Skip to main content

Advertisement

We’d like to understand how you use our websites in order to improve them. Register your interest.

The existence of solutions for perturbed fractional differential equations with impulses via Morse theory

Abstract

In the present paper, the existence of nontrivial solutions of impulsive fractional differential equations with Dirichlet boundary conditions is studied. We apply Morse theory coupled with local linking arguments to solve the topic, and we prove the existence of at least one nontrivial solution for the impulsive fractional differential equations.

Introduction

Fractional calculus is a powerful tool for describing the genetic properties and memory processes of various materials [13]. Fractional differential equations (FDEs) have been widely used in the field of medical, physical, economic and technological sciences in recent times. Though fractional differential equations containing Riemann–Liouville fractional derivatives or Caputo fractional derivatives have got more and more attentions, the fixed point theorems, coincidence degree theory and monotone iteration methods are still the main approaches. For the critical point theory, we refer to [46] and the references therein.

In [7], the fractional boundary-value problems considered by Jiao and Zhou is listed as follows:

$$ \textstyle\begin{cases} - \frac{1}{2}\frac{d}{dt}({}_{0}D_{t}^{ - \beta } + {}_{t}D_{T}^{ - \beta } )u'(t) = \nabla F(t,u(t)), &\mbox{a.e. }t \in [0,T], \\ u(0) = u(T) = 0, \end{cases} $$
(1.1)

where \(\beta \in [0,1)\), \({}_{0}D_{t}^{ - \beta } \) and \({}_{t}D_{T}^{ - \beta } \) are the left and right Riemann–Liouville fractional derivatives respectively. \(F:[0,T] \times R^{N} \to R\) (with \(N \ge 1\)) is a suitable given function and \(\nabla F(t,x)\) is the gradient of F with respect to x. In this paper, the sufficient conditions for the existence of solutions are obtained by using the least action principle and the mountain path theorem. Since then, the variational methods have been applied to study fractional differential equations; see [710].

The problem (1.1) arose from the phenomenon of advection dispersion and was first scrutinized by Ervin and Loop in [11]. From then on, the existence and multiplicity of solutions for the above problem (1.1) or related problems were further studied by the authors in [1215] with the critical point theory.

The impulsive differential equations originated from the real world problems to describe the dynamics of processes in which sudden, discontinuous jumps occur. Due to their significance, many researchers established the solvability of impulsive differential equations. If you are interested in the general theory and applications of such equations, please refer to [1618] and the references therein.

Up to now, there are few papers that use variational methods and critical point theory to study the fractional boundary-value problems with impulses [1924].

In [22], the authors use variational methods and critical point theory to study the following fractional differential systems with impulsive effects:

$$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha _{i}}(a_{i}(t){}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}(t)) = \lambda F_{u_{i}}(t,u) + h_{i}(u_{i}(t)), \quad 0 < t < T, t \ne t_{j}, \\ \Delta ({}_{t}D_{T}^{\alpha _{i} - 1}({}_{0}^{c}D_{t}^{\alpha _{i}}u _{i})(t_{j}) = I_{ij}(u_{i}(t_{j})),\quad j = 1,2, \ldots ,m, \\ u_{i}(0) = u_{i}(T) = 0,\quad 1 \le i \le N. \end{cases} $$

In [23], the authors have considered the following boundary-value problems of impulsive fractional differential equations:

$$ \textstyle\begin{cases} - \frac{1}{2}\frac{d}{dt}({}_{0}D_{t}^{ - \beta _{i}} + {}_{t}D_{T} ^{ - \beta _{i}})u_{i}^{\prime } (t) \\ \quad = a_{i}(t)u_{i}(t) + F_{u_{i}}(t,u(t)), \quad 1 \le i \le N,t \ne t_{j}, \mbox{a.e. }t \in [0,T], \\ \Delta (D_{t}^{\alpha _{i}}u_{i})(t_{j}) = I_{ij}(u_{i}(t_{j})),\quad t_{j} \in (0,T), j = 1,2, \ldots ,l, \\ u_{i}(0) = u_{i}(T) = 0,\quad 1 \le i \le N, \end{cases} $$
(1.2)

where \(u = (u_{1}, \ldots ,u_{N})\), \(\vert u \vert = \sqrt{ \sum_{i = 1}^{N} u_{i}^{2}} \), \(\beta _{i} \in [0,1)\), \(\alpha _{i} = 1 - \frac{ \beta _{i}}{2} \in (\frac{1}{2},1]\) for \(1 \le i \le N\), \({}_{0}D_{t} ^{ - \beta _{i}}\), \({}_{t}D_{T}^{ - \beta _{i}}\) are the left and right Riemann–Liouville fractional integrals of order \(\beta _{i}\), \({}_{0}^{c}D_{t}^{\alpha _{i}}\) and \({}_{t}^{c}D_{T}^{\alpha _{i}}\) are the left and right Caputo fractional derivative of order \(a_{i}\), \(a_{i} \in L^{\infty } [0,T]\), \(0 = t_{0} < t_{1} < t_{2} < \cdots < t_{l} < t_{l + 1} = T\), \(I_{ij} \in C([0,T],R)\), \(F:[0,T] \times R^{N} \to R\) is measurable, continuously differentiable, \(F_{u_{i}}\) denotes the partial derivative of F with respect to \(u_{i}\) for \(1 \le i \le N\), and

$$\begin{aligned}& \bigl(D_{t}^{\alpha _{i}}u_{i}\bigr) (t_{j}) = \frac{1}{2} \bigl\{ {}_{0}D_{t} ^{\alpha _{i} - 1}\bigl({}_{0}^{c}D_{t}^{\alpha _{i}}u_{i} \bigr) - {}_{t}D_{T} ^{\alpha _{i} - 1} \bigl({}_{t}^{c}D_{T}^{\alpha _{i}}u_{i} \bigr) \bigr\} (t _{j}), \\& \Delta \bigl(D_{t}^{\alpha _{i}}u_{i}\bigr) (t_{j}) = \frac{1}{2} \bigl\{ {}_{0}D _{t}^{\alpha _{i} - 1}\bigl({}_{0}^{c}D_{t}^{\alpha _{i}}u_{i} \bigr) - {}_{t}D _{T}^{\alpha _{i} - 1} \bigl({}_{t}^{c}D_{T}^{\alpha _{i}}u_{i} \bigr) \bigr\} \bigl(t _{j}^{ +} \bigr) \\& \hphantom{\Delta (D_{t}^{\alpha _{i}}u_{i})(t_{j}) =}{} - \frac{1}{2} \bigl\{ {}_{0}D_{t}^{\alpha _{i} - 1} \bigl({} _{0}^{c}D_{t}^{\alpha _{i}}u_{i} \bigr) - {}_{t}D_{T}^{\alpha _{i} - 1}\bigl({} _{t}^{c}D_{T}^{\alpha _{i}}u_{i} \bigr) \bigr\} \bigl(t_{j}^{ -} \bigr), \\& \bigl\{ {}_{0}D_{t}^{\alpha _{i} - 1} \bigl({}_{0}^{c}D_{t}^{\alpha _{i}}u _{i}\bigr) - {}_{t}D_{T}^{\alpha _{i} - 1} \bigl({}_{t}^{c}D_{T}^{\alpha _{i}}u _{i}\bigr) \bigr\} \bigl(t_{j}^{ +} \bigr) = \lim_{t \to t_{j}^{ +}} \bigl\{ {}_{0}D_{t}^{\alpha _{i} - 1} \bigl({}_{0}^{c}D_{t}^{\alpha _{i}}u_{i} \bigr) - {}_{t}D_{T}^{\alpha _{i} - 1} \bigl({}_{t}^{c}D_{T}^{\alpha _{i}}u_{i} \bigr) \bigr\} (t), \\& \bigl\{ {}_{0}D_{t}^{\alpha _{i} - 1} \bigl({}_{0}^{c}D_{t}^{\alpha _{i}}u _{i}\bigr) - {}_{t}D_{T}^{\alpha _{i} - 1} \bigl({}_{t}^{c}D_{T}^{\alpha _{i}}u _{i}\bigr) \bigr\} \bigl(t_{j}^{ -} \bigr) = \lim_{t \to t_{j}^{ -}} \bigl\{ {}_{0}D_{t}^{\alpha _{i} - 1} \bigl({}_{0}^{c}D_{t}^{\alpha _{i}}u_{i} \bigr) - {}_{t}D_{T}^{\alpha _{i} - 1} \bigl({}_{t}^{c}D_{T}^{\alpha _{i}}u_{i} \bigr) \bigr\} (t), \end{aligned}$$

for \(j = 1, \ldots ,l\), \(1 \le i \le N\).

