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Existence on positive solutions for boundary value problems of nonlinear fractional differential equations with pLaplacian
Advances in Difference Equations volume 2013, Article number: 30 (2013)
Abstract
In this paper, we study the existence of positive solutions for the nonlinear fractional boundary value problem with a pLaplacian operator
where 2<\alpha \u2a7d3, 1<\beta \u2a7d2, {D}_{0+}^{\alpha}, {D}_{0+}^{\beta} are the standard RiemannLiouville fractional derivatives, {\varphi}_{p}(s)={s}^{p2}s, p>1, {\varphi}_{p}^{1}={\varphi}_{q}, 1/p+1/q=1, and f(t,u)\in C([0,1]\times [0,+\mathrm{\infty}),[0,+\mathrm{\infty})). By the properties of Green’s function, the GuoKrasnosel’skii fixedpoint theorem, the LeggettWilliams fixedpoint theorem, and the upper and lower solutions method, some new results on the existence of positive solutions are obtained. As applications, examples are presented to illustrate the main results.
MSC:34A08, 34B18, 35J05.
1 Introduction
Recently, fractional differential equations have been of great interest. The motivation for those works stems from both the intensive development of the theory of fractional calculus itself and the applications such as economics, engineering and other fields [1–7]. Many people pay attention to the existence and multiplicity of solutions or positive solutions for boundary value problems of nonlinear fractional differential equations by means of some fixedpoint theorems [8–24] (such as the Schauder fixedpoint theorem, the GuoKrasnosel’skii fixedpoint theorem, the LeggettWilliams fixedpoint theorem) and the upper and lower solutions method [25–27].
To the best of our knowledge, there are few papers devoted to the study of fractional differential equations with a pLaplacian operator [22–24, 26–29]. Its theories and applications seem to be just being initiated.
Wang et al.[26] considered the following pLaplacian fractional differential equations boundary value problems:
where 1<\alpha ,\gamma \u2a7d2, 0\u2a7da,b\u2a7d1, 0<\xi ,\eta <1, and {D}_{0+}^{\alpha} is the standard RiemannLiouville fractional derivative. {\varphi}_{p}(s)={s}^{p2}s, p>1, {\varphi}_{p}^{1}={\varphi}_{q}, 1/p+1/q=1. They obtained the existence of at least one positive solution by means of the upper and lower solutions method.
Wang et al.[24] investigated the existence and multiplicity of concave positive solutions of a boundary value problem of a fractional differential equation with a pLaplacian operator as follows:
where 2<\alpha <3, 0<\gamma <1, 0<\rho \u2a7d1, {D}_{0+}^{\alpha} is the Caputo derivative. By using a fixedpoint theorem, some results for multiplicity of concave positive solutions are obtained.
Chen et al.[28] considered the boundary value problem for a fractional differential equation with a pLaplacian operator at resonance
where 0<\alpha ,\beta \u2a7d1, 1<\alpha +\beta \u2a7d2, and {D}_{0+}^{\alpha} is the Caputo fractional derivative. By using the coincidence degree theory, a new result on the existence of solutions is obtained.
Guoqing Chai [29] investigated the existence and multiplicity of positive solutions for a class of boundary value problems of fractional differential equations with a pLaplacian operator
where 1<\alpha \le 2, 0<\beta \le 1, 0<\gamma \le 1, 0\le \alpha \gamma 1, σ is a positive constant number, {D}_{0+}^{\alpha}, {D}_{0+}^{\beta}, {D}_{0+}^{\gamma} are the standard RiemannLiouville derivatives. By means of the fixedpoint theorem on cones, some existence and multiplicity results of positive solutions are obtained.
Motivated by all the works above, in this paper, we deal with the following pLaplacian fractional differential equation boundary value problem:
where 2<\alpha \u2a7d3, 1<\beta \u2a7d2, {D}_{0+}^{\alpha}, {D}_{0+}^{\beta} are the standard RiemannLiouville fractional derivatives and f(t,u)\in C([0,1]\times [0,+\mathrm{\infty}),[0,+\mathrm{\infty})). By the properties of Green’s function, the GuoKrasnosel’skii fixedpoint theorem, the LeggettWilliams fixedpoint theorem, and the upper and lower solutions method, some new results on the existence of positive solutions are obtained for the fractional differential equation boundary value problem (1.1) and (1.2).
