Abstract
By using the continuation theorem of coincidence degree theory, we study the existence of solutions of Duffing type fractional differential equations with a p-Laplacian operator. Under certain nonlinear growth conditions of the nonlinearity, we obtain a new result on the existence of solutions.
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1 Introduction
Fractional calculus is a generalization of ordinary differentiation and integration on an arbitrary order that may be noninteger. Fractional differential equations have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. We can see numerous applications in viscoelasticity, neurons, electrochemistry, control, etc. (see [1–6]). Recently, with the intensive development of the theory of fractional calculus itself and its applications, there have many important results of fractional differential equations on initial value problems, and boundary value problems at nonresonance and resonance (see [7–12]).
In the study of the turbulent flow in a porous medium, Leibenson (see [13]) introduced the p-Laplacian equation as follows:
where \(\phi_{p}(s)=|s|^{p-2}s\), \(p>1\). Obviously, \(\phi_{p}\) is invertible and its inverse operator is \(\phi_{q}\), where \(q>1\) is a constant such that \(1/p+1/q=1\). In the past few decades, many important results relative to (1.1) with certain boundary value conditions have been obtained. See the papers [14–20] and the references therein. However, as far as we know, work on the existence of solutions for periodic boundary value problems (PBVPs for short) of fractional differential equations was discussed less.
The aim of this paper is to concentrate on the following periodic boundary value problem for Duffing type fractional differential equations with p-Laplacian operator:
where \(0<\alpha,\beta\leq1\), \(D_{0^{+}}^{\alpha},D_{0^{+}}^{\beta}\) are Caputo fractional derivatives, \(T>0\) is a given constant, and \(g:[0,T]\times \mathbb{R}\rightarrow\mathbb{R}\), \(e:[0,T]\rightarrow\mathbb{R}\) are continuous. Throughout this paper, we assume that
The choice of periodic boundary conditions is motivated by the difficulty in the study of the PBVP
As we know, PBVP (1.3) is not solvable for each \(h\in C([0,T],\mathbb{R})\), and, when solvable, has no unique solution because \(x(t)+c\), \(\forall c\in\mathbb{R}\) is a solution together with \(x(t)\). A trivial necessary condition for the solvability of PBVP (1.3) is that \(\int_{0}^{T}(T-s)^{\beta-1}h(s)\,ds=0\).
Notice that \(D_{0^{+}}^{\beta}(\phi_{p}(D_{0^{+}}^{\alpha}) )\) is a nonlinear operator, so the coincidence degree theory for linear differential operators is invalid in the direct application to it.
The rest of this paper is organized as follows. In Section 2, we describe the fractional differential operator and some lemmas. In Section 3, some sufficient conditions for the existence of solutions for PBVP (1.2) are established, and a new result on the existence of solutions is obtained. Finally, in Section 4, an example is given to illustrate the main result.
2 Preliminaries
Some definitions of the fractional derivative have emerged over the years (see [21, 22]), and in this paper we restrict our attention to the use of the Caputo fractional derivative. In this section, we introduce some basic definitions and lemmas which will be used in what follows. For details, we refer the reader to [21–25].
Definition 2.1
The Riemann-Liouville fractional integral operator of order \(\alpha>0\) of a function \(u:(0,+\infty )\rightarrow\mathbb{R}\) is given by
provided that the right side integral is pointwise defined on \((0,+\infty)\), where \(\Gamma(\cdot) >0\) is the Gamma function.
Definition 2.2
The Caputo fractional derivative of order \(\alpha>0\) of a continuous function \(u:(0,+\infty)\rightarrow \mathbb{R}\) is given by
where n is the smallest integer greater than or equal to α, provided that the right side integral is pointwise defined on \((0,+\infty)\).
Lemma 2.1
(see [23])
Let \(\alpha>0\). Assume that u, \(D_{0^{+}}^{\alpha}u\in L([0,T],\mathbb{R})\). Then the following equality holds:
where \(c_{i}\in{\mathbb{R}}\), \(i=0,1,\ldots,n-1\), here n is the smallest integer greater than or equal to α.
Lemma 2.2
(see [24])
For any \(u,v\geq0\),
Now we briefly recall some notations and an abstract existence result, which can be found in [25].
Let X, Y be real Banach spaces, \(L: \operatorname{dom}L\subset X\rightarrow Y\) be a Fredholm operator with index zero, and \(P: X\rightarrow X\), \(Q:Y\rightarrow Y \) be projectors such that
It follows that
is invertible. We denote the inverse by \(K_{P}\).
If Ω is an open bounded subset of X such that \(\operatorname{dom}L\cap \bar{\Omega}\neq\varnothing\), then the map \(N:X\rightarrow Y\) will be called L-compact on \(\bar{\Omega}\) if \(QN(\bar{\Omega})\) is bounded and \(K_{P}(I-Q)N:\bar{\Omega}\rightarrow X\) is compact.
