Abstract
We consider a class of Caputo fractional p-Laplacian differential equations with integral boundary conditions which involve two parameters. By using the Avery–Peterson fixed point theorem, we obtain the existence of positive solutions for the boundary value problem. As an application, we present an example to illustrate our main result.
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1 Introduction
In this paper, we investigate the following integral boundary value problem (short for BVP) of Caputo fractional differential equations with p-Laplacian operator and parameters:
where \(1< n-1<\alpha<n\), \(1< m-1<\beta< m\), \(\alpha-\beta>1 \), \(D_{{0^{+}}}^{\alpha}\) and \(D_{{0^{+}}}^{\beta}\) are the Caputo fractional derivatives. \(\varphi_{p}\) is the p-Laplacian operator, \(\varphi _{p}(s)=\vert s\vert^{p-2}s\), \(p>1\), \(\varphi_{p}^{-1}=\varphi_{q}\), \(1/p+1/q=1\). \(g_{0},g_{1} \in C([0,1], [0,+\infty)) \), \(f \in C([0,1] \times[0,+\infty) \times[0,+\infty), [0,+\infty)) \) are given functions. \(a,b>0\) are disturbance parameters.
As we all know, fractional differential equation theory is becoming more and more perfect because of its extensive application, and many significant achievements have been made; see [1–12]. As one of many applications, turbulence problem can be well characterized by the p-Laplacian operator; see [13]. Fractional p-Laplacian equations are becoming more and more important, they can be used to describe a class of diffusion phenomena, which have been widely used in the fields of fluid mechanics, material memory, biology, plasma physics, finance and chemistry. Many important results related to the boundary value problems of fractional differential equations with p-Laplacian operator have been obtained; see [14–24]. But in practical problems, disturbance is objective. As a boundary value problem with disturbance parameter can describe real problems better, many scholars turn their attention to it.
In [6], Jia et al. consider the fractional-order differential equation integral boundary value problem with disturbance parameters
where \(J=[0,1]\), \(1 < \delta\leq2\), \(f \in C( [0,1]\times[0,+\infty) , [0,+\infty) )\), \(m_{i} \geq0\), \(n_{i} \geq0\), \(m^{2}_{i}+n^{2}_{i}>0\), \({i=1,2}\), \(g \in C( [0,1] , [0,+\infty))\), disturbance parameter \(a >0\), and \(^{C}{D^{\delta}}\) is the Caputo fractional derivative of order δ. By using an upper and lower solution method, the fixed point index theorem and the Schauder fixed point theorem, sufficient conditions are obtained for the problem to have at least one positive solution, two positive solutions and no solution.
In [25], Wang et al. consider a class of fractional differential equations with integral boundary conditions which involve two disturbance parameters. By using the Guo–Krasnoselskii fixed point theorem, new results on the existence and nonexistence of positive solutions for the boundary value problem are obtained. The problem is given by
where \(D_{0+}^{\alpha} \) is the standard Riemann–Liouville fractional derivative with \(3<\alpha\leq4 \), \({f:[0,1]\times[0,+\infty) \rightarrow[0,+\infty)}\) is a continuous function, \(g_{1},g_{2}\in L^{1}[0,1]\) and \(a,b\geq0\).
In [26] Hao et al. consider the existence of positive solutions of higher order fractional integral boundary value problem with a parameter
where \(D^{\eta-2}_{0+}\), \(D^{k-2}_{0+}\) are the standard Riemann–Liouville fractional derivative, \(n-1 < \eta\leq n\), \(\eta \geq4\), \(2 \leq k \leq n-2\), \(\alpha, \beta, \gamma, \delta>0 \). \(\int_{0}^{1}u(s)\,d A(s)\) and \(\int_{0}^{1}u(s)\,d B(s)\) denote the Riemann–Stieltjes integrals of u with respect to A and B. \(A(t)\), \(B(t)\) are nondecreasing on \([0, 1]\), \(f : [0, 1] \times [0, +\infty) \rightarrow[0, +\infty)\) is continuous, \(\lambda> 0\) is a parameter. By using the Guo–Krasnoselskii fixed point theorem on cones, under different conditions of nonlinearity, existence and nonexistence, results for positive solutions are derived in terms of different parameter intervals.