On the other hand, in recent years, Morse theory has been used to discuss the existence of solutions of differential equations [25, 26]. However, to the best of our knowledge, Morse theory is rarely applied to the impulsive fractional boundary-value problems.

In [27], based on Morse theory coupled with local linking arguments, the authors studied the following impulsive fractional differential equation:

$$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha } ({}_{0}^{c}D_{t}^{\alpha } u(t)) + k(t)u(t) = f(t,u),\quad 0 < t < T, t \ne t_{j}, \\ \Delta ({}_{t}D_{T}^{\alpha - 1}({}_{0}^{c}D_{t}^{\alpha _{i}}u))(t _{j}) = I_{j}(u(t_{j})),\quad j = 1,2, \ldots ,m, \\ u(0) = u(T) = 0, \end{cases} $$

Motivated by the work above, we will investigate the existence of at least one nontrivial weak solution of problem (1.2) by Morse theory. Compared with the research in [23], the method of this paper is different.

To investigate problem (1.2), we make the following assumptions.

\(( I0 )\):

\(I_{ij} \in C ( [ 0,T ],R )\), \(I _{ij} ( 0 ) = 0\), \(I_{ij} ( u_{i} )u_{i} \ge 0\) and there exist constants \(a _{j},b_{j} > 0\) and \(c_{j},\gamma _{j} \in [ 0,1 )\), such that \(\vert I_{ij} ( u ) \vert \le a_{j} \vert u_{i} \vert ^{\gamma _{j}}\), \(\lim_{ \vert u_{i} \vert \to 0}\frac{ \vert I_{j} ( u_{i} ) \vert }{ \vert u_{i} \vert ^{c_{j}}} = b_{j}\), \(i = 1,\ldots, N\), \(j = 1,\ldots, l\);

\(( I1 )\):

there exists \(\theta _{1} \ge 1\) such that \(\theta _{1}I_{ij}^{*} ( u ) \ge I_{ij}^{*} ( \zeta _{1}u_{i} )\), \(\forall u_{i} \in R\) and \(\zeta _{1} \in [ 0,1 ]\), where

$$ I_{ij}^{*} ( u ): = 2 \int _{0}^{u_{i}} I_{ij} ( s )\,ds - I_{ij} ( u_{i} )u_{i}. $$

We introduce the following conditions on the nonlinearity function \(F_{u_{i}}(t,u)\):

\(( F0 )\):

\(F_{u_{i}}(t,0) = 0\), \(\lim_{ \vert u \vert \to 0}\sup \frac{ \vert F ( t,u ) \vert }{ \vert u \vert ^{2}} < \sum_{i = 1}^{N} \frac{\varGamma ^{2} ( \alpha _{i} + 1 )}{2T^{2\alpha _{i}}} \vert \cos ( \pi \alpha _{i} ) \vert \) uniformly for \(t \in ( 0,T )\), and there are constants \(C > 0\), r, \(r_{0}\), γ with \(\gamma \in ( 1,\max_{j \in \{ 1,2,\ldots,l\}} \{ \gamma _{j} + 1 \} )\) such that

$$\begin{aligned}& F ( t,u ) \ge C \vert u_{i} \vert ^{\gamma },\quad r \le \vert u_{i} \vert \le r_{0}\mbox{ a.e. }t \in [ 0,T ]; \end{aligned}$$
\(( F1 )\):

there exists \(\theta _{2} \ge 1\) such that \(\theta _{2}F_{*} ( t,u ) \ge F_{*} ( t,\zeta _{2}u )\), \(\forall ( t,u ) \in [ 0,T ] \times R\), \(\zeta _{2} \in [ 0,1 ]\), where \(F_{*} ( t,u ): = F_{u_{i}} ( t,u )u _{i} - 2F ( t,u )\);

\(( F2 )\):

\(F_{u_{i}} ( t,u )u_{i} \ge 0\), \(\forall ( t,u ) \in [ 0,T ] \times R\); \(\lim_{ \vert u \vert \to \infty } \frac{F_{u_{i}} ( t,u )}{u} = + \infty\) uniformly for \(t \in ( 0,T )\);

\(( F3 )\):

\(\lim_{ \vert u \vert \to \infty } \frac{ \vert F ( t,u ) \vert }{ \vert u \vert ^{2}} = + \infty\) uniformly for \(t \in ( 0,T )\).

Theorem 1.1

Assume that\(( I0 )\), \(( I1 )\), \(( F0 )\), \(( F1 )\), \(( F2 )\)hold. Then problem (1.2) has at least one nontrivial weak solution.

Theorem 1.2

Assume that\(( I0 )\), \(( I1 )\), \(( F0 )\), \(( F1 )\), \(( F3 )\)are satisfied. Then problem (1.2) has at least one nontrivial weak solution.

Preliminaries

As discussed in [23], we can transfer problem (1.2) to the following problem:

$$ \textstyle\begin{cases} \frac{d}{dt} \{ \frac{1}{2}{}_{0}D_{t}^{\alpha _{i} - 1}({}_{0} ^{c}D_{t}^{\alpha _{i}}u_{i}(t)) - \frac{1}{2}{}_{t}D_{T}^{\alpha _{i} - 1}({}_{t}^{c}D_{T}^{\alpha _{i}}u_{i}(t)) \} + a_{i}(t)u_{i}(t) + F_{u_{i}}(t,u(t)) = 0, \\ \quad 1 \le i \le N,t \ne t_{j}, \mbox{a.e.}, t \in [0,T], \\ \Delta (D_{t}^{\alpha _{i}}u_{i})(t_{j}) = I_{ij}(u_{i}(t_{j})),\quad t_{j} \in (0,T), j = 1,2, \ldots ,l, \\ u_{i}(0) = u_{i}(T) = 0,\quad 1 \le i \le N, \end{cases} $$
(2.1)

The problem (1.2) is equivalent to problem (2.1). Therefore, a solution of problem (2.1) corresponds to a solution of the BVP (1.2).

A variational structure is established to transform the existence of solutions to problem (2.1) into the existence of corresponding functional critical points. We construct the following appropriate function spaces.

Let us recall that, for any fixed \(t \in [0,T]\) and \(1 \le p \le \infty \),

$$ \Vert u \Vert _{\infty } = \max_{t \in [0,T]} \bigl\vert u(t) \bigr\vert , \qquad \Vert u \Vert _{L^{p}} = \biggl( \int _{0}^{T} \bigl\vert u(s) \bigr\vert ^{p}\,ds \biggr)^{\frac{1}{p}}. $$

For \(\alpha _{i} \in [0,1)\), \(1 \le i \le N\), we define the fractional derivative spaces \(E_{0}^{\alpha _{i}}\) by the closure of \(C_{0}^{ \infty } ([0,T],R^{N})\) with \(u_{i}(0) = u_{i}(T)\) under the norm

$$ \Vert u_{i} \Vert _{\alpha _{i}} = \biggl( \int _{0}^{T} \bigl\vert {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}(t) \bigr\vert ^{2}\,dt + \int _{0}^{T} \bigl\vert u_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}, \quad \forall u_{i} \in E_{0}^{\alpha _{i}}. $$

Obviously, the fractional derivative space \(E_{0}^{\alpha _{i}}\) is the space of functions \(u_{i} \in L^{2}(0,T)\) having \(\alpha _{i}\)-order Caputo left and right fractional derivatives and Riemann–Liouville left and right fractional derivatives, , and \(u_{i}(0) = u _{i}(T) = 0\).

