*_{2 }∈ *I *with |*t*_{1 }- *t*_{2}| < *δ*, the following inequality

holds for all *s *∈ *I*. Therefore,

Thus, *AB *is equicontinuous. Consequently, the operator is compact on *P *by Arzel*à*-Ascoli theorem.

Step 2. The operator *A *is continuous.

Let {*u*_{n}} be an arbitrary sequence in *P *with *u*_{n }→ *u*_{0 }∈ *P*. Then exists an *L *> 0 such that

Thus,

On the other hand, the uniform continuity of *f *combined with the fact that ∥*u*_{n }- *u*_{0}∥ → 0 yields that there exists a *N *≥ 1 such that the following estimate

holds for *n *≥ *N*.

(1) If 1 < *q *≤ 2, then from Lemma 2.7 (ii), we have

Hence, by Lemmas 2.3 and 2.4, from (3.1), we obtain

Thus,

(2) If *q *> 2, then from Lemma 2.7 (i), we have

Thus, we have

and so

From (3.2)-(3.3), it follows that ∥*Au*_{n }- *Au*_{0}∥ → 0(*n *→ ∞).

Summing up the above analysis, we obtain that the operator *A *is completely continuous on *P*.

We are now in a position to state and prove the first theorem in this article.

**Theorem 3.1**. Let (*H*_{1}), (*H*_{2}), and (*H*_{3}) hold. Then BVP (1.5) has at least one positive solution.

**Proof**. By Lemma 2.8, it is easy to know that BVP (1.5) has a nonnegative solution if and
only if the operator *A *has a fixed point in *P*. Also, we know that *A *: *P *→ *P *is completely continuous by Lemma 3.1.

The following proof is divided into two steps.

Step 1. From (*H*_{2}), we can choose a *ε*_{0 }∈ (0, *l*) such that

Therefore, there exists a *R*_{0 }> 0 such that the inequality

holds for *x *≥ *R*_{0}.

Let

From the fact that (*l *- *ε*_{0})^{q-1 }< *l*^{q-1}, we can choose a *k *> 0 such that (*l *- *ε*_{0})^{q-1 }< *l*^{q-1 }- *k*.

Set

where *D *is as (2.20). Take
_{R }= {*u *∈ *P *: ∥*u*∥ < *R*}. We shall show that the relation

holds.

In fact, if not, then there exists a *u*_{0 }∈ *∂*Ω_{R }and a *μ*_{0 }≥ 1 with

By (3.5), we have

Therefore, in view of Lemmas 2.3, 2.4, from (3.1), it follows that

Also, keeping in mind that (*p *- 1)(*q *- 1) = 1, by Lemma 2.6, we have

Hence, from (3.6), (3.8), and (3.9), it follows that

By definition of *l*, we have *D*_{1}*l*^{q-1 }= 1. From (3. 10), it follows that *R *= ∥*u*_{0}∥ ≤ (1 - *E*)*R *+ *G*, and so
*R*. Hence, the condition (3.7) holds. By virtue of the fixed point index theorem, we
have

Step 2. By (*H*_{2}), we can choose a *ε*_{0 }> 0 such that

Hence, there exists a *r*_{1 }∈ (0, *r*_{0}) such that

where *r*_{0 }is given by (*H*_{3}).

Take 0 < *r *< min {*R*, *r*_{1}}, and set Ω_{r }= {*u *∈ *P *: ∥*u*∥ < *r*}. Now, we show that

(i)

(ii) A*u *≠ *μu*, ∀*u *∈ *∂*Ω_{r}, *μ *∈ [0, 1].

We first prove that (i) holds. In fact, for any *u *∈ *∂*Ω_{r}, we have 0 ≤ *u*(*t*) ≤ *r*. By (*H*_{3}), the function *f*(*t*, *x*) is nonincreasing relative to *x *on [0, *r*] for any *t *∈ *I*, and so

from (3.12).