Lemma 2.3
(see [25])
Let X and Y be two Banach spaces, \(L:\operatorname{dom}L\subset X\rightarrow Y\) be a Fredholm operator with index zero, \(\Omega\subset X\) be an open bounded set, and \(N:\bar {\Omega}\rightarrow Y\) be L-compact on \(\bar{\Omega}\). Suppose that all of the following conditions hold:
-
(1)
\(Lx\neq\lambda Nx\), \(\forall x\in\partial\Omega\cap\operatorname{dom}L\), \(\lambda\in(0,1)\);
-
(2)
\(QNx\neq0\), \(\forall x\in\partial\Omega\cap\operatorname{Ker}L\);
-
(3)
\(\operatorname{deg}(JQN,\Omega\cap\operatorname{Ker}L,0)\neq0\), where \(J: \operatorname{Im}Q\rightarrow\operatorname{Ker}L\) is an isomorphism map.
Then the equation \(Lx=Nx\) has at least one solution on \(\bar{\Omega}\cap\operatorname{dom}L\).
3 Existence result
For making use of the continuation theorem to study the existence of solutions for PBVP (1.2), we consider a system as follows:
Clearly, if \(x(\cdot)=(x_{1}(\cdot),x_{2}(\cdot))^{\mathrm{T}}\) is a solution of PBVP (3.1), then \(x_{1}(\cdot)\) must be a solution of PBVP (1.2). So, to prove PBVP (1.2) has solutions, we only need to show that PBVP (3.1) has solutions.
In this paper, we take \(X=\{x=(x_{1},x_{2})^{\mathrm{T}}\mid x_{1},x_{2}\in C([0,T],\mathbb{R})\}\) with the norm \(\|x\|=\max\{\|x_{1}\|_{0},\|x_{2}\|_{0}\} \), where \(\|x_{i}\|_{0}=\max_{t\in[0,T]}|x_{i}(t)|\) (\(i\in\{1,2\}\)). By means of the linear functional analysis theory, we can prove X is a Banach space.
Define the operator \(L:\operatorname{dom}L\subset X\rightarrow X\) by
where
Let \(N:X\rightarrow X\) be defined by
It is easy to see that PBVP (3.1) can be converted to the operator equation
Now let us introduce some lemmas.
Lemma 3.1
Let L be defined by (3.2), then
Proof
Obviously, from Lemma 2.1, we can see that (3.4) holds.
If \(y\in\operatorname{Im}L\), then there exists \(x\in\operatorname{dom}L\) such that \(y=Lx\). That is, \(y_{1}(t)=D_{0^{+}}^{\alpha}x_{1}(t)\), \(y_{2}(t)=D_{0^{+}}^{\beta}x_{2}(t)\). By using Lemma 2.1, we have
From conditions \(x_{1}(0)=x_{1}(T)\), \(x_{2}(0)=x_{2}(T)\), we obtain
On the other hand, suppose \(y\in X\) and satisfies (3.6). Let \(x_{1}(t)=I_{0^{+}}^{\alpha}y_{1}(t)\), \(x_{2}(t)=I_{0^{+}}^{\beta}y_{2}(t)\). Obviously \(x_{1}(0)=x_{1}(T)\), \(x_{2}(0)=x_{2}(T)\). Hence \(x=(x_{1},x_{2})^{\mathrm{T}}\in\operatorname {dom}L\) and \(Lx=y\). So \(y\in\operatorname{Im}L\). The proof is complete. □
Lemma 3.2
Let L be defined by (3.2), then L is a Fredholm operator of index zero. The projectors \(P:X \rightarrow X\) and \(Q:X \rightarrow X\) can be defined as
Furthermore, the operator \(K_{P}: \operatorname{Im}L\rightarrow{\operatorname{dom}L\cap \operatorname{Ker}P}\) can be written as
which is \((L|_{\operatorname{dom}L\cap\operatorname{Ker}P})^{-1}\).
Proof
For any \(y \in X\), we have
Let \(y^{*}=y-Qy=\bigl( {\scriptsize\begin{matrix}{}y_{1}^{*}\cr y_{2}^{*} \end{matrix}} \bigr)\), then we get
Similarly, we have \(\int_{0}^{T}(T-s)^{\beta-1}y_{2}^{*}(s)\,ds=0\). So \(y^{*}\in \operatorname{Im}L\). Hence \(X=\operatorname{Im}L+\operatorname{Im}Q\). Since \(\operatorname{Im}L\cap \operatorname{Im}Q=\{0\}\), we have \(X=\operatorname{Im}L\oplus\operatorname{Im}Q\). Thus
This means that L is a Fredholm operator of index zero.