Definition 2.1

([23])

We denote \(u = (u_{1}, \ldots ,u_{N})\), \(u_{i} \in E_{0}^{\alpha _{i}}\), (\(i = 1,\ldots, N\)) this being a weak solution of the problem (2.1) if the following identity:

$$\begin{aligned}& \sum_{i = 1}^{N} \int _{0}^{T} \biggl\{ - \frac{1}{2} \bigl[{}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}(t){}_{t}^{c}D_{T}^{\alpha _{i}}v_{i}(t) + {}_{t}^{c}D_{T} ^{\alpha _{i}}u_{i}(t){}_{0}^{c}D_{t}^{\alpha _{i}}v_{i}(t) \bigr] - a_{i}(t)u _{i}(t)v_{i}(t)\biggr\} \,dt \\& \quad {}+ \sum_{i = 1}^{N} \sum _{j = 1}^{l} I_{ij} \bigl(u_{i}(t_{j})\bigr)v_{i}(t_{j}) - \sum_{i = 1}^{N} \int _{0}^{T} F_{u_{i}}\bigl(t,u(t) \bigr)v_{i}(t)\,dt = 0 \end{aligned}$$

holds for all \(\forall v_{i} \in E_{0}^{\alpha _{i}}\).

Consider the functional Φ: \(E_{0}^{\alpha _{1}} \times \cdots \times E_{0}^{\alpha _{N}} \to R\) defined by

$$\begin{aligned} \varPhi ( u ) =& \sum_{i = 1}^{N} \int _{0}^{T} \biggl\{ - \frac{1}{2} {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}(t){}_{t}^{c}D_{T}^{ \alpha _{i}}u_{i}(t) - \frac{1}{2}a_{i}(t)u^{2}_{i}(t) \biggr\} \,dt - \int _{0} ^{T} F\bigl(t,u(t)\bigr)\,dt \\ &{}+ \sum_{i = 1}^{N} \sum _{j = 1}^{l} \int _{0}^{u_{i} ( t_{j} )} I_{ij} ( s )\,ds. \end{aligned}$$

From \(( I0 )\) and \(( F0 )\), we can infer that Φ is continuous, differentiable and for all \(u = (u_{1}, \ldots ,u_{N})\), \(v = (v_{1}, \ldots ,v_{N})\), \(u_{i},v_{i} \in E_{0}^{\alpha _{i}}\) (\(i = 1,\ldots, N\)), and we have

$$\begin{aligned} \bigl\langle \varPhi '(u),v \bigr\rangle =& \sum _{i = 1}^{N} \int _{0} ^{T} \biggl\{ - \frac{1}{2} \bigl[{}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}(t){}_{t}^{c}D _{T}^{\alpha _{i}}v_{i}(t) + {}_{t}^{c}D_{T}^{\alpha _{i}}u_{i}(t){} _{0}^{c}D_{t}^{\alpha _{i}}v_{i}(t) \bigr] - a_{i}(t)u_{i}(t)v_{i}(t)\biggr\} \,dt \\ &{}+ \sum_{i = 1}^{N} \sum _{j = 1}^{l} I_{ij} \bigl(u_{i}(t_{j})\bigr)v_{i}(t_{j}) - \sum_{i = 1}^{N} \int _{0}^{T} F_{u_{i}}\bigl(t,u(t) \bigr)v_{i}(t)\,dt. \end{aligned}$$
(2.2)

Then, the critical point of Φ is the weak solution of (2.1).

Lemma 2.2

([7])

Let\(\frac{1}{2} < \alpha \le 1\)and\(1 < p < \infty \), for all\(u \in E_{0}^{\alpha } \), one has

$$ \Vert u \Vert _{L^{p}} \le \frac{T^{\alpha }}{\varGamma (\alpha + 1)} \bigl\Vert {}_{0}^{c}D_{t}^{\alpha } u \bigr\Vert _{L^{p}}. $$
(2.3)

Moreover, if \(\alpha > \frac{1}{p}\) and \(\frac{1}{p} + \frac{1}{q} = 1\), then

$$ \Vert u \Vert _{\infty } \le \frac{T^{\alpha - 1/p}}{\varGamma (\alpha )((\alpha - 1)q + 1)^{\frac{1}{q}}} \bigl\Vert {}_{0}^{c}D_{t} ^{\alpha } u \bigr\Vert _{L^{p}}. $$
(2.4)

It is easy to verify that the norm \(\Vert u_{i} \Vert _{ \alpha _{i}} = (\int _{0}^{T} \vert {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}(t) \vert ^{2}\,dt + \int _{0}^{T} \vert u_{i}(t) \vert ^{2}\,dt )^{ \frac{1}{2}}\) is equivalent to \(\Vert u_{i} \Vert _{\alpha _{i}} = (\int _{0}^{T} \vert {}_{0}^{c}D_{t}^{\alpha {}_{i}}u_{i}(t) \vert ^{2}\,dt )^{\frac{1}{2}}\), \(\forall u_{i} \in E_{0}^{\alpha _{i}}\). In the following, we will consider the fractional derivative spaces \(E_{0}^{\alpha _{i}}\) with respect to the norm \(\Vert u_{i} \Vert _{\alpha _{i}} = (\int _{0}^{T} \vert {}_{0}^{c}D_{t}^{ \alpha _{i}}u_{i}(t) \vert ^{2}\,dt )^{\frac{1}{2}}\).

Lemma 2.3

([23])

For\(\alpha _{i} \in [\frac{1}{2},1)\), \(1 \le i \le N\), one has

$$ \sum_{i = 1}^{N} \Vert u_{i} \Vert _{{L^{p}}}^{p} \le A_{p} \sum_{i = 1}^{N} \Vert u_{i} \Vert _{\alpha _{i}}^{p},\qquad \sum _{i = 1}^{N} \Vert u_{i} \Vert _{{\infty }}^{2} \le B \sum _{i = 1}^{N} \Vert u_{i} \Vert _{\alpha _{i}}^{2}, $$

where\(A_{p} = \max \{ \frac{T^{p\alpha _{i}}}{\varGamma ^{p}(\alpha _{i} + 1)},1 \le i \le N\}\), \(B = \max \{ \frac{T^{2\alpha _{i} - 1}}{\varGamma ^{2}(\alpha _{i})(2\alpha _{i} - 1)},1 \le i \le N\}\).

Lemma 2.4

([23])

Let\(\frac{1}{2} < \alpha _{i} \le 1\)for\(1 \le i \le N\). Assume the sequence\(\{ x_{n}\}\)converges weakly toxin\(E_{0}^{\alpha _{i}}\). Then\(x_{n} \to x\)strongly in\(C([0,T],R)\), i.e., \(\Vert x_{n} - x \Vert _{\infty } \to 0\), as\(n \to \infty \).

Lemma 2.5

([23])

Let\(\frac{1}{2} < \alpha _{i} \le 1\)for\(1 \le i \le N\). For any\(u_{i} \in E_{0}^{\alpha _{i}}\), one has

$$ \sum_{i = 1}^{N} \bigl\vert \cos (\pi \alpha _{i}) \bigr\vert \Vert u _{i} \Vert _{\alpha _{i}}^{2} \le \sum_{i = 1}^{N} - \int _{0}^{T} {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}(t){}_{t}^{c}D_{T}^{\alpha _{i}}u_{i}(t)\,dt \le \sum_{i = 1}^{N} \frac{1}{ \vert \cos (\pi \alpha _{i}) \vert } \Vert u_{i} \Vert _{\alpha _{i}}^{2}. $$

In the sequel, we denote \(X = E_{0}^{\alpha _{1}} \times \cdots \times E_{0}^{\alpha _{N}}\), then X is a reflexive and separable Banach space with the norm

$$ \Vert u \Vert _{X} = \bigl\Vert (u_{1}, \ldots ,u_{N}) \bigr\Vert _{X} = \sum _{i = 1}^{N} \Vert u_{i} \Vert _{\alpha _{i}} = \sum_{i = 1}^{N} \biggl( \int _{0}^{T} \bigl\vert {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}}. $$

Definition 2.6

The sequence \(\{ u^{ ( k )} \} \subset X\) is said to be a C sequence of the functional Φ if for \(u^{ ( k )} = ( u_{1}^{ ( k )},\ldots, u _{i}^{ ( k )},\ldots, u_{N}^{ ( k )} )\), \(1 \le k < \infty \), \(c \in R\), one has \(\varPhi ( u^{ ( k )} ) \to c\), \(\Vert u^{ ( k )} \Vert _{X} \to + \infty \) and \(\langle \varPhi ' ( u^{ ( k )} ),u^{ ( k )} \rangle \to 0\), as \(k \to \infty \). The functional Φ satisfies the C-condition if ever the C sequence of Φ has a convergent subsequence.

