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On the existence of three solutions of Dirichlet fractional systems involving the p-Laplacian with Lipschitz nonlinearity

Boundary Value Problems volume 2020, Article number: 131 (2020) Cite this article

Abstract

A class of perturbed fractional nonlinear systems is studied. The dynamical system possesses two control parameters and a Lipschitz nonlinearity order of \(p-1\). The multiplicity of the weak solutions is proved by means of the variational method and by Ricceri critical points theorems. An illustrative example is analyzed in order to highlight the obtained result.

Introduction

Fractional differential equations have proved to be promising tools in the modeling of diverse phenomena in various fields, such as physics, chemistry, biology, engineering and economics. In recent years, there was a significant development in fractional differential equations due to the possibility of accounting for a larger class of memory properties. For instance, consider the studies of Miller and Ross [1], Boulaaras et al. [2, 3], Podlubny [4], Hilfer [5], Kilbas et al. [6], and the related papers [1, 7–16] and the references therein.

Critical point theory was very useful in determining the existence of solutions to complete differential equations with certain boundary conditions; see, for example, in the extensive literature on the subject, the classical books [17–19], and the references therein. However, so far, some problems were created for fractional boundary value problems (F-BVP) by exploiting this approach, where it is often very difficult to create suitable spaces and functions for fractional problems.

In [20], the authors investigated the following nonlinear fractional differential equation depending on two parameters:

$$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha _{i}} ( a_{i} ( t ) { }_{0}D_{t}^{ \alpha _{i}}u_{i} ( t ) ) \\ \quad =\lambda F_{u_{i}} ( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) ) \\ \qquad {}+\mu G_{u_{i}} ( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) ) +h_{i} ( u_{i} ) \quad \text{a.e } [ 0,T ] , \\ u_{i} ( 0 ) =u_{i} ( T ) =0,\end{cases} $$
(1.1)

for \(1\leq i\leq n\), where \(\alpha _{i}\in (0;1]\), \({}_{0}D_{t}^{\alpha _{i}}\) and \({}_{t}D_{T}^{\alpha _{i}}\) are the left and right Riemann–Liouville fractional derivatives of order \(\alpha _{i}\), respectively, with \(a_{i}\in L^{\infty } ( [ 0,T ] ) \) for

$$ a_{i0}=\operatorname{ess}\inf_{ [ 0,T ] }a_{i}(t)>0 \quad \text{for }1\leq i \leq n, \lambda , \mu $$

are positive parameters, \(F,G: [ 0,T ] \times \mathbb{R} ^{n}\longrightarrow \mathbb{R} \) are measurable functions with respect to \(t\in [ 0,T ] \) for every \(( x_{1},\ldots,x_{n} ) \in \mathbb{R} ^{n}\) and are \(C^{1}\) with respect to \(( x_{1},\ldots,x_{n} ) \in \mathbb{R} ^{n}\) for a.e. \(t\in [ 0,T ] \), \(F_{u_{i}}\) and \(G_{u_{i}}\) denote the partial derivative of F and G with respect to \(u_{i}\), respectively, and \(h_{i}: \mathbb{R} \rightarrow \mathbb{R} \) are Lipschitz continuous functions with the Lipschitz constants \(L_{i}>0\) for \(1\leq i\leq n\), i.e.,

$$ \bigl\vert h_{i} ( x_{1} ) -h_{i} ( x_{2} ) \bigr\vert \leq L_{i} \vert x_{1}-x_{2} \vert $$

for every \(x_{1},x_{2}\in \mathbb{R} \), and \(h_{i} ( 0 ) =0\) for \(1\leq i\leq n\). Motivated by [21] and [22], using a three critical points theorem obtained in [23], which is recalled in the next section (Theorem 2), the existence of at least three solutions for this system is demonstrated.

For example, according to some assumptions, in [24], by using variational methods the authors obtained the existence of at least one weak solution for the following p-Laplacian fractional differential equation [24]:

$$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha } ( \phi _{p} ( _{0}D_{t}^{\alpha }u ( t ) ) ) =\lambda f ( t,u ( t ) ) \quad \text{a.e. }t\in [ 0,T ] , \\ u ( 0 ) =u ( T ) =0,\end{cases} $$
(1.2)

where \({}_{0}D_{t}^{\alpha }\) and \({}_{t}D_{T}^{\alpha }\) are the left and right Riemann–Liouville fractional derivatives with \(0<\alpha \leq 1\), respectively, the function \(\phi _{p} ( s ) = \vert s \vert ^{p-2}s\), \(p>1\). Taking a class of fractional differential equation with p-Laplacian operator as a model, Li et al. investigated the following equation recently [25]:

$$ \textstyle\begin{cases} _{t}D_{T}^{\alpha } ( \frac{1}{w ( t ) ^{p-2}} \varphi _{p} ( _{0}D_{t}^{\alpha }u ( t ) ) ) +\lambda u ( t ) =f ( t,u,_{0}^{c}D_{t}^{\alpha }u ( t ) ) +h ( u ( t ) ) \quad \text{a.e. }t\in [ 0,T ] , \\ u ( 0 ) =u ( T ) =0,\end{cases} $$
(1.3)

with \(\frac{1}{p}<\alpha \leq 1\), λ a non-negative real parameter.

The function

$$ \varphi _{p} ( s ) = \vert s \vert ^{p-2}s, \quad p \geq 2,f: [ 0;T ] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, $$

is continuous and \(h:\mathbb{R} \rightarrow \mathbb{R} \) is a Lipschitz continuous function.

