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Multiplicity results for impulsive fractional differential equations with p-Laplacian via variational methods
- Yulin Zhao^{1} and
- Liang Tang^{1}Email author
Received: 18 January 2017
Accepted: 10 August 2017
Published: 25 August 2017
Abstract
In this paper, we apply critical point theory and variational methods to study the multiple solutions of boundary value problems for an impulsive fractional differential equation with p-Laplacian. Some new criteria guaranteeing the existence of multiple solutions are established for the considered problem.
Keywords
1 Introduction and main results
Fractional calculus is a generalization of classical derivatives and integrals to an arbitrary (non-integer) order. It represents a powerful tool in applied mathematics to deal with a myriad of problems from different fields such as physics, mechanics, electricity, control theory, rheology, signal and image processing, aerodynamics, electricity, etc. (see [1–3] and the references therein). Recently the theory and application of fractional differential equations have been rapidly developed. The existence and multiplicity of solutions to such problems have been extensively studied by many mathematicians; see the monographs of Podlubny [1], Kilbas [4], Diethelm [5], and also [6–12] and the references therein. The main classical techniques to study fractional differential equations are degree theory, the method of upper and lower solutions, and fixed point theorems. For some related work on the theory and application of fractional differential equations, we refer the interested reader to [11–17] and the references therein.
In recent years, variational methods and critical point theory have already been applied successfully to investigate the existence of solutions for nonlinear fractional boundary value problems [18–27]. By establishing a corresponding variational structure and using the Mountain Pass theorem, the authors [18] first dealt with the existence of solutions for a class of fractional boundary value problems. Since then the variational methods are applied to discuss the existence of solutions for fractional differential equations. The literature on this approach has been extended by many authors as [20–26]. Moreover, the p-Laplacian introduced by Leibenson (see [28]) often occurs in non-Newtonian fluid theory, nonlinear elastic mechanics and so on. Note that, when \(p=2\), the nonlinear and nonlocal differential operator \({{}_{t}D}_{T}^{\alpha}\Phi_{p}({}_{0}^{c}D_{t}^{\alpha})\) reduces to the linear differential operator \({{}_{t}D}_{T}^{\alpha}{{}_{0}^{c}D_{t}^{\alpha}}\), and further reduces to the local second-order differential operator \(-d^{2}/dt^{2}\) when \(\alpha=1\).
Motivated by the described work, our goal is to apply variational methods to problem (1.1) and prove the existence of weak solutions under some suitable assumptions. With the impulsive effects and p-Laplacian operator taken into consideration, the corresponding variational functional φ will be more complicated. To the best of our knowledge, with exception of [38], little work is done on the existence and multiplicity of solutions for impulsive fractional differential problems with p-Laplacian by using variational methods. The main results of this paper are different from the aforementioned results, and extend the recent results studied in [22, 25, 29–35] in the sense that we deal with the case \(p\neq2\). The effectiveness of our results is illustrated by some examples.
- (H1)
There exists a constant \(\mu>p\) such that \(I_{j}(u)u\leq\mu \int_{0}^{u}I_{j}(s)\,ds<0\) for any \(u\in E^{\alpha,p}\setminus\{0\}\), \(j=1,2,\ldots,m\), where \(E^{\alpha,p}\) will be introduced in Definition 2.3.
- (H2)
There exists a constant \(\vartheta\in(p,\mu]\) such that \(\vartheta F(t,u)\leq f(t,u)u\) for all \(u\in E^{\alpha,p}\), \(t\in[0,T]\), where \(F(t,u)=\int_{0}^{u}f(t,s)\,ds\).
- (H3)There exist constants \(\delta,\gamma>0\) such that \(F^{0}\leq \delta\) and \(F_{\infty}\geq\gamma\), where$$F^{0}=\lim_{|u|\rightarrow0}\sup\frac{F(t,u)}{|u|^{\vartheta}},\quad\quad F_{\infty}=\lim_{|u|\rightarrow\infty}\inf\frac {F(t,u)}{|u|^{\vartheta}}. $$
- (H4)
There exist constants \(\delta_{j}>0\) such that \(\int _{0}^{u}I_{j}(s)\,ds\geq-\delta_{j}|u|^{\mu}\), for all \(u\in{E^{\alpha ,p}}\setminus\{0\}\), where \(j=1,\ldots,m\).
Here are our main results.