Let E be a real Banach space and \(\varPhi \in C^{1} [ E,R )\), \(Q = \{ u \in E:\varPhi ' ( u ) = 0 \} \).

Definition 2.7

([28])

For \(c \in R\), we define u as an isolated critical point of Φ with \(\varPhi ( u ) = c\), and define U as a neighborhood of u such that Φ has the only u as a critical point in U. We call

$$ C_{q} ( \varPhi ,u ): = H_{q} \bigl( \varPhi ^{c} \cap U,\varPhi ^{c} \cap U\backslash \{ u \} \bigr),\quad \bigl( q \in N: = \{ 0,1, 2,\ldots \} \bigr) $$

the qth critical group of Φ at u, where \(\varPhi ^{c}: = \{ u \in E:\varPhi ( u ) \le c \} \) is the c-sublevel set, and \(H_{q}\) is the singular relative homology group with coefficients in an Abelian group G.

Lemma 2.8

([28])

If\(Q = \{ 0 \} \), then\(C_{q} ( \varPhi ,\infty ) = C_{q} ( \varPhi ,0 )\), \(\forall q \in N\). It follows that if\(C_{q} ( \varPhi ,\infty ) \ne C_{q} ( \varPhi ,0 )\)for some\(q \in N\), thenΦmust have a nontrivial critical point.

Lemma 2.9

([29])

Let 0 be a critical point ofΦwith\(\varPhi ( 0 ) = 0\). Suppose thatΦhas a local linking at 0 with respect to\(E = V \oplus W\), \(k = \dim V < \infty \), that is, there exists\(\rho > 0\)small such that

$$ \varPhi ( u ) \le 0,\quad \forall u \in V, \Vert u \Vert \le \rho \quad \textit{and}\quad \varPhi ( u ) > 0,\quad \forall u \in W,0 < \Vert u \Vert \le \rho . $$

Then \(C_{k} ( \varPhi ,0 )\ncong 0\), hence 0 is a homological nontrivial point of Φ.

Proofs of main results

Lemma 3.1

Assume that\((I0)\), \((F0)\), \(( F2 )\)hold, thenΦsatisfies theC-condition.

Proof

Assume \(\{ u^{ ( k )} \} \) is a C sequence in X, where \(u^{ ( k )}=(u^{ ( k )}_{1},\ldots,u^{ ( k )}_{i},\ldots,u^{ ( k )}_{N})\).

First, we address the boundedness of C sequence \(\{ u^{ ( k )} \} \).

Assume that C sequence \(\{ u^{ ( k )} \} \) is unbounded. Up to a subsequence we have

$$ \varPhi \bigl( u^{ ( k )} \bigr) \to c,\qquad \bigl\Vert u^{ ( k )} \bigr\Vert _{X} \to + \infty ,\qquad \bigl\langle \varPhi ' \bigl( u^{ ( k )} \bigr),u^{ ( k )} \bigr\rangle \to 0,\quad k \to \infty ,c \in R. $$
(3.1)

Set \(v_{i}^{ ( k )}: = \Vert u_{i}^{ ( k )} \Vert _{\alpha _{i}}^{ - 1}u_{i}^{ ( k )} ( t ) \in E_{0}^{\alpha _{i}}\backslash \{ 0 \} \), then \(\Vert v_{i}^{ ( k )} \Vert _{\alpha _{i}} = 1\) for all \(n \in N\), where \(v_{0} = ( v_{01},\ldots, v_{0i},\ldots, v_{0N} )\), \(v^{ ( k )} = ( v_{1}^{ ( k )},\ldots,v _{i}^{ ( k )},\ldots, v_{N}^{ ( k )} )\), \(1 \le i \le n\), \(1 \le k < \infty \). By Lemma 2.4, we have

$$ v_{i}^{ ( k )} \to v_{0i} \quad \text{in } L^{p} \bigl( [ 0,T ] \bigr), \qquad v_{i}^{ ( k )} \to v_{0i} \quad \text{a.e. } t \in [ 0,T ]. $$

Obvious \(v_{0i} \ne 0\), set \(\varSigma _{1}: = \{ t \in [ 0,T ]:v ( t ) \ne 0 \} \) and \(\varSigma _{2}: = [ 0,T ]\backslash \varSigma _{1}\). Then \([ 0,T ] = \varSigma _{1} \cup \varSigma _{2}\) and \(\varSigma _{1} \cap \varSigma _{2} = \emptyset \). So \(\operatorname{meas} ( \varSigma _{1} ) > 0\). By (3.1), we obtain

$$\begin{aligned}& \sum_{i = 1}^{N} \int _{0}^{T} \bigl\{ - {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i} ^{ ( k )}(t){}_{t}^{c}D_{T}^{\alpha _{i}}u_{i}^{ ( k )}(t) - a_{i}(t) \bigl( u_{i}^{ ( k )} \bigr) ^{2}(t)\bigr\} \,dt + \sum_{i = 1}^{N} \sum_{j = 1}^{l} I_{ij} \bigl(u_{i}^{ ( k )}(t_{j})\bigr)u_{i}^{ ( k )}(t_{j}) \\& \quad {}- \sum_{i = 1}^{N} \int _{0}^{T} F_{u_{i}^{ ( k )}}\bigl(t,u(t) \bigr)u _{i}^{ ( k )}(t)\,dt = \bigl\langle \varPhi ' ( u ),u \bigr\rangle = o ( 1 ). \end{aligned}$$
(3.2)

Combining \(( I0 )\) and \(\Vert u^{ ( k )} \Vert _{X} \to + \infty \), as \(k \to \infty \), we can derive

$$ \lim_{k \to \infty } \frac{\sum_{i = 1}^{N} \sum_{j = 1}^{l} I_{ij} (u _{i}^{ ( k )}(t_{j}))u_{i}^{ ( k )}(t_{j})}{ \sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \Vert u_{i}^{ ( k )} \Vert _{\alpha _{i}}^{2}} \to 0. $$
(3.3)

From \(( F2 )\), (3.2), (3.3) and Lemma 2.5, we have

$$\begin{aligned} 1 =& \lim_{k \to \infty } \frac{\sum_{i = 1}^{N} \int _{0}^{T} \{ - {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}^{ ( k )}(t){}_{t}^{c}D _{T}^{\alpha _{i}}u_{i}^{ ( k )}(t) - a_{i}(t) ( u _{i}^{ ( k )} )^{2}(t)\}\,dt - o ( 1 )}{ \sum_{i = 1}^{N} \int _{0}^{T} \{ - {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i} ^{ ( k )}(t){}_{t}^{c}D_{T}^{\alpha _{i}}u_{i}^{ ( k )}(t) - a_{i}(t) ( u_{i}^{ ( k )} ) ^{2}(t)\}\,dt} \\ \ge& \lim_{k \to \infty } \frac{\sum_{i = 1}^{N} \int _{0}^{T} \{ - {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}^{ ( k )}(t){}_{t}^{c}D _{T}^{\alpha _{i}}u_{i}^{ ( k )}(t) - a_{i}(t) ( u _{i}^{ ( k )} )^{2}(t)\}\,dt - o ( 1 )}{ \sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \Vert u_{i}^{ ( k )} \Vert _{\alpha _{i}}^{2}} \\ \ge& \lim_{k \to \infty } \int _{\sum _{2}} \frac{\sum_{i = 1}^{N} F_{u _{i}^{ ( k )}}(t,u(t))u_{i}^{ ( k )}(t)\,dt}{ \sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \vert u_{i}^{ ( k )} \vert ^{2}} \bigl\vert v _{i}^{ ( k )} ( t ) \bigr\vert ^{2}\,dt \\ &{}+ \lim_{k \to \infty } \int _{\sum _{1}} \frac{\sum_{i = 1}^{N} F_{u _{i}^{ ( k )}}(t,u(t))u_{i}^{ ( k )}(t)\,dt}{ \sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \vert u_{i}^{ ( k )} \vert ^{2}} \bigl\vert v _{i}^{ ( k )} ( t ) \bigr\vert ^{2}\,dt \\ &{} - \lim_{k \to \infty } \frac{\sum_{i = 1}^{N} \sum_{j = 1}^{l} I_{ij} (u _{i}^{ ( k )}(t_{j}))u_{i}^{ ( k )}(t_{j})}{ \sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \Vert u_{i}^{ ( k )} \Vert _{ \alpha _{i}}^{2}} \\ \ge& \lim_{k \to \infty } \int _{\sum _{1}} \frac{\sum_{i = 1}^{N} F_{u _{i}^{ ( k )}}(t,u(t))u_{i}^{ ( k )}(t)\,dt}{ \sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \vert u_{i}^{ ( k )} \vert ^{2}} \bigl\vert v _{i}^{ ( k )} ( t ) \bigr\vert ^{2}\,dt. \end{aligned}$$
(3.4)