From the definition of \(K_{P}\), for \(y\in\operatorname{Im}L\), we have
On the other hand, for \(x\in\operatorname{dom}L\cap\operatorname{Ker}P\), we have \(x_{1}(0)=x_{2}(0)=0\). By Lemma 2.1, we get
So, we know that \(K_{P}=(L_{\operatorname{dom}L\cap\operatorname{Ker}P})^{-1}\). The proof is complete. □
Lemma 3.3
Let N be defined by (3.3). Assume \(\Omega\subset X\) is an open bounded subset such that \(\operatorname{dom}L\cap \bar{\Omega}\neq\varnothing\), then N is L-compact on \(\bar{\Omega}\).
Proof
By the continuity of \(\phi_{q}\), e, g, we find that \(QN(\bar{\Omega})\) and \(K_{P} (I-Q)N(\bar{\Omega})\) are bounded. Moreover, there exists a constant \(M>0\) such that \(\|(I-Q)Nx\|\leq M\), \(\forall x\in\bar{\Omega}\), \(t\in[0,T]\). Thus, in view of the Arzelà-Ascoli theorem, we only need prove that \(K_{P}(I-Q)N(\bar{\Omega})\subset X\) is equicontinuous.
For \(0\leq t_{1}< t_{2}\leq T\), \(x\in\bar{\Omega}\), we have
From \(\|(I-Q)Nx\|\leq M\), \(\forall x\in\bar{\Omega}\), \(t\in[0,T]\), we can see that
Since \(t^{\alpha}\) is uniformly continuous on \([0,T]\), we find that \((K_{P}(I-Q)N(\bar{\Omega}))_{1}\subset C([0,T],\mathbb{R})\) is equicontinuous. A similar proof can show that \((K_{P}(I-Q)N(\bar{\Omega}))_{2}\subset C([0,T],\mathbb{R})\) is equicontinuous. So we find that \(K_{P}(I-Q)N:\bar{\Omega}\rightarrow X\) is compact. The proof is complete. □
Now we give the main result as regards the existence of solutions for PBVP (1.2).
Theorem 3.1
Assume that:
- (H1):
-
there exists a constant \(d_{1}>0\) such that
$$ (-1)^{i}xg(t,x)>0 \quad\bigl(i\in\{1,2\}\bigr), \forall t\in[0,T], |x|>d_{1}; $$ - (H2):
-
there exist a constant \(d_{2}>0\) and nonnegative functions \(a,b\in C([0,T],\mathbb{R})\) such that
$$ \bigl\vert g(t,x)\bigr\vert \leq a(t)|x|^{p-1}+b(t),\quad \forall t \in[0,T],|x|>d_{2}. $$Then PBVP (1.2) has at least one solution, provided that
$$ \begin{aligned} &\gamma_{1}:=\frac{2^{p} T^{\beta+{\alpha p-\alpha}} \|a\|_{0}}{\Gamma (\beta+1)(\Gamma(\alpha+1))^{p-1}}< 1, \quad\textit{if } p< 2; \\ & \gamma_{2}:=\frac{2^{2p-2} T^{\beta+{\alpha p-\alpha}} \|a\|_{0}}{\Gamma (\beta+1)(\Gamma(\alpha+1))^{p-1}}< 1, \quad\textit{if } p\geq2. \end{aligned} $$(3.7)
Proof
Set
For \(x\in\Omega'\), we get \(Nx\in\operatorname{Im}L\). So by (3.5), we have
From the integral mean value theorem and \(\int_{0}^{T}(T-s)^{\beta -1}e(s)\,ds=0\), there exist constants \(\zeta,\eta\in(0,T)\) such that \(x_{2}(\zeta)=0\), \(g(\eta,x_{1}(\eta))=0\). Together with the condition (H1), we have \(|x_{1}(\eta)|\leq d_{1}\). By Lemma 2.1, we have
which, together with
and \(|x_{1}(\eta)|\leq d_{1}\), yields
On the other hand, if \(x\in\Omega'\), we have
From the first equation of (3.9), we get \(x_{2}(t)=\phi_{p}(\lambda ^{-1} D_{0^{+}}^{\alpha}x_{1}(t))\). By substituting it into the second equation of (3.9), we get
Thus, by Lemma 2.1, we obtain
Then we have
By the boundary condition \(x_{1}(0)=x_{1}(T)\), we get
Obviously, there exists a constant \(\xi\in(0,T)\) such that \(\phi _{q}(c_{0}+\lambda^{p}I_{0^{+}}^{\beta}N_{g}x_{1}(\xi))=0\), which implies that \(c_{0}=-\lambda^{p}I_{0^{+}}^{\beta}N_{g}x_{1}(\xi)\). By substituting it into (3.10), we have
From the hypothesis (H2), we get
where \(G_{d_{2}}=\max \{|g(t,x)|\mid t\in[0,T],|x|\leq d_{2}\} \). Together with (3.8), (3.11), and
we have
If \(p<2\), by using Lemma 2.2, we get
where \(A_{1}=\frac{2T^{\beta}}{\Gamma(\beta+1)}(\|e\|_{0}+\|b\| _{0}+G_{d_{2}})+\frac{2T^{\beta}\|a\|_{0}}{\Gamma(\beta+1)}d_{1}^{p-1}\). Then, from (3.7), we have