Theorem 1.1
Suppose that (H1)-(H4) hold. Then problem (1.1) admits at least two weak solutions.
Theorem 1.2
Suppose that (H1)-(H4) hold. Moreover, \(f(t,u)\) and \(I_{j}(u)\) are odd about u, where \(j=1,\ldots,m\). Then problem (1.1) admits infinitely many weak solutions.
The rest of this paper is organized as follows. In Section 2, we present some basic definitions, lemmas and a variational setting. In Section 3, we give the proofs of our main results.
2 Variational setting and preliminaries
To apply critical point theory to discuss the existence of solutions for problem (1.1), we present some basic notations and lemmas and construct a variational framework, which will be used in the proof of our main results.
Suppose that X is a real Banach space and the functional \(\phi :X\rightarrow\mathbb{R}\) is differentiable. The functional ϕ satisfies the Palais-Smale condition if each sequence \(\{u_{n}\}\) in the space X such that \(\{\phi(u_{n})\}\) is bounded and \(\lim_{n\rightarrow\infty}\phi^{\prime}(u_{n})=0\) admits a convergent subsequence.
Lemma 2.1
Mountain pass theorem; see [39]
Lemma 2.2
Theorem 38.A in [40]
- (i)
X is a real reflexive Banach space;
- (ii)
B is bounded and weak sequentially closed;
- (iii)
ϕ is weakly sequentially lower semi-continuous in B, i.e., by definition, for every sequence \(\{u_{n}\}\) in B such that \(u_{n}\rightharpoonup u\) as \(n\rightarrow\infty\), one has \(\phi(u)\leq\underline{\lim}_{n\rightarrow\infty}\phi(u_{n})\).
Lemma 2.3
Theorem 9.12 in [41]
- (i)
there exist constants \(\rho,\eta>0\) such that \(\phi |_{{ \partial B_{\rho}}\cap Z}\geq\eta\), and
- (ii)
for each finite dimensional subspace \(W\subset E\), there is an \(r=r(W)\) such that \(\phi\leq0\) on \(W\setminus{B_{r(W)}}\),
Now we present some definitions and notations of the fractional calculus as follows (for details, see [1, 4, 5, 13, 18]):
Denote by \(\operatorname{AC}([a,b])\) the space of absolutely continuous functions on \([a,b]\).
Definition 2.1
Definition 2.2
Let \(C_{0}^{\infty}([0,T],\mathbb{R})\) be the set of all functions \(x\in C^{\infty}([0,T],\mathbb{R})\) with \(x(0)=x(T)=0\) and the norm \(\| x\|_{\infty}=\max_{[0,T]}|x(t)|\). Denote the norm of the space \(L^{p}([0,T],\mathbb{R})\) for \(1\leq p<\infty\) by \(\|x\|_{L^{p}}= (\int_{0}^{T}|x(s)|^{p}\,ds )^{1/p}\).
Definition 2.3
According to [18], Proposition 3.1, it is well known that the space \(E^{\alpha,p}\) is a reflexive and separable Banach space. For \(u\in E^{\alpha,p}\), we have \(u,{{}_{0}^{c}D_{t}^{\alpha}}u\in L^{p}([0,T],R)\), \(u(0)=u(T)=0\).
Proposition 2.1
[13]
Proposition 2.2
[13]
Let \(\frac{1}{p}<\alpha\leq1\) and \(1< p<\infty\). For any \(u\in E^{\alpha,p}\), the imbedding of \(E^{\alpha,p}\) in \(C([0,T],\mathbb{R})\) is compact.
Proposition 2.3
[18], Proposition 3.3
Assume that \(\frac{1}{p}<\alpha\leq1\) and \(1< p<\infty\), and the sequence \(\{ u_{k}\}\) converges weakly to u in \(E^{\alpha,p}\), i.e., \(u_{k}\rightharpoonup u\). Then \(u_{k}\rightarrow u\) in \(C([0,T],\mathbb {R})\), i.e., \(\|u_{k}-u\|_{\infty}\rightarrow0\), as \(k\rightarrow\infty\).