By \(( F2 )\), we have, for any \(t \in [ 0,T ]\),

$$ \lim_{k \to \infty } \frac{\sum_{i = 1}^{N} \int _{0}^{T} F_{u_{i}^{ ( k )}}(t,u(t))u_{i}^{ ( k )}(t)\,dt}{\sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \vert u_{i}^{ ( k )} \vert ^{2}} \bigl\vert v_{i}^{ ( k )} ( t ) \bigr\vert ^{2} = + \infty . $$

From Fatou’s lemma, we can get

$$ \lim_{k \to \infty } \int _{\sum _{1}} \frac{\sum_{i = 1}^{N} F_{u_{i} ^{ ( k )}}(t,u(t))u_{i}^{ ( k )}(t)\,dt}{ \sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \vert u_{i}^{ ( k )} \vert ^{2}} \bigl\vert v _{i}^{ ( k )} ( t ) \bigr\vert ^{2}\,dt = + \infty , $$

which is contradictory with (3.4), thus \(\{ u^{ ( k )} \} \) is bounded in X.

Second, we verify C sequence \(\{ u^{ ( k )} \} \) have convergent subsequence in X.

Since X is reflexive, we know that \(\{ u^{ ( k )} \} \) have a weakly convergent subsequence in X. Hence, we have

$$\begin{aligned}& u_{i}^{ ( k )}\rightharpoonup u_{0i} \in E_{0}^{\alpha _{i}}, \quad i = 1,\ldots, N\text{, as } k\to \infty , u_{0}=(u_{01}, \ldots,u_{0i},\ldots,u_{0N}), \\& u_{i}^{ ( k )} \to u_{0i} \quad \text{in } C \bigl( [ 0,T ] \bigr), \quad i = 1,\ldots, N, \text{a.e. } t \in [ 0,T ]. \end{aligned}$$

Thus \(\Vert u_{i}^{ ( k )} - u_{01} \Vert _{ \infty } \to 0\), as \(k \to \infty \). According to (2.2), it is easy to prove

$$ \bigl\langle \varPhi ' \bigl( u^{ ( k )} \bigr) - \varPhi ' ( u_{0} ),u^{ ( k )} - u_{0} \bigr\rangle \to 0,\quad k \to \infty . $$

From Lemma 2.5, we can get

$$\begin{aligned} 0 \leftarrow& \bigl\langle \varPhi ' \bigl( u^{ ( k )} \bigr) - \varPhi ' ( u_{0} ),u^{ ( k )} - u_{0} \bigr\rangle \\ =& \sum_{i = 1}^{N} \int _{0}^{T} \bigl\{ - \bigl[ {}_{0}^{c}D_{t}^{\alpha _{i}} \bigl( u_{i}^{ ( k )} - u_{0i} \bigr){}_{t}^{c}D _{T}^{\alpha _{i}} \bigl( u_{i}^{ ( k )} - u_{0i} \bigr) \bigr] - a_{i}(t) \bigl( u_{i}^{ ( k )} - u_{0i} \bigr) \bigl( u_{i}^{ ( k )} - u_{0i} \bigr)\bigr\} \,dt \\ &{}+ \sum_{i = 1}^{N} \sum _{j = 1}^{l} \bigl[ I_{ij} \bigl(u_{i}^{ ( k )}(t _{j}) - u_{0i}(t_{j})\bigr) \bigr] \bigl(u_{i}^{ ( k )}(t_{j}) - u _{0i}(t_{j})\bigr) \\ &{}- \sum_{i = 1}^{N} \int _{0}^{T} \bigl[ F_{u_{i}^{ ( k )}} \bigl(t,u(t)\bigr) - F_{u_{0i}} \bigl(t,u_{0}(t)\bigr) \bigr] \bigl(u_{i}^{ ( k )} - u_{0i} \bigr)\,dt \\ \ge& \sum_{i = 1}^{N} \bigl\vert \cos ( \pi \alpha _{i} ) \bigr\vert \bigl\Vert u_{i}^{ ( k )} - u_{0i} \bigr\Vert _{\alpha _{i}} ^{2} + \Biggl\vert \sum_{i = 1}^{N} \sum _{j = 1}^{l} \bigl[ I_{ij} \bigl(u_{i} ^{ ( k )}(t_{j}) - u_{0i}(t_{j})\bigr) \bigr] \Biggr\vert \bigl\Vert u_{i}^{ ( k )} - u_{0i} \bigr\Vert _{\infty } \\ &{}- \Biggl\vert \sum_{i = 1}^{N} \int _{0}^{T} \bigl[ F_{u_{i}^{ ( k )}} \bigl(t,u(t)\bigr) - F_{u_{0i}}\bigl(t,u_{0}(t)\bigr) \bigr] \Biggr\vert \bigl\Vert u_{i}^{ ( k )} - u_{0i} \bigr\Vert _{\infty }. \end{aligned}$$

Combining \(( I0 )\), \(( F0 )\) and \(\Vert u _{i}^{ ( k )} - u_{0i} \Vert _{\infty } \to 0\), we know \(\sum_{i = 1}^{N} \Vert u_{i}^{ ( k )} - u_{0i} \Vert _{\alpha _{i}} \to 0\) as \(k \to \infty \) and \(u_{i}^{ ( k )} \to u_{0i}\) in \(E_{0}^{\alpha _{i}}\), \(i = 1,\ldots, N\). Thus, \(\{ u^{ ( k )} \} \) admits a convergent subsequence, which implies that Φ satisfies the C-condition. □

Corollary 3.2

Assume\((I0)\), \((F0)\), \(( F3 )\)hold, thenΦsatisfies theC-condition.

Lemma 3.3

Assume\(( I0 )\), \(( I1 )\), \(( F1 )\), \(( F3 )\)hold, then\(C_{q} ( \varPhi ,\infty ) = 0\)for every\(q \in N\).

Proof

Let \(\varOmega = \{ u_{i} \in E_{0}^{\alpha _{i}}: \Vert u_{i} \Vert _{\alpha _{i}} = 1 \} \). For \(u_{i} \in \varOmega \), by \(( I0 )\) we have

$$ \biggl\vert \int _{0}^{\tau u_{i} ( t_{j} )} I_{ij} ( s )\,ds \biggr\vert \le \frac{a_{j}}{\gamma _{j} + 1} \bigl\vert \tau u _{i} ( t_{j} ) \bigr\vert ^{\gamma _{j} + 1} \le \frac{A_{0} ^{\gamma _{j} + 1}a_{j}}{\gamma _{j} + 1} \vert \tau \vert ^{\gamma _{j} + 1} \Vert u_{i} \Vert _{\alpha _{i}}^{\gamma _{j} + 1} \le \frac{A_{0}^{\gamma _{j} + 1}a_{j}}{\gamma _{j} + 1} \vert \tau \vert ^{\gamma _{j} + 1}, $$

where \(A_{0} = \frac{T^{\alpha _{i} - \frac{1}{2}}}{\varGamma ( \alpha _{i} )\sqrt{2\alpha _{i} - 1}} \). According to Fatou’s lemma and \(( F3 )\), we can get

$$ \lim_{\tau \to \infty } \int _{0}^{T} \frac{F ( t,\tau u )}{ \vert \tau \vert ^{2}}\,dt \ge \int _{0}^{T} \lim_{\tau \to \infty } \frac{F ( t,\tau u )}{ \vert \tau u \vert ^{2}} \vert u \vert ^{2}\,dt = + \infty . $$
(3.5)