The purpose of this paper is to establish conditions ensuring the existence of three positive solutions of BVP (1) and give an estimate of these solutions by using the Avery–Peterson fixed point theorem. Our supposed problem is different from the problems studied before and mentioned above. Our result is new and our work extends the application of the theorem.
In this paper, a positive solution \(x=x(t)\) of BVP (1) means a solution of (1) satisfying \(x(t)>0\), \(t\in[0,1]\).
Throughout this paper, we always assume that the following condition is satisfied:
- (L0):
\(0< a< b<2a<+\infty\), \(0\leq g_{0}(t) \leq g_{1}(t) \leq 2g_{0}(t)\), \(0\leq\int_{0}^{1}g_{0}(s)\,ds\), \(\int_{0}^{1}g_{1}(s)\,ds < 1\).
2 Preliminaries and lemmas
The basic theory of fractional-order differential equation and boundary value problem can be obtained from many places in the literature, which will not be repeated here; see [1–9]. Here we present some necessary basic results that will be used.
Lemma 2.1
(see [2])
The Caputo fractional derivative of order \(n-1<\alpha<n \) for \(t^{\beta}\) is given by
Lemma 2.2
(see [19])
Let \(h\in C[0,1] \)and \(1< m-1<\beta<m\). Then the BVP
has an unique solution
where
Denote
From (L0), we know, for \(t \in(0,1) \),
Thus
Thus, the following lemma holds.
Lemma 2.3
Let (L0) hold, \(y\in C[0,1] \)and \(1< n-1<\alpha<n\), \(1< m-1<\beta <m\), then the following boundary value problem:
has an unique solution
and
where
Proof
Consider BVP (5), we have
In view of \(x^{(j)}(0)=0\) (\(j=2, 3,\ldots,n-1\)), we know \(C_{2}=C_{3}=\cdots=C_{n-1}=0\), and
so that
From the boundary condition of BVP (5), by methods similar to Lemma 2.4 in [19], through traditional analytical calculation and integration techniques, we have
where \(\omega(t)\), \(G_{1}(t,s)\) and \(G_{2}(t,s)\) are given by (3), (9) and (10).
On the other hand, in view of (11), because \(1< m-1<\beta<\alpha -1<n-1\), by Lemma 2.1, we have
□
Lemma 2.4
The BVP (1) is equivalent to the following integral equation:
and
where \(H(t,s)\), \(\omega(t) \)and \(G(t,s)\)are given by (2), (3) and (8).
Proof
From Lemma 2.2 and Lemma 2.3, let \(y(t)=\varphi_{q}(u(t))\), \(h(t)=-f (t,x(t),D_{0+}^{\beta}x(t) )\), we have
Immediately we obtain
On the other hand, if \(x(t)\) satisfies (13), we can easily prove that \(x(t)\) satisfies BVP (1). □
Lemma 2.5
Assume (L0) hold, then the function \(H(t,s)\)defined by (2), the function \(G(t,s)\)defined by (8), and then the function \(\omega(t)\)defined by (3) satisfies
- (1)
\(H(t,s)\geq0\)is continuous for all \(t,s\in[0,1]\);
- (2)
\(H(t,s)\leq H(s,s)\)for all \(t,s\in[0,1]\);
- (3)
\(\int_{0}^{1}H(t,s)\,ds=\frac{1-t^{\beta}}{\varGamma(\beta +1)} \leq\frac{1}{\varGamma(\beta+1)}\)for all \(t\in[0,1]\);
- (4)
\(G(t,s)\geq0\)is continuous for all \(t,s\in[0,1]\);
- (5)
\(\omega(t)>0\)for all \(t\in[0,1]\).
Proof
(1) and (2) are proved in [19], we omit the proofs.
(3) For \(t \in[0,1]\), by a simple integral operation, we can obtain
(4) From (9), we know \(G_{1}(t,s)\geq0\), \(t,s\in[0,1]\), and \(G_{1}(t,s)>0\), \(t,s\in(0,1)\). Combined with (L0), for \(t\in[0,1]\), we have
so that \(G_{2}(t,s)\) monotonically increase with respect to t.
As a consequence, from (10), we get
Hence, \(G(t,s)\geq0\).
(5) From (3), for \(t\in[0,1]\), we know
so that
□
Lemma 2.6
Let \(\eta\in(0,\frac{1}{2})\), then
where
Proof
Step 1: We prove
For \(0\leq s < t\leq1\) and \(t\in[0,\eta]\),
so that
and
For \(s\geq t\) and \(t \in[0,\eta]\),
so that
and
Therefore, (16) holds.