The rest of this paper is organized as follows. In Section 2.7, we introduce some definitions and lemmas to prove our main results. In Section 3, we investigate the existence of a single positive solution for boundary value problems (1.1) and (1.2) by the upper and lower solutions method. In Section 4, we establish the existence of single and multiple positive solutions for boundary value problems (1.1) and (1.2) by fixedpoint theorems. As applications, examples are presented to illustrate our main results in Section 3 and Section 4, respectively.
2 Preliminaries and lemmas
For the convenience of the reader, we give some background material from fractional calculus theory to facilitate the analysis of problem (1.1) and (1.2). These materials can be found in the recent literature, see [7, 8, 26, 27, 30–33].
Definition 2.1[7]
The fractional integral of order \alpha >0 of a function y:(0,+\mathrm{\infty})\to \mathbb{R} is given by
provided the righthand side is pointwise defined on (0,+\mathrm{\infty}).
Definition 2.2[7]
The fractional derivative of order \alpha >0 of a continuous function y:(0,+\mathrm{\infty})\to \mathbb{R} is given by
where n is the smallest integer greater than or equal to α, provided that the righthand side is pointwise defined on (0,+\mathrm{\infty}).
Lemma 2.1[7]
Let\alpha >0. If we assumeu\in {D}_{0+}^{\alpha}u\in {L}^{1}(0,1), then the fractional differential equation
has
where n is the smallest integer greater than or equal to α.
Lemma 2.2[7]
Assume that{D}_{0+}^{\alpha}u\in {L}^{1}(0,1)with a fractional derivative of order\alpha >0. Then
for some{c}_{i}\in \mathbb{R}, i=1,2,\dots ,n, where n is the smallest integer greater than or equal to α.
Lemma 2.3[27]
Lety\in C[0,1]and2<\alpha \u2a7d3. Then fractional differential equation boundary value problem
has a unique solution
where
Lemma 2.4 Lety\in C[0,1]and2<\alpha \u2a7d3, 1<\beta \u2a7d2. Then the fractional differential equation boundary value problem
has a unique solution
where
G(t,s)is defined as (2.3).
Proof From Lemma 2.2 and 1<\beta \u2a7d2, we have
In view of (2.4), we obtain
Therefore,
that is,
By the boundary conditions {D}_{0+}^{\alpha}u(0)={D}_{0+}^{\alpha}u(1)=0, we have
Therefore, the solution u(t) of fractional differential equation boundary value problem (2.4) and (2.5) satisfies
Consequently, {D}_{0+}^{\alpha}u(t)+{\varphi}_{q}({\int}_{0}^{1}H(t,\tau )y(\tau )\phantom{\rule{0.2em}{0ex}}d\tau )=0. Thus, fractional differential equation boundary value problem (2.4) and (2.5) is equivalent to the following problem:
Lemma 2.3 implies that fractional differential equation boundary value problem (2.4) and (2.5) has a unique solution
The proof is complete. □
Lemma 2.5 Let2<\alpha \u2a7d3, 1<\beta \u2a7d2. The functionsG(t,s)andH(t,s)defined by (2.3) and (2.6), respectively, are continuous on[0,1]\times [0,1]and satisfy

(1)
G(t,s)\u2a7e0, H(t,s)\u2a7e0 for t,s\in [0,1];

(2)
G(t,s)\u2a7dG(1,s), H(t,s)\u2a7dH(s,s) for t,s\in [0,1];

(3)
G(t,s)\u2a7e{t}^{\alpha 1}G(1,s) for t,s\in (0,1);

(4)
there exist two positive functions {\delta}_{1},{\delta}_{2}\in C[0,1] such that
(2.7)(2.8)
Proof Observing the expression of G(t,s) and H(t,s), it is easy to see that G(t,s)\u2a7e0 and H(t,s)\u2a7e0 for s,t\in [0,1].