Hence \(\forall u_{i} \in \varOmega \), by (3.5) and Lemma 2.5, we obtain

$$\begin{aligned} \varPhi ( \tau u ) =& \sum_{i = 1}^{N} \int _{0}^{T} - \frac{1}{2} {}_{0}^{c}D_{t}^{\alpha _{i}} ( \tau u_{i} ) (t) {}_{t}^{c}D_{T}^{\alpha _{i}} ( \tau u_{i} ) (t)\,dt - \sum_{i = 1}^{N} \frac{1}{2} a_{i}(t) \bigl( \tau ^{2}u^{2}_{i} \bigr) (t) \\ &{}- \int _{0}^{T} F\bigl(t,\tau u(t)\bigr)\,dt + \sum _{i = 1}^{N} \sum _{j = 1}^{l} \int _{0}^{\tau u ( t_{j} )} I_{ij} ( s )\,ds \\ \le& \frac{\tau ^{2}}{2}\sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \Vert u_{i} \Vert _{\alpha _{i}}^{2} - \int _{0}^{T} \frac{F(t,\tau u)}{ \vert \tau \vert ^{2}} \vert \tau \vert ^{2}\,dt + \sum_{i = 1}^{N} \sum_{j = 1}^{l} \frac{A_{0}^{\gamma _{j} + 1}a _{j}}{\gamma _{j} + 1} \vert \tau \vert ^{\gamma _{j} + 1} \\ =& \tau ^{2} \Biggl( \frac{1}{2}\sum _{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \Vert u_{i} \Vert _{\alpha _{i}}^{2} - \int _{0}^{T} \frac{F(t,\tau u)}{ \vert \tau \vert ^{2}}\,dt + \sum _{i = 1}^{N} \sum _{j = 1}^{l} \frac{A_{0}^{\gamma _{j} + 1}a_{j}}{ ( \gamma _{j} + 1 ) \vert \tau \vert ^{1 - \gamma _{j}}} \Biggr) \\ \to& - \infty , \end{aligned}$$

as \(\tau \to \infty \).

Let \(a < \min \{ \inf \Vert u_{i} \Vert _{\alpha _{i} \le 1}\varPhi ( u ),0 \} \), for any \(u_{i} \in \varOmega \), then there exists \(\tau _{0} > 1\) such that \(\varPhi ( \tau u ) \le a\) for \(\tau > \tau _{0}\). We can derive

$$\begin{aligned}& \sum_{i = 1}^{N} \int _{0}^{T} - {}_{0}^{c}D_{t}^{\alpha _{i}} ( \tau u_{i} ){}_{t}^{c}D_{T}^{\alpha _{i}} ( \tau u_{i} )\,dt \\& \quad \le 2a + \sum_{i = 1}^{N} a_{i}(t)\tau ^{2}u^{2}_{i} + 2 \int _{0}^{T} F(t,\tau u)\,dt - 2\sum _{i = 1}^{N} \sum_{j = 1}^{l} \int _{0} ^{\tau u_{i} ( t_{j} )} I_{ij} ( s )\,ds. \end{aligned}$$
(3.6)

Combining \(( I1 )\), \(( F1 )\) with (3.6), we obtain

$$ F_{*} ( t,x ) \ge 0, \quad \text{for } \forall ( t,x ) \in [ 0,T ] \times R, \quad \text{and}\quad I_{j}^{*} ( x ) \ge 0, \quad \text{for } \forall x \in R,j = 1,2,\ldots, l. $$

It follows (3.6) that

$$\begin{aligned}& \frac{d}{d\tau } \varPhi ( \tau u ) \\& \quad = \frac{1}{\tau } \Bigg( \sum_{i = 1}^{N} \int _{0}^{T} - {}_{0}^{c}D_{t}^{\alpha _{i}} ( \tau u_{i} ){}_{t}^{c}D_{T}^{\alpha _{i}} ( \tau u_{i} )\,dt - \sum_{i = 1}^{N} a_{i}(t) \bigl( \tau ^{2}u^{2}_{i} ( t ) \bigr)\,dt \\& \qquad {} + \sum_{i = 1}^{N} \sum _{j = 1}^{l} I_{ij} \bigl( \tau u_{i}(t_{j}) \bigr) \tau u_{i}(t_{j}) \\& \qquad {} - \sum_{i = 1}^{N} \int _{0}^{T} F_{u_{i}}(t,\tau u)\tau u_{i}\,dt \Bigg) \\& \quad \le \frac{1}{\tau } \Bigg( 2a + 2 \int _{0}^{T} F ( t,\tau u )\,dt - \sum _{i = 1}^{N} \int _{0}^{T} F_{u_{i}}(t,\tau u)\tau u _{i}\,dt + \sum_{i = 1}^{N} \sum_{j = 1}^{l} I_{ij} \bigl( \tau u_{i}(t_{j}) \bigr) \tau u_{i}(t_{j}) \\& \qquad {}- 2\sum_{i = 1}^{N} \sum_{j = 1}^{l} \int _{0}^{\tau u_{i} ( t_{j} )} I_{ij} ( s )\,ds \Bigg) \\& \quad \le \frac{1}{\tau } \Biggl( 2a - \int _{0}^{T} F_{*} ( t,\tau u )\,dt - \sum_{i = 1}^{N} \sum _{j = 1}^{l} I^{*}_{ij} \bigl( \tau u_{i}(t_{j}) \bigr) \Biggr) < 0. \end{aligned}$$

According to the implicit function theorem, there exists a unique \(S \in C ( \varOmega ,R )\), such that \(\varPhi ( S ( u )u ) = a\). Similarly to discussing in [29], there exists a strong deformation retract from \(E_{0}^{\alpha _{i}}\backslash \{ 0 \} \) to \(\varPhi ^{a _{i}}\). Thus

$$ C_{q} ( \varPhi ,\infty ) = H_{q} \bigl( E_{0}^{\alpha _{i}},E _{0}^{\alpha _{i}}\backslash \{ 0 \} \bigr) = 0, \quad \forall q \in N. $$

So we completed the conclusion. □

Corollary 3.4

Assume\(( I0 )\), \(( I1 )\), \(( F1 )\), \(( F2 )\)hold, then\(C_{q} ( \varPhi ,\infty ) = 0\)for every\(q \in N\).

Since \(E_{0}^{\alpha _{i}}\) (\(i = 1,\ldots, N\)) is a reflexive and separable Banach space, there exists an orthogonal basis \(\{ e_{ik} \} \) of \(E_{0}^{\alpha _{i}}\) such that \(E_{0}^{\alpha _{i}} = \overline{ \operatorname{span} \{ e_{ik}:k = 1,2,\ldots \} }\). For \(m = 1,2,\ldots \) , denote

$$\begin{aligned}& Y_{ik}: = \operatorname{span} \{ e_{ik} \} ,\qquad V_{im} = \bigoplus_{k = 1}^{m}Y_{ik}, \qquad Z_{im} = \overline{ \bigoplus_{k = m}^{\infty } Y_{ik}}, \\& V = V_{1m} \times \cdots \times V_{Nm}, \qquad W = Z_{1m} \times \cdots \times Z_{Nm}. \end{aligned}$$

Then \(E_{0}^{\alpha _{i}} = V_{im} \oplus Z_{im}\), \(i = 1,\ldots, N\), \(X = V \oplus W\).

Lemma 3.5

Assume\(( I0 )\), \(( F0 )\)hold, then there exists\(k_{0} \in N\)such that\(C_{k_{0}} ( \varPhi ,0 ) \ne 0\).

Proof

From \(( I0 )\), \(( F0 )\) we know, \(F_{u_{i}} ( t,0 ) = 0\), \(I_{ij} ( 0 ) = 0\) (\(i = 1,\ldots,N\); \(j = 1,\ldots,l \)). Then we found that the functional Φ has a trivial critical point at zero. So it has a local linking at zero in X.