Thus, from (3.8), we get
If \(p\geq2\), similar to the above argument, let \(A_{2}=\frac{2T^{\beta}}{\Gamma(\beta+1)}(\|e\|_{0}+\|b\|_{0}+G_{d_{2}})+\frac{2^{p-1}T^{\beta}\|a\| _{0}}{\Gamma(\beta+1)}d_{1}^{p-1}\), we obtain
where \(M_{2}=(\frac{A_{2}}{1-\gamma_{2}})^{q-1}\). So, combining (3.12) with (3.13), we get
From the second equation of (3.9) and Lemma 2.1, we have
which together with \(x_{2}(\zeta)=0\) yields
Then we have
where \(G_{M}=\max\{|g(t,x)| \mid t\in[0,T],|x|\leq M\}\). Together with (3.14), we obtain
Let \(\Omega=\{x\in X\mid\|x\|< M_{0}+1\}\). From the above argument, we know that the equation
has no solution on \(\partial\Omega\cap\operatorname{dom}L \). So the condition (1) of Lemma 2.3 is satisfied. Next the other two conditions of Lemma 2.3 are to be verified.
For \(x\in\operatorname{Ker}L\), we have \(x_{1}(t)=c_{1}\), \(x_{2}(t)=c_{2}\), \(\forall t\in [0,T]\), \(c_{1},c_{2}\in\mathbb{R}\). If \(QNx=0\), we obtain
From the first equality, we get \(c_{2}=0\). From the second equality and (H1), we have \(|c_{1}|\leq d_{1}\). So \(\|x\|=\max\{|c_{1}|,|c_{2}|\} \leq d_{1}< M_{0}+1\). Then the condition (2) of Lemma 2.3 is satisfied.
Define the operators \(J:\operatorname{Im}Q\rightarrow\operatorname{Ker}L\) by
and \(F: [0,1]\times\bar{\Omega}\rightarrow X\) by
where \(i\in\{1,2\}\). Let \(x\in\operatorname{Ker}L\) satisfying \(F(\mu,x)=0\), we get \(x_{1}(t)=c_{1}\), \(x_{2}(t)=c_{2}\), \(\forall t\in[0,T]\), \(c_{1},c_{2}\in\mathbb{R}\), and
From (3.16), we get \(c_{2}=0\) because \(c_{2}\) and \(\phi_{q}(c_{2})\) have the same sign. From (3.15), if \(\mu=0\), we get \(|c_{1}|\leq d_{1}\) because of (H1). If \(\mu\in(0,1]\), we also get \(|c_{1}|\leq d_{1}\). In fact, if \(|c_{1}|>d_{1}\), in view of (H1), one has
which contradicts (3.15). From the argument above, we obtain \(\|x\| < M_{0}+1\). Thus
Hence, by the homotopy property of the degree, we have
So the condition (3) of Lemma 2.3 is satisfied.
Consequently, by using Lemma 2.3, the operator equation \(Lx=Nx\) has at least one solution \(x(\cdot)=(x_{1}(\cdot),x_{2}(\cdot))^{\mathrm{T}} \) on \(\bar{\Omega}\cap\operatorname{dom}L\). Namely, PBVP (1.2) has at least one solution \(x_{1}(\cdot)\). The proof is complete. □
4 An example
In this section, we will give an example to illustrate our main result.
Example 4.1
Consider the following PBVP for a fractional p-Laplacian equation:
Corresponding to PBVP (1.2), we get \(p=4\), \(\alpha=1/2\), \(\beta =3/4\), \(T=1\), \(e(t)=(1-t)^{\frac{1}{4}}\sin2\pi t\), and
Choose \(a(t)=\frac{1}{120}\), \(b(t)=1\). By a simple calculation, we obtain
Obviously, PBVP (4.1) satisfies all assumptions of Theorem 3.1. Hence, PBVP (4.1) has at least one solution.
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Acknowledgements
The authors would like to thank the anonymous referee for his/her valuable comments, which have improved the correctness and presentation of the manuscript. This work is supported by the National Natural Science Foundation of China (11271364).
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Jin, H., Liu, W. On the periodic boundary value problem for Duffing type fractional differential equation with p-Laplacian operator. Bound Value Probl 2015, 144 (2015). https://doi.org/10.1186/s13661-015-0408-3
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DOI: https://doi.org/10.1186/s13661-015-0408-3