Thus, in view of Lemma 2.4 combined with (3.1) and (3.13), we have

where *Q *is as (2.21). Consequently,

Thus

(ii) Suppose on the contrary that there exists a *u*_{0 }∈ *∂*Ω_{r }and *μ*_{0 }∈ [0, 1] such that

Then, by similar arguments to (3.14), we have

where

By (3.15)-(3.16), we obtain

The hypothesis
*B *> 1, and so *r *> *r *from above inequality, which is a contradiction. That means that (ii) holds.

Hence, applying fixed point index theorem, we have

By (3.11) and (3.17), we have

and so, there exists
*u*_{* }= *u*_{*}, ∥*u*_{*}∥ > *r*. Hence, *u*_{* }is a nonnegative solution of BVP (1.5) satisfying ∥*u*_{*}∥ > *r*. Now, we show that *u*_{*}(*t*) > 0, *t *∈ (0, 1).

In fact, since ∥*u*_{*}∥ > *r*, *u*_{* }∈ *P*, *G*(*t*, *s*) > 0, *t*, *s *∈ (0, 1), from (3.1), we have

and so

from the fact that *G*(*t*, *s*) > 0 and
*s *∈ [0, 1]. That is, *u*_{* }is a positive solution of BVP (1.5).

The proof is complete.

Now, we state another theorem in this article. First, let me introduce some notations which will be used in the sequel.

Let
*D *is as (2.20).

Let

Set *P*_{r }= {*u *∈ *P *: ∥*u*∥ < *r*}, for *r *> 0. Let
*u *∈ *P*. Obviously, *ω *is a nonnegative continuous concave functional on *P*.

**Theorem 3.2**. Let (*H*_{1}) hold. Assume that there exist constants *a*, *b*, *c*, *l*_{1}, *l*_{2 }with 0 < *a *< *b *< *c *and *l*_{1 }∈ (0, *M*_{1}), *l*_{2 }∈ (*M*_{2}, ∞) such that

(*D*_{1}) *f*(*t*, *x*) ≤ *l*_{1}*c*^{p-1}, *x *∈ [0,*c*], *t *∈ *I*; *f*(*t*, *x*)≤ *l*_{1}*a*^{p-1}, *x *∈ [0, *a*], *t *∈*I*,

(*D*_{2}) *f*(*t*, *x*) ≥ *l*_{1}*b*^{p-1}, *x *∈ [*b*,*c*], *t *∈ [*η*_{0}, 1].

Then BVP (1.5) has at least one nonnegative solution *u*_{1 }and two positive solutions *u*_{2}, *u*_{3 }with

**Proof**. By Lemmas 2.3 and 2.4, for
*D*_{1}), it follows that

and so

from the hypothesis *l*_{1 }< *M*_{1}.

Thus, we obtain
*D*_{1}). Take
*ω*(*u*_{0}) > *b*, and so

For any *u *∈ *P*(*ω*, *b*, *c*), we have that *u*(*t*) ≥ *b*, *t *∈ [*η*_{0}, 1] and ∥*u*∥ ≤ *c*. Consequently, by Lemma 2.3, 2.4 and the formula (3.1), for any *t *∈ [*η*_{0}, 1], it follows from condition (*D*_{2}) that

Also, by changing the variable

where *B*_{2 }is given by (3.19).

Substituting (3.21) into (3.20), we obtain

and so *ω*(A*u*) > *b *from the hypothesis *l*_{2 }> *M*_{2}.

Summing up the above analysis, we know that all the conditions of Lemma 2.9 with *c *= *d *are satisfied, and so BVP (1.5) has at least three nonnegative solutions *u*_{1}, *u*_{2}, *u*_{3 }with

By similar argument to (3.18), we can deduce that *u*_{2 }and *u*_{3 }are two positive solutions.

The proof is complete.

**Example 3.1**. Consider the following BVP