Owing to the fact that all norms of a finite dimensional normed space are equivalent, there exist positive constants \(M_{1}\), \(M_{2}\), \(M'_{1}\), \(M'_{2}\), such that

$$ M_{1} \Vert u_{i} \Vert _{\alpha _{i}} \le \Vert u_{i} \Vert _{\infty } \le M_{2} \Vert u_{i} \Vert _{\alpha _{i}}, \qquad M'_{1} \Vert u_{i} \Vert _{\alpha _{i}} \le \Vert u_{i} \Vert _{L^{\gamma }} \le M'_{2} \Vert u_{i} \Vert _{ \alpha _{i}}, \quad u_{i} \in V_{ik}. $$

First, we verify there exists \(0 < \rho _{1} < 1\), such that

$$ \varPhi ( u ) < 0 \quad \text{for } \Vert u \Vert _{X} < \rho _{1}, \forall u = (u_{1}, \ldots ,u_{N}), u_{i} \in V_{im}, i = 1,\ldots,N. $$

Because \(V_{im}\) is finite dimensional, then, for given \(r_{0}\), such that

$$\begin{aligned}& \bigl\vert u_{i} ( t ) \bigr\vert \le \frac{r_{0}}{N}, \quad \text{for } \Vert u_{i} \Vert _{\alpha _{i}} \le \frac{\rho _{1}}{N}, \forall 1 \le i \le N, \\& \vert u \vert \le r_{0},\quad \text{for } \Vert u \Vert _{X} < \rho _{1} \end{aligned}$$
(3.7)

For any \(r \in ( 0,r_{0} )\), we set

$$\begin{aligned}& \varOmega _{1} = \bigl\{ t \in [ 0,T ]: \vert u \vert \le r \bigr\} , \\& \varOmega _{2} = \bigl\{ t \in [ 0,T ]:r \le \vert u \vert \le r_{0} \bigr\} , \\& \varOmega _{3} = \bigl\{ t \in [ 0,T ]:r_{0} \le \vert u \vert \bigr\} . \end{aligned}$$

Then \([ 0,T ] = \bigcup_{i = 1}^{3} \varOmega _{i}\) and \(\varOmega _{i}\) (\(i = 1,2,3 \)) are pairwise disjoint.

Set \(F^{*} ( t,u ) = F ( t,u ) - C \vert u _{i} \vert ^{\gamma } \), it follows from \(( I0 )\), \(( F0 )\) and Lemma 2.5 that

$$\begin{aligned} \varPhi ( u ) =& \sum_{i = 1}^{N} \int _{0}^{T} \biggl\{ - \frac{1}{2} {}_{0}^{c}D_{t}^{\alpha _{i}}u_{i}(t){}_{t}^{c}D_{T}^{ \alpha _{i}}u_{i}(t) - \frac{1}{2}a_{i}(t)u^{2}_{i}(t) \biggr\} \,dt + \sum_{i = 1}^{N} \sum _{j = 1}^{l} \int _{0}^{u_{i} ( t_{j} )} I_{ij} ( s )\,ds \\ &{}- \int _{0}^{T} C \vert u_{i} \vert ^{\gamma }\,dt - \int _{\varOmega _{1}} F^{*} \bigl( t,u ( t ) \bigr)\,dt - \int _{\varOmega _{2}} F^{*} \bigl( t,u ( t ) \bigr)\,dt - \int _{\varOmega _{3}} F^{*} \bigl( t,u ( t ) \bigr)\,dt \\ \le& \sum_{i = 1}^{N} \int _{0}^{T} \biggl\{ - \frac{1}{2} {}_{0}^{c}D_{t} ^{\alpha _{i}}u_{i}(t){}_{t}^{c}D_{T}^{\alpha _{i}}u_{i}(t) - \frac{1}{2}a_{i}(t)u^{2}_{i}(t) \biggr\} d + \sum_{i = 1}^{N} \sum _{j = 1} ^{l} \frac{a_{j}}{\gamma _{j} + 1} \bigl\vert u_{i} ( t_{j} ) \bigr\vert ^{\gamma _{j} + 1} \\ &{}- \int _{0}^{T} C \vert u_{i} \vert ^{\gamma }\,dt - \int _{\varOmega _{1}} F^{*} \bigl( t,u ( t ) \bigr)\,dt - \int _{\varOmega _{2}} F^{*} \bigl( t,u ( t ) \bigr)\,dt \\ &{}- \int _{\varOmega _{3}} F^{*} \bigl( t,u ( t ) \bigr)\,dt. \end{aligned}$$
(3.8)

According to (3.7) and the definition of \(\varOmega _{3}\), we have \(\int _{\varOmega _{3}} F^{*} ( t,u ( t ) )\,dt = 0\), \(\forall u_{i} \in V_{im}\). From \(( F0 )\), we have \(F_{*} > 0\) on \(\varOmega _{2}\), \(\vert u \vert < r\) on \(\varOmega _{1}\). From \(( F0 )\), one has

$$ F^{*} ( t,u ) \le \sum_{i = 1}^{N} \biggl[ \frac{\varGamma ^{2} ( \alpha _{i} + 1 )}{2T^{2\alpha _{i}}} \bigl\vert \cos ( \pi \alpha _{i} ) \bigr\vert \vert u \vert ^{2} - C \vert u_{i} \vert ^{\gamma } \biggr] \to 0,\quad \text{as } r \to 0. $$

Hence, we can obtain \(\int _{\varOmega _{1}} F^{*} ( t,u ( t ) )\,dt \to 0\). Then, \(\forall u \in X\), \(\Vert u \Vert _{X} \le \rho _{1} \le 1\), \(1 < \gamma < \max \{ \gamma _{j} + 1 \} < 2\), \(r \in ( 0,r_{0} )\), from (3.8), we can get

$$\begin{aligned} \varPhi ( u ) \le& \frac{1}{2}\sum_{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \Vert u_{i} \Vert _{\alpha _{i}}^{2} + \sum_{i = 1}^{N} \sum _{j = 1}^{l} \frac{a _{j}A_{o}^{\gamma _{j} + 1}}{\gamma _{j} + 1} \Vert u_{i} \Vert _{\alpha _{i}}^{\gamma _{j} + 1} - M_{1}^{\gamma } C \Vert u_{i} \Vert _{\alpha _{i}}^{\gamma } \\ \le& \max_{1 \le i \le N} \Vert u_{i} \Vert _{\alpha _{i}}^{ \gamma } \Biggl( \frac{1}{2}\sum _{i = 1}^{N} \frac{1}{ \vert \cos ( \pi \alpha _{i} ) \vert } \Vert u_{i} \Vert _{\alpha _{i}}^{2 - \gamma } + \sum _{i = 1}^{N} \sum_{j = 1}^{l} \frac{a _{j}A_{o}^{\gamma _{j} + 1}}{\gamma _{j} + 1} \Vert u_{i} \Vert _{\alpha _{i}}^{\gamma _{j} + 1 - \gamma } - M_{1}^{\gamma } C \Biggr) \\ \le& 0. \end{aligned}$$

Next, we will prove that there exists \(0 < \rho _{2} < 1\), \(\forall u \in X\), such that \(\Vert u \Vert _{X} < \rho _{2}\), and we have \(\varPhi ( u ) \ge 0\).

Because the continuous embedding \(X \to C_{0}^{\infty } ( [ 0,T ],R^{N} )\) is compact. \(\forall u \in X\), then, for given \(\varepsilon > 0\), there exists \(0 < \rho _{2} < 1\) such that \(\vert u \vert < \Vert u \Vert _{\infty } < \varepsilon \), for \(\Vert u \Vert _{X} \le \rho _{2}\), \(t \in [ 0,T ]\).