Step 2: We prove
From Lemma 2.5, we know that \(G_{2}(t,s)\) is a monotone increasing function with respect to \(t \in[0,1]\), so that
By (L0) and (4), we have
Obviously, \(\rho>0 \) and
so that \(0<\rho<\frac{1}{2}\) and (17) hold.
Finally, from (16) and (17), we can easily show that the following results hold:
and
□
Lemma 2.7
Assume (L0) hold, then the function \(\omega(t)\)satisfies the following properties:
- (1)
\(\omega(t)\leq\omega(1)=\max_{t\in[0,1]}\omega(t)\);
- (2)
\(\min_{t\in[0,\eta]}\omega(t)\geq\rho\max_{t\in [0,1]}\omega(t)\), whereρis given by (15).
Proof
From Lemma 2.5 and (3), we have
and
Hence,
□
To finish this section, we present the well-known Avery–Peterson fixed point theorem as follows.
Let γ and θ be nonnegative continuous convex functionals on P, φ be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P. For \(A,B,C,D>0\), we define the following convex set:
and a closed set
Lemma 2.8
(see [27])
LetPbe a cone in a real Banach spaceE. Letγandθbe nonnegative continuous convex functionals onP, φbe a nonnegative continuous concave functional onP, andψbe a nonnegative continuous functional onPsatisfying \(\psi(\lambda x)\leq\lambda\psi(x)\)for \(0\leq\lambda\leq1\), such that, for some positive numbersMandD, \(\varphi(x)\leq\psi(x)\), \(\Vert x \Vert\leq M\gamma(x)\)for all \(x\in\overline{P(\gamma;D)}\). Suppose
is completely continuous and there exist positive numbersA, B, andCwith \(A < B\)such that
- \((H1)\):
\(\{x\in P(\gamma,\theta,\varphi;B,C,D):\varphi(x)>B\} \neq{\O}\), and \(\varphi(x)>B \)for \(x\in P(\gamma,\theta,\varphi;B,C,D)\);
- \((H2)\):
\(\varphi(Tx)>B\)for \(x\in P(\gamma,\varphi;B,D)\)with \(\theta(Tx)>A\);
- \((H3)\):
\(0\notin P(\gamma,\psi;A,D)\)and \(\psi(Tx)< A\)for \(x\in P(\gamma,\psi;A,D)\)with \(\psi(x)=A\).
Then T has at least three fixed point \(x_{1},x_{2},x_{3}\in\overline {P(\gamma;D)}\) such that
3 Main results
In this section, we prove the existence of positive solution of BVP (1) by applying the following Avery–Peterson fixed point theorem.
We consider the Banach space \(E=\{x\in C[0,1]:D_{0+}^{\beta}x\in C[0,1]\}\) with the norm
Let
then P is a cone in E.
Define the operator \(T:P \rightarrow E\) by
Lemma 3.1
Assume (L0) hold, then \(T:P \rightarrow P\)is a completely continuous operator.
Proof
For \(x \in P\), it is easy to see that T is continuous operator and \(Tx(t)\geq0\). By (14), we have
From Lemma 2.5 and Lemma 2.6 and Lemma 2.7, similar to Lemma 3.1 in [19], we can easily prove that T is a completely continuous operator. □
Define continuous nonnegative convex functionals as
Define continuous nonnegative concave functionals as
Thus
where \(M=1\).
Let
and
Theorem 3.1
Suppose (L0) hold, and there exist constants \(A, B, D \geq\omega (1) \)with \(A < B <\rho D \min\{\frac{J_{3}}{J_{1}},\frac {J_{3}}{J_{2}}\} \)and \(C=\frac{B}{\rho}\), such that
- \((L_{1})\):
\(f(t,x,y) \leq\min \{\varphi_{p}(\frac{D-\omega (1)}{J_{1}}), \varphi_{p}(\frac{D}{J_{2}}) \}\), \((t,x,y)\in[0,1] \times [0,D] \times[0,D]\);
- \((L_{2})\):
\(f(t,x,y) > \varphi_{p}(\frac{B-\rho\omega(1)}{\rho J_{3}})\), \((t,x,y)\in[0,\eta]\times[B,\frac{B}{\rho}]\times[0,D]\);
- \((L_{3})\):
\(f(t,x,y) < \varphi_{p}(\frac{A-\omega(1)}{J_{1}})\), \((t,x,y)\in[0,1]\times[0,A]\times[0,D]\).