From Lemma 3.1 in [27] and Lemma 2.4 in [8], we obtain (2) and (3).
In the following, we consider the existence of {\delta}_{1}(s) and {\delta}_{2}(s). Firstly, for given s\in (0,1), G(t,s) is increasing with respect to t for t\in (0,1). Consequently, setting
we have
Secondly, with the use of the monotonicity of G(t,s), we have
Thus, setting
then (2.7) holds.
Similar to Lemma 2.4 in [8], we choose
The proof is complete. □
Lemma 2.6 Let2<\alpha \u2a7d3. Ify(t)\in C[0,1]andy(t)\u2a7e0, then fractional differential equation boundary value problem (2.1) and (2.2) has a unique solutionu(t)\u2a7e0, t\in [0,1].
Proof From Lemma 2.3, the fractional differential equation boundary value problem (2.1) and (2.2) has a unique solution
In view of Lemma 2.5, we know G(t,s) is continuous on [0,1]\times [0,1] and G(t,s)\u2a7e0 for t,s\in [0,1]. If y(t)\in C[0,1] and y(t)\u2a7e0, we obtain u(t)\u2a7e0. The proof is complete. □
Let {E}_{0}=\{u:u\in {C}^{3}[0,1],{\varphi}_{p}({D}_{0+}^{\alpha}u)\in {C}^{2}[0,1]\}. Now, we introduce definitions about the upper and lower solutions of fractional differential equation boundary value problem (1.1) and (1.2).
Definition 2.3[26]
A function \eta (t) is called an upper solution of fractional differential equation boundary value problem (1.1) and (1.2) if \eta (t)\in {E}_{0} and \eta (t) satisfies
Definition 2.4[26]
A function \xi (t) is called a lower solution of fractional differential equation boundary value problem (1.1) and (1.2) if \xi (t)\in {E}_{0} and \xi (t) satisfies
Definition 2.5[8]
The map θ is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E provided that \theta :P\to [0,+\mathrm{\infty}) is continuous and
for all x,y\in P and 0\u2a7dt\u2a7d1.
Lemma 2.7[8]
Let E be a Banach space, P\subseteq Ebe a cone, and{\mathrm{\Omega}}_{1}, {\mathrm{\Omega}}_{2}be two bounded open balls of E centered at the origin with{\overline{\mathrm{\Omega}}}_{1}\subset {\mathrm{\Omega}}_{2}. Suppose that\mathcal{A}:P\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})\to Pis a completely continuous operator such that either

(i)
\parallel \mathcal{A}x\parallel \u2a7d\parallel x\parallel, x\in P\cap \partial {\mathrm{\Omega}}_{1} and \parallel \mathcal{A}x\parallel \u2a7e\parallel x\parallel, x\in P\cap \partial {\mathrm{\Omega}}_{2} or

(ii)
\parallel \mathcal{A}x\parallel \u2a7e\parallel x\parallel, x\in P\cap \partial {\mathrm{\Omega}}_{1} and \parallel \mathcal{A}x\parallel \u2a7d\parallel x\parallel, x\in P\cap \partial {\mathrm{\Omega}}_{2}
holds. Then\mathcal{A}has a fixed point inP\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}).
Let a,b,c>0 be constants, {P}_{c}=\{u\in P:\parallel u\parallel <c\}, P(\theta ,b,d)=\{u\in P:b\u2a7d\theta (u),\parallel u\parallel \u2a7dd\}.