From \(( F0 )\), \(\forall \vert u \vert < \varepsilon \), for \(\Vert u \Vert _{X} \le \rho _{2}\), \(t \in [ 0,T ]\), there exists \(\zeta \in ( 0,1 )\) we know that

$$ F ( t,u ) \le \sum_{i = 1}^{N} ( 1 - \zeta )\frac{ \varGamma ^{2} ( \alpha _{i} + 1 )}{2T^{2\alpha _{i}}} \bigl\vert \cos ( \pi \alpha _{i} ) \bigr\vert \vert u \vert ^{2}. $$
(3.9)

Let \(c = \min_{1 \le j \le l}b_{j}\), \(c = \max_{1 \le j \le l}c_{j}\). By \(( I0 )\), we know \(e \in [ 0,1 )\). \(\forall u \in X\), \(\vert u \vert < \varepsilon \) combining \(( I0 )\), Lemma 2.3 and (3.9), we obtain

$$\begin{aligned} \varPhi ( u ) \ge& \frac{1}{2}\sum_{i = 1}^{N} \bigl\vert \cos ( \pi \alpha _{i} ) \bigr\vert \Vert u_{i} \Vert _{\alpha _{i}}^{2} - \frac{1}{2}\sum_{i = 1}^{N} ( 1 - \zeta ) \bigl\vert \cos ( \pi \alpha _{i} ) \bigr\vert \Vert u_{i} \Vert _{\alpha _{i}}^{2} \\ &{} - \frac{1}{2}AT \bigl\vert u_{i} ( \xi ) \bigr\vert ^{2} + \frac{1}{2}\frac{bl}{c + 1} \vert u_{i} \vert ^{c + 1} \\ \ge& \frac{1}{2} \Biggl( \sum_{i = 1}^{N} \bigl\vert \cos ( \pi \alpha _{i} ) \bigr\vert \zeta \Vert u_{i} \Vert _{\alpha _{i}}^{2} + \vert u_{i} \vert ^{c + 1} \biggl( \frac{bl}{e + 1} - AT \vert u_{i} \vert ^{1 - c} \biggr) \Biggr) > 0. \end{aligned}$$

Then,

$$ \varPhi ( u ) > 0, \quad \forall u \in X,\qquad \Vert u \Vert _{X} \le \rho _{2} \le 1,\quad t \in [ 0,T ]. $$
(3.10)

Let \(\rho = \min \{ \rho _{1},\rho _{2} \} \), according to (3.8), (3.10), we can get

$$ \varPhi ( u ) \le 0,\quad \forall u \in V, \Vert u \Vert _{X} \le \rho ;\qquad \varPhi ( u ) > 0,\quad \forall u \in W, 0 < \Vert u \Vert _{X} \le \rho , $$

From Lemma 2.9, we have, \(C_{k} ( \varPhi ,0 )\ncong 0\). □

Proof of Theorem 1.1

It follows from Lemma 3.1 that Φ satisfies the C-condition. By Corollary 3.4 and Lemma 3.5, we have \(C_{k_{0}} ( \varPhi ,\infty ) \ne C_{k_{0}} ( \varPhi ,0 )\) for some \(k_{0} \in N\). Then we can conclude Φ has a nontrivial critical point from Lemma 2.8. Hence, problem (1.2) has at least one nontrivial weak solution. □

Proof of Theorem 1.1

It follows from Corollary 3.2 that Φ satisfies the C-condition. By Lemma 3.3 and Lemma 3.5, we have \(C_{k_{0}} ( \varPhi ,\infty ) \ne C_{k_{0}} ( \varPhi ,0 )\) for some \(k_{0} \in N\). Then we can conclude Φ have a nontrivial critical point from Lemma 2.8. Hence, problem (1.2) has at least one nontrivial weak solution. □

References

  1. 1.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

  2. 2.

    Lakshmikantham, V., Leela, S., Vasundhara, D.J.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009)

  3. 3.

    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

  4. 4.

    Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Appl. Math. Sci., vol. 74. Springer, New York (1989)

  5. 5.

    Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS, vol. 65. Am. Math. Soc., Providence (1986)

  6. 6.

    Corvellec, J.N., Motreanu, V.V., Saccon, C.: Doubly resonant semilinear elliptic problems via nonsmooth critical point theory. J. Differ. Equ. 248, 2064–2091 (2010)

  7. 7.

    Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, 1181–1199 (2011)

  8. 8.

    Chen, J., Tang, X.H.: Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory. Abstr. Appl. Anal. 2012, Article ID 2012 (2012)

  9. 9.

    Hu, Z.G., Liu, W.B., Liu, J.Y.: Ground state solutions for a class of fractional differential equations with Dirichlet boundary value condition. Abstr. Appl. Anal. 2014, Article ID 958420 (2014)

  10. 10.

    Jiao, F., Zhou, Y.: Existence results for fractional boundary value problem via critical point theory. Int. J. Bifurc. Chaos Appl. Sci. Eng. 22, 1250086 (2012)

  11. 11.

    Erwin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)

  12. 12.

    Li, Y., Sun, H., Zhang, Q.: Existence of solutions to fractional boundary-value problems with a parameter. Electron. J. Differ. Equ. 2013, 141 (2013)

  13. 13.

    Sun, H., Zhang, Q.: Existence of solutions for a fractional boundary value problem via the Mountain Pass method and an iterative technique. Comput. Math. Appl. 64, 3436–3443 (2012)

  14. 14.

    Hu, Z., Liu, W., Liu, J.: Ground state solutions for a class of fractional differential equations with Dirichlet boundary value condition. Abstr. Appl. Anal. 2014, Article ID 958420 (2014)

  15. 15.

    Ge, B.: Multiple solutions for a class of fractional boundary value problems. Abstr. Appl. Anal. 2012, Article ID 468980 (2012)

  16. 16.

    Mophou, G.M.: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 72, 1604–1615 (2010)

  17. 17.

    Tai, Z., Wang, X.: Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces. Appl. Math. Lett. 22, 1760–1765 (2009)

  18. 18.

    Shu, X., Lai, Y., Chen, Y.: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. 74, 2003–2011 (2011)

  19. 19.

    Bonanno, G., Rodríguez-López, R., Tersian, S.: Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17(3), 717–744 (2014)

  20. 20.

    Rodríguez-López, R., Tersian, S.: Multiple solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal. 17(4), 1016–1038 (2014)

  21. 21.

    Nyamoradi, N., Rodríguez-López, R.: On boundary value problems for impulsive fractional differential equations. Appl. Math. Comput. 271, 874–892 (2015)

  22. 22.

    Zhao, Y.L., Zhao, Y.L.: Nontrivial solutions for a class of perturbed fractional differential systems with impulsive effects. Bound. Value Probl. 2016, 129 (2016)

  23. 23.

    Li, P.L., Ma, J.W., Wang, H., Li, Z.Q.: Infinitely many nontrivial solutions for fractional boundary value problems with impulses and perturbation. J. Nonlinear Sci. Appl. 10, 2283–2295 (2017)

  24. 24.

    Zhao, Y., Luo, C., Chen, H.: Existence Results for Non-instantaneous Impulsive Nonlinear Fractional Differential Equation Via Variational Methods. Bull. Malays. Math. Sci. Soc., 1–19 (2019). https://doi.org/10.1007/s40840-019-00797-7

  25. 25.

    Agarwal, R.P., Bhaskar, T.G., Perera, K.: Some results for impulsive problems via Morse theory. J. Math. Anal. Appl. 409, 752–759 (2014)

  26. 26.

    Shi, H., Chen, H., Liu, H.: Morse theory and local linking for a class of boundary value problems with impulsive effects. J. Appl. Math. Comput. 51, 353–365 (2016)

  27. 27.

    Zhao, Y.L., Chen, H.B., Xu, C.J.: Nontrivial solutions for impulsive fractional differential equations via Morse theory. Appl. Math. Comput. 307, 170–179 (2017)

  28. 28.

    Chang, K.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)

  29. 29.

    Liu, Q., Sun, B.: Remarks on multiple nontrivial solutions for quasi-linear resonant problems. J. Math. Anal. Appl. 258, 209–222 (2001)

Download references

Acknowledgements

The authors much appreciate the referees for their careful reading of the manuscript and for their insightful comments, which helped to improve the quality of the paper. We also want to thank the editors for their contributions. It is their valuable comments and suggestions that vastly contributed to the perfection of the paper.

Availability of data and materials

Not applicable.

Funding

This work is supported by National Natural Science Foundation of China (No. 61673008), the Young Backbone Teacher Funding Scheme of Henan (No. 2019GGJS079).

Author information

Affiliations

Authors

Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Peiluan Li.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Abbreviations

Not applicable.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, P., Xu, L., Wang, H. et al. The existence of solutions for perturbed fractional differential equations with impulses via Morse theory. Bound Value Probl 2020, 21 (2020). https://doi.org/10.1186/s13661-020-01330-7

Download citation

MSC

  • 26A33
  • 34B37
  • 34K10

Keywords

  • Morse theory
  • Perturbed fractional differential equations
  • Impulses
  • Nontrivial solutions