Definition 2.4
Definition 2.5
Proposition 2.4
- (i)Let \(\alpha>0\), \(p\geq 1\), \(p\geq1\), and \(\frac{1}{p}+\frac{1}{q}\leq1+\alpha\) (\(p\neq 1\), \(q\neq1\), in the case when \(\frac{1}{p}+\frac{1}{q}=1+\alpha\)). If \(u\in L^{p}(a,b)\) and \(v\in L^{q}(a,b)\), then$$ \int_{a}^{b} \bigl({{}_{a}D^{-\alpha}_{t}}u(t) \bigr)v(t)\,dt= \int_{a}^{b}u(t) \bigl({{}_{t}D^{-\alpha}_{b}}v(t) \bigr)\,dt. $$(2.6)
- (ii)Let \(0<\alpha<1\), \(u\in \operatorname{AC}([a,b])\) and \(v\in L^{p}(a,b)\) (\(1\leq p<\infty \)). Then$$ \int_{a}^{b}u(t) \bigl({{}_{a}^{c}D^{\alpha}_{t}}v(t) \bigr)\,dt={{}_{t}D^{\alpha-1}_{b}}u(t)v(t)\big|_{t=a}^{t=b} + \int_{a}^{b}{{}_{t}D^{\alpha}_{b}}u(t)v(t) \,dt. $$(2.7)
Proposition 2.5
Proof
Proposition 2.6
If the function \(u\in E^{\alpha,p}\) is a weak solution of (1.1), then u is a classical solution of (1.1).
Proof
3 Proof of main results
In this section, we will study the existence and multiplicities of problem (1.1). First, we give a Lemma.
Lemma 3.1
Assume that (H1) and (H2) hold. Then the function \(\varphi:E^{\alpha,p}\rightarrow\mathbb{R}\) defined by (2.5) is continuous differentiable and weakly sequentially lower semi-continuous. Moreover, it satisfies the Palais-Smale condition.
Proof
Now we prove Theorem 1.1 and Theorem 1.2.
Proof of Theorem 1.1
Step I: Obviously, \(\varphi(0)=0\), and Lemma 3.1 has shown that φ satisfies the Palais-Smale condition. For any \(r>0\), take \(\Omega_{r}=\{u\in E^{\alpha,p}:\|u\| _{E^{\alpha,p}}< r\}\). It is easy to show that \(\overline{\Omega}_{r}\) is bounded and weakly sequentially closed. Indeed, if we let \(\{u_{n}\} \subseteq\overline{\Omega}_{r}\) and \(u_{n}\rightharpoonup u\) as \(n\rightarrow\infty\), by the Mazur Theorem [10], there is a sequence of convex combinations \(v_{n}=\sum_{i=1}^{n}\beta_{n_{i}}u_{i}\) with \(\sum_{i=1}^{n}\beta_{n_{i}}=1\), \(\beta_{n_{i}}\geq0\), \(i\in\textbf{N}\) such that \(v_{n}\rightarrow u\) in \(E^{\alpha,p}\). Since \(\overline{\Omega}_{r}\) is a closed convex set, we have \(\{v_{n}\}\subseteq\overline{\Omega}_{r}\) and \(u\in\overline{\Omega}_{r}\).
Step II: We will verify that there exists \(u_{1}\) with \(\| u_{1}\|_{E^{\alpha,p}}>r_{0}\) such that \(\varphi(u_{1})<\inf\{\varphi (u):u\in \partial\Omega_{r_{0}}\}\), where \(r_{0}\) is given above.
Example 3.1
Proof of Theorem 1.2
We will apply Lemma 2.3 to finish the proof. Obviously, \(\varphi\in C^{1}(E^{\alpha,p},\mathbb{R})\) is even and \(\varphi(0)=0\). Moreover, Lemma 3.1 shows that φ satisfies the Palais-Smale condition.
Example 3.2
4 Conclusion
In this paper, we have proved the existence and multiplicity of the solutions for an impulsive fractional differential equation with p-Laplacian operator. Our approach is based on the well-known mountain pass theorem and minimax methods in critical point. With the impulsive effects and p-Laplacian operator taken into consideration, the corresponding variational functional is more complicated. Therefore, the existence of solutions for impulsive fractional differential problems with p-Laplacian is interesting. As applications, two examples are presented to illustrate the main results. In the future, we will consider the existence of solutions for the impulsive fractional differential equation with p-Laplacian via Morse theory.
Declarations
Acknowledgements
The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by a Project of the Department of Education in Hunan Province (12C0088), and the Aid program for Science and Technology Innovative Research Team in Higher Educational Instituions of Hunan Province.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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