Lemma 2.8[8]
Let P be a cone in a real Banach space E, {P}_{c}=\{x\in P\mid \parallel x\parallel \u2a7dc\}, θ be a nonnegative continuous concave functional on P such that\theta (x)\u2a7d\parallel x\parallelfor allx\in {\overline{P}}_{c}, andP(\theta ,b,d)=\{x\in P\mid b\u2a7d\theta (x),\parallel x\parallel \u2a7dd\}. Suppose\mathcal{B}:{\overline{P}}_{c}\to {\overline{P}}_{c}is completely continuous and there exist constants0<a<b<d\u2a7dcsuch that
(C1) \{x\in P(\theta ,b,d)\mid \theta (x)>b\}\ne \mathrm{\varnothing}and\theta (\mathcal{B}x)>bforx\in P(\theta ,b,d);
(C2) \parallel \mathcal{B}x\parallel <aforx\u2a7da;
(C3) \theta (\mathcal{B}x)>bforx\in P(\theta ,b,c)with\parallel \mathcal{B}x\parallel >d.
Then ℬ has at least three fixed points{x}_{1}, {x}_{2}, and{x}_{3}with
Let E=C[0,1] be endowed with \parallel u\parallel ={max}_{0\u2a7dt\u2a7d1}u(t). Define the cone P\subset E by
Let the nonnegative continuous concave functional θ on the cone P be defined by
Lemma 2.9 Let T:P\to E be the operator defined by
ThenT:P\to Pis completely continuous.
Proof Let u\in P, in view of the nonnegativeness and continuity of G(t,s), H(t,s), and f(t,u(t)), we have T:P\to P is continuous.
Let \mathrm{\Omega}\subset P be bounded, i.e., there exists a positive constant M>0 such that \parallel u\parallel \u2a7dM for all u\in \mathrm{\Omega}. Let L={max}_{0\u2a7dt\u2a7d1,0\u2a7du\u2a7dM}f(t,u)+1, then for u\in \mathrm{\Omega}, we have
Hence, T(\mathrm{\Omega}) is uniformly bounded.
On the other hand, since G(t,s) is continuous on [0,1]\times [0,1], it is uniformly continuous on [0,1]\times [0,1]. Thus, for fixed s\in [0,1] and for any \epsilon >0, there exists a constant \delta >0 such that any {t}_{1},{t}_{2}\in [0,1] and {t}_{1}{t}_{2}<\delta,
Then, for all u\in \mathrm{\Omega},
that is to say, T(\mathrm{\Omega}) is equicontinuous. By the ArzelaAscoli theorem, we have T:P\to P is completely continuous. The proof is complete. □
3 Existence of a single positive solution
In this section, for the sake of simplicity, we assume that
(H_{1}) f(t,u) is nonincreasing to u;
(H_{2}) There exists a continuous function p(t)\u2a7e0, t\in [0,1] such that
Theorem 3.1 Assume that (H_{1}) and (H_{2}) hold. Then the fractional differential equation boundary value problem (1.1) and (1.2) has at least one positive solution\gamma (t).
Proof From Lemma 2.9, we obtain T(P)\subseteq P. By direct computations, we have
Now, we prove that the functions \eta (t)=Tp(t), \xi (t)=Tq(t) are upper and lower solutions of fractional differential equation boundary value problem (1.1) and (1.2), respectively.
From (H_{1}) and (H_{2}), we have
Hence, \xi (t)\u2a7d\eta (t). By T(P)\subseteq P, we know \xi (t),\eta (t)\in P. From (3.3)(3.5) we have
that is, \eta (t) and \xi (t) are upper and lower solutions of fractional differential equation boundary value problem (1.1) and (1.2), respectively.
Next, we show that the fractional differential equation boundary value problem
has a positive solution, where
Thus, we consider the operator B:P\to E defined as follows:
where G(t,s) and H(t,s) are defined as (2.3) and (2.6), respectively. It is clear that Bu(t)\u2a7e0 for all u\in P and a fixed point of the operator B is a solution of the fractional differential equation boundary value problem (3.8) and (3.9).
Similar to Lemma 2.9, we know that B is a compact operator. By the Schauder fixedpoint theorem, the operator B has a fixed point, that is, the fractional differential equation boundary value problem (3.8) and (3.9) has a positive solution.
Finally, we will prove that fractional differential equation boundary value problem (1.1) and (1.2) has at least one positive solution.