Then BVP (1) has at least three positive solutions \(x_{1}\), \(x_{2}\), \(x_{3}\), satisfying
Proof
Obviously, the function x is a positive solution of BVP (1) if and only if x is a fixed point of the operator T in P.
For \(x \in\overline{P(\gamma;D)}\), we get
this implies
From \((L_{1})\), we get
and
so that
Therefore \(T:\overline{P(\gamma;D)} \rightarrow\overline{P(\gamma;D)}\).
From \(\frac{B}{\rho} \in P(\gamma,\theta,\varphi;B,C,D)\) and \(\varphi (\frac{B}{\rho}) > B \), we have
For \(x \in P(\gamma,\theta,\varphi;B,C,D)\), we know that \(B \leq x(t) \leq C=\frac{B}{\rho} \) for \(t\in[0,\eta]\) and \(0 \leq{ D_{0+}^{\beta}x(t) \leq D}\).
By \((L_{2})\),
So \(\varphi(Tx)>B\) for all \(x \in P(\gamma,\theta,\varphi;B,C,D)\). Hence, the condition \((H1)\) of Lemma 2.8 is satisfied.
For all \(x \in P(\gamma,\varphi;B,D) \) with \(\theta(Tx)>C=\frac{B}{\rho }\), we have
Thus, the condition \((H2)\) of Lemma 2.8 holds.
Because of \(\psi(0) = 0 < A \), then \(0 \notin P(\gamma,\psi;A,D)\). For \(x \in P(\gamma,\psi;A,D)\) with \(\psi(x)=A\), we know \(\gamma(x) \leq D \). It means that \(\max_{t \in[0,1]}x(t)=A\) and \(0 \leq D_{0+}^{\beta }x(t) \leq D\).
From \((L_{3})\), we can obtain
Therefore, the condition \((H3)\) of Lemma 2.8 holds.
To sum up, the conditions of Lemma 2.8 are all verified and we notice that \(x_{i}(t)\geq{\omega(0)>0}\). Hence, BVP (1) has at least three positive solutions \(x_{1}\), \(x_{2}\), \(x_{3}\) satisfying (18) and (19). □
4 Example
Consider the following boundary value problem:
where \(\alpha=\frac{11}{3}\), \(\beta=\frac{7}{3}\), \(p=\frac{3}{2}\), \(g_{0}(t)=t\), \(g_{1}(t)=t^{2}+t\), and
Choose \(A=3\), \(B=65\), \(D=25{,}000\), \(\eta=\frac{1}{4}\). By simple computation, we have
We can check that the nonlinear term \(f(t,x,y)\) satisfies
- \((L_{1})\):
\(\max f(t,x,y)\approx383.413 \leq\min \{\varphi _{p}(\frac{D-\omega(1)}{J_{1}}), \varphi_{p}(\frac{D}{J_{2}}) \}\approx 412.216\), \((t,x,y)\in[0,1] \times [0,25{,}000] \times[0,25{,}000]\);
- \((L_{2})\):
\(\min f(t,x,y) \approx381.413 > \varphi_{p}(\frac {B-\rho\omega(1)}{\rho J_{3}})\approx228.283\), \((t,x,y)\in[0,\frac {1}{3}]\times[65,\frac{B}{\rho}]\times [0,25{,}000]\);
- \((L_{3})\):
\(\max f(t,x,y) \approx1.02108 < \varphi_{p}(\frac {A-\omega(1)}{J_{1}})\approx1.05195\), \((t,x,y)\in[0,1]\times[0,3]\times [0,25{,}000]\).
Thus, from Theorem 3.1, we know that BVP (20) has at least three positive solutions \(x_{1}\), \(x_{2}\), \(x_{3}\), satisfying
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Zhang, L., Zhang, W., Liu, X. et al. Positive solutions of fractional p-Laplacian equations with integral boundary value and two parameters. J Inequal Appl 2020, 2 (2020). https://doi.org/10.1186/s13660-019-2273-6
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DOI: https://doi.org/10.1186/s13660-019-2273-6