Abstract
In this article, the author investigates the existence and multiplicity of positive solutions for boundary value problem of fractional differential equation with pLaplacian operator
where and are the standard RiemannLiouville derivatives with 1 < α ≤ 2, 0 < β ≤ 1, 0 < γ ≤ 1, 0 ≤ α  γ  1, the constant σ is a positive number and pLaplacian operator is defined as φ_{p}(s) = s^{p2}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; pLaplacian operator; positive solution; multiplicity of solutions1 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 [15]). 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, [1028] and the references therein. Especially, in [15], by means of GuoKrasnosel'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 LeggettWilliams 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 LeraySchauder' 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 multipoint BVP of integrodifferential equations of fractional order q ∈ (1, 2]
On the other hand, integerorder pLaplacian boundary value problems have been widely studied owing to its importance in theory and application of mathematics and physics, see for example, [2933] 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 thirdorder SturmLiouville boundary value problems with pLaplacian operator
where φ_{p}(s) = s^{p2}s.
However, there are few articles dealing with the existence of solutions to boundary value problems for fractional differential equation with pLaplacian 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 LeggettWilliams 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 pLaplacian operator
where and are the standard RiemannLiouville derivative with 1 < α ≤ 2, 0 < β ≤ 1, 0 < γ ≤ 1, 0 ≤ α  γ  1, the constant σ is a positive number, the pLaplacian operator is defined as φ_{p}(s) = s^{p2}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 wellknown SturmLiouville boundary value conditions u(0) + bu'(0) = 0, u(1) + σu'(1) = 0 (such as BVP (1.1)) with b = 0. It is a wellknown fact that the boundary value problems with SturmLiouville 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), show the relations between the derivatives of same order and . By contrast with that, the condition in BVP (1.5) shows that relation between the derivatives of different order u(1) and is regarded as the derivative value of zero order of 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 thirdorder BVP and the fourthorder 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 , where p is given by (1. 5).
Definition 2.1. [6] The RiemannLiouville fractional integral of order α > 0 of a function y : (a, b] → ℝ is given by
Definition 2.2. [6] The RiemannLiouville 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 , it follows that
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 . Denote , s ∈ [0, 1]. Set g(s) = G(s, s), s ∈ [0, 1]. From 0 < γ ≤ 1, σ > 0, 1 < α ≤ 2 and , we know that 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
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 . By Lemma 2.1, the solution of initial value problem
is given by
From the relations v(0) = 0, 0 < β ≤ 1, it follows that C_{1 }= 0, and so
Noting that , from (2.16), we know that the solution of (2.15) satisfies
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
.
(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^{γ1 }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.
By simple calculation, we know that
In this article, the following hypotheses will be used.
(H_{1}) f ∈ C(I × ℝ_{+}, ℝ_{+}).
(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 be completely continuous and α be a nonnegative continuous concave function on P such that α(y) ≤ ∥y∥ for all . Suppose that there exist a, b and d with 0 < a < b < d ≤ c such that
(C1) and 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 . The following proof will be divided into two steps.
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, by Lemmas 2.3 and 2.4, it follows that
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 ArzelàAscoli 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 . It follows from (3.4) that
From the fact that (l  ε_{0})^{q1 }< l^{q1}, we can choose a k > 0 such that (l  ε_{0})^{q1 }< l^{q1 } k.
Set
where D is as (2.20). Take . Set Ω_{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^{q1 }= 1. From (3. 10), it follows that R = ∥u_{0}∥ ≤ (1  E)R + G, and so , which contradicts the choice of 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
(ii) Au ≠ 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,
(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
By (3.15)(3.16), we obtain
The hypothesis implies that 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 with Au_{* }= 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
Set P_{r }= {u ∈ P : ∥u∥ < r}, for r > 0. Let , for 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^{p1}, x ∈ [0,c], t ∈ I; f(t, x)≤ l_{1}a^{p1}, x ∈ [0, a], t ∈I,
(D_{2}) f(t, x) ≥ l_{1}b^{p1}, 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 , from (3.1) and condition (D_{1}), it follows that
and so
from the hypothesis l_{1 }< M_{1}.
Thus, we obtain . Similarly, we can also obtain by condition (D_{1}). Take . Then 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 , we have
where B_{2 }is given by (3.19).
Substituting (3.21) into (3.20), we obtain
and so ω(Au) > 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
where 1 < α < 2, 0 < β < 1, 0 < γ < 1, 0 ≤ α  γ  1, σ > 0 and the pLaplacian operator is defined as φ_{p}(s) = s^{p2}s, p > 1.
It is easy to verify that all assumptions of Theorem 3.1 are satisfied. Hence, by the conclusion of Theorem 3.1, BVP (3.22) has at least one positive solution on [0, 1].
Example 3.2. Consider the following BVP
where relative to Theorem 3.2. With the aid of computation we have that . Take l_{1 }= 1.5, l_{2 }= 4. Then l_{1 }∈ (0, M_{1}), l_{2 }∈ (M_{2}, ∞). Again choosing , b = 1, c = 196, and setting
for t ∈ [0, 1], then we see that f satisfies the following relations:
So, all the assumptions of Theorem 3.2 are satisfied. By Theorem 3.2, we arrive at BVP (3.23) has at least one nonnegative solution u_{1 }and two positive solutions u_{2}, u_{3 }with
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author sincerely thanks the anonymous referees for their valuable suggestions and comments which have greatly helped improve this article. Supported by the Natural Science Foundation of Educational Committee of Hubei Province (D200722002).
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