Suppose that \gamma (t) is a solution of (3.8) and (3.9). Now, to complete the proof, it suffices to show that \xi (t)\u2a7d\gamma (t)\u2a7d\eta (t), t\in [0,1].
Let \gamma (t) be a solution of (3.8) and (3.9). We have
From (H_{1}), we have
By (H_{2}) and (3.5), we obtain
By p(t)\in P and (3.3), we can get
Combining (3.4), (3.11)(3.14), we have
Let x(t)={\varphi}_{p}({D}_{0+}^{\alpha}\eta (t)){\varphi}_{p}({D}_{0+}^{\alpha}\gamma (t)). By (3.16), we obtain x(0)=x(1)=0.
By Lemma 2.6, we know x(t)\u2a7d0, t\in [0,1], which implies that
Since {\varphi}_{p} is monotone increasing, we obtain {D}_{0+}^{\alpha}\eta (t)\u2a7d{D}_{0+}^{\alpha}\gamma (t), that is, {D}_{0+}^{\alpha}(\eta \gamma )(t)\u2a7d0. By Lemma 2.6, (3.15) and (3.16), we have (\eta \gamma )(t)\u2a7e0. Therefore, \eta (t)\u2a7e\gamma (t), t\in [0,1].
In a similar way, we can prove that \xi (t)\u2a7d\gamma (t), t\in [0,1]. Consequently, \gamma (t) is a positive solution of fractional differential equation boundary value problem (1.1) and (1.2). This completes the proof. □
Example 3.1
We consider the following fractional differential equation boundary value problem:
Clearly, f(t,u)={t}^{2}+\frac{1}{\sqrt{u}} is nonincreasing relative to u. This shows that (H_{1}) holds.
Let m(t)={t}^{3/2}. From Lemma 2.5, we have
and Tn(t)={T}^{2}m(t)\in P, there exist positive numbers {d}_{1} and {d}_{2} such that Tm(t)\u2a7e{d}_{1}m(t) and {T}^{2}m(t)\u2a7e{d}_{2}m(t).
Choosing a positive number {d}_{0}\u2a7d\{1,{d}_{1}\} and combining the monotonicity of T, we have
Taking p(t)={d}_{0}{t}^{3/2}, then we have
That is, the condition (H_{2}) holds. By Theorem 3.1, the fractional differential equation boundary value problem (3.17) and (3.18) has at least one positive solution.
4 Existence of single and multiple positive solutions
In this section, for convenience, we denote
Theorem 4.1 Letf(t,u)be continuous on[0,1]\times [0,+\mathrm{\infty}). Assume that there exist two positive constants{a}_{2}>{a}_{1}>0such that
(A1) f(t,u)\u2a7e{\varphi}_{p}(N{a}_{1})for(t,u)\in [0,1]\times [0,{a}_{1}];
(A2) f(t,u)\u2a7d{\varphi}_{p}(M{a}_{2})for(t,u)\in [0,1]\times [0,{a}_{2}].
Then the fractional differential equation boundary value problem (1.1) and (1.2) has at least one positive solution u such that{a}_{1}\u2a7d\parallel u\parallel \u2a7d{a}_{2}.
Proof From Lemmas 2.3, 2.4, and 2.9, we get that T:P\to P is completely continuous and fractional differential equation boundary value problem (1.1) and (1.2) has a solution u=u(t) if and only if u solves the operator equation u=Tu(t). In order to apply Lemma 2.7, we divide our proof into two steps.
Step 1. Let {\mathrm{\Omega}}_{1}:=\{u\in P\mid \parallel u\parallel <{a}_{1}\}. For u\in \partial {\mathrm{\Omega}}_{1}, we have 0\u2a7du(t)\u2a7d{a}_{1} for all t\in [0,1]. It follows from (A1) that for t\in [1/4,3/4],
So,
Step 2. Let {\mathrm{\Omega}}_{2}:=\{u\in P\mid \parallel u\parallel <{a}_{2}\}. For u\in \partial {\mathrm{\Omega}}_{2}, we have 0\u2a7du(t)\u2a7d{a}_{2} for all t\in [0,1]. It follows from (A2) that for t\in [0,1],
Therefore,
Then, by (ii) of Lemma 2.7, we complete the proof. □
Example 4.1
We consider the following fractional differential equation boundary value problem:
Let p=2. By a simple computation, we obtain M=11.25, N\approx 736.6099. Choosing {a}_{1}=0.003, {a}_{2}=0.25, therefore
With the use of Theorem 4.1, the fractional differential equation boundary value problem (4.1) and (4.2) has at least one solution u such that 0.003\u2a7d\parallel u\parallel \u2a7d0.25.
Theorem 4.2 Letf(t,u)be continuous on[0,1]\times [0,+\mathrm{\infty}). Assume that there exist constants0<a<b<csuch that the following assumptions hold:
(B1) f(t,u)<{\varphi}_{p}(Ma)for(t,u)\in [0,1]\times [0,a];
(B2) f(t,u)\u2a7e{\varphi}_{p}(Nb)for(t,u)\in [1/4,3/4]\times [b,c];
(B3) f(t,u)\u2a7d{\varphi}_{p}(Mc)for(t,u)\in [0,1]\times [0,c].
Then the fractional differential equation boundary value problem (1.1) and (1.2) has at least three positive solutions{u}_{1}, {u}_{2}, and{u}_{3}with
Proof From Lemmas 2.3, 2.4, and 2.9, we have T:P\to P is completely continuous and fractional differential equation boundary value problem (1.1) and (1.2) has a solution u=u(t) if and only if u satisfies the operator equation u=Tu(t).
We show that all the conditions of Lemma 2.8 are satisfied. If u\in {\overline{P}}_{c}, then \parallel u\parallel \u2a7dc. By (B3), we have
Hence, T:{\overline{P}}_{c}\to {\overline{P}}_{c}. In the same way, if u\in {\overline{P}}_{a}, then assumption (B1) yields \parallel Tu\parallel <a. Therefore, condition (C2) of Lemma 2.8 is satisfied.
To check condition (C1) of Lemma 2.6, we choose u(t)=(b+c)/2, 0\u2a7dt\u2a7d1. It is easy to see that u(t)=(b+c)/2\in P(\theta ,b,c), \theta (u)=\theta ((b+c)/2)>b; consequently, \{u\in P(\theta ,b,c)\mid \theta (u)>b\}\ne \mathrm{\varnothing}. Hence, if u\in P(\theta ,b,c), then b\u2a7du(t)\u2a7dc for 1/4\u2a7dt\u2a7d3/4. From assumption (B2), we have f(t,u(t))\u2a7e{\varphi}_{p}(Nb) for 1/4\u2a7dt\u2a7d3/4. So,
i.e., \theta (Tu)>b for all u\in P(\theta ,b,c). Choosing d=c, this shows that condition (C1) of Lemma 2.8 is also satisfied.
In the same way, if u\in P(\theta ,b,c) and \parallel Tu\parallel >c=d, we also obtain \theta (Tu)>b. Then condition (C3) of Lemma 2.8 is also satisfied.
By Lemma 2.8, the fractional differential equation boundary value problem (1.1) and (1.2) has at least three positive solutions {u}_{1}, {u}_{2}, and {u}_{3}, satisfying
The proof is complete. □
Example 4.2
We consider the following fractional differential equation boundary value problem:
where
Let p=2. We obtain M=11.25, N\approx 736.6099. Choosing a=0.01, b=1, c=72, therefore
With the use of Theorem 4.2, the fractional differential equation boundary value problem (4.3) and (4.4) has at least three positive solutions {u}_{1}, {u}_{2}, and {u}_{3} with
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Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2010AL002, ZR2011AL007), also supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).
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Lu, H., Han, Z., Sun, S. et al. Existence on positive solutions for boundary value problems of nonlinear fractional differential equations with pLaplacian. Adv Differ Equ 2013, 30 (2013). https://doi.org/10.1186/16871847201330
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DOI: https://doi.org/10.1186/16871847201330