By using the mountain pass theorem combined with iterative technique, the authors obtained the existence of at least one solution for problem (1.3). In addition, in [20], three weak solutions for a new class of fractional p-Laplacian for boundary value Systems were established by using variational methods and critical point theory. In contrast, motivated by [21] and [22], the existence of at least three solutions for system (1.4) is demonstrated in the present paper, by means of the three critical points theorem obtained in [23], which is recalled in the next section (Theorem 2). This theorem has been successfully employed to establish the existence of at least three solutions for the case of perturbed boundary value problems; see [7, 26–28] and [29].

Consider the following fractional nonlinear system:

$$ \textstyle\begin{cases} _{t}D_{T}^{\alpha _{i}}\phi _{p} ( _{0}D_{t}^{\alpha _{i}}u_{i} ( t ) ) \\ \quad =\lambda F_{u_{i}} ( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) ) \\ \qquad {}+\mu G_{u_{i}} ( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) ) \\ \qquad {}+h_{i} ( u_{i} ( t ) ) \quad \text{a.e. }t\in [ 0,T ] , \text{ for }1\leq i\leq n \\ u_{i} ( 0 ) =u_{i} ( T ) =0,\end{cases} $$
(1.4)

where \(\alpha _{i}\in (0;1]\), \(\phi _{p} ( t ) = \vert t \vert ^{p-2}t\), \(t\neq 0\), \(\phi _{p} ( 0 ) =0\), \(p>1,_{0}D_{t}^{\alpha _{i}}\) and \({}_{t}D_{T}^{\alpha _{i}}\) are the left and right Riemann–Liouville fractional derivatives of order \(\alpha _{i}\), respectively, for \(1\leq i\leq n,\lambda \) and μ are positive parameters, and \(F,G: [ 0,T ] \times \mathbb{R} ^{n}\rightarrow \mathbb{R} \) are measurable functions with respect to \(t\in [ 0,T ] \) for every \(( x_{1},x_{2},\ldots,x_{n} ) \in \mathbb{R} ^{n}\) and are \(C^{1}\) with respect to \(( x_{1},x_{2},\ldots,x_{n} ) \in \mathbb{R} ^{n}\).for a.e. \(t\in [ 0,T ] \), \(F_{u_{i}}\), \(G_{u_{i}}\) denote the partial derivative of F and G with respect to \(u_{i}\), respectively, and \(h_{i}:\mathbb{R} \rightarrow \mathbb{R} \) are Lipschitz continuous functions of order \(( p-1 ) \) with Lipschizian constants \(L_{i}>0\) for \(1\leq i\leq n\), i.e.,

$$ \bigl\vert h_{i} ( x_{1} ) -h_{i} ( x_{2} ) \bigr\vert \leq L_{i} \vert x_{1}-x_{2} \vert ^{p-1} $$
(1.5)

for every \(x_{1},x_{2}\in \mathbb{R} \), and \(h_{i} ( 0 ) =0\) for \(1\leq i\leq n\).

In this paper, the following conditions are assumed:

\((H_{0})\):

\(\alpha _{i}\in (\frac{1}{p};1]\) for \(1\leq i\leq n\).

\((F1)\):

for every \(M>0\) and every \(1\leq i\leq n\),

$$ \sup_{ \vert ( x_{1},x_{2},\ldots,x_{n} ) \vert \leq M} \bigl\vert F_{u_{i}} ( t,x_{1},x_{2},\ldots,x_{n} ) \bigr\vert \in L^{1} \bigl( [ 0,T ] \bigr) . $$
\((F2)\):

\(F ( t,0,0,\ldots,0 ) =0\) for a.e. \(t\in [ 0,T ] \).

\((G)\):

for every \(M>0\) and every \(1\leq i\leq n\),

$$ \sup_{ \vert ( x_{1},x_{2},\ldots,x_{n} ) \vert \leq M} \bigl\vert G_{u_{i}} ( t,x_{1},x_{2},\ldots,x_{n} ) \bigr\vert \in L^{1} \bigl( [ 0,T ] \bigr) . $$

This rest of this paper is organized as follows. The next section presents the necessary preliminary to develop the main contribution of this paper. In Sect. 3, the main result (Theorem 2) is derived, and meaningful consequences (Corollaries 1 and 2 and Example 1) are presented.

Preliminaries

For the sake of clarity, the necessary definitions and properties of fractional calculus are presented below.

Definition 1

([6])

Let u be a function defined on \([ a,b ] \). The left and right Riemann–Liouville fractional derivatives of order \(\alpha >0\) for a function u are defined by

$$ {}_{a}D_{t}^{\alpha }u ( t ) := \frac{d^{n}}{dt^{n}} {}_{a}D_{t}^{\alpha -n}u ( t ) = \frac{1}{\varGamma ( n-\alpha ) }\,\frac{d^{n}}{dt^{n}} \int _{a}^{t} ( t-s ) ^{n-\alpha -1}u ( s ) \,ds $$

and

$$ _{t}D_{b}^{\alpha }u ( t ) := ( -1 ) ^{n} \frac{d^{n}}{dt^{n}}{ }_{t}D_{b}^{\alpha -n}u ( t ) = \frac{ ( -1 ) ^{n}}{\varGamma ( n-\alpha ) }\frac{d^{n}}{dt^{n}}\int _{t}^{b} ( t-s ) ^{n-\alpha -1}u ( s ) \,ds, $$

for every \(t\in [ a,b ] \), provided the right-hand sides are pointwise defined on \([ a,b ] \), where \(n-1\leq \alpha < n\) and \(n\in \mathbb{N} \).

Here, \(\varGamma ( \alpha ) \) is the gamma function, given by

$$ \varGamma ( \alpha ) := \int _{0}^{+\infty }z^{ \alpha -1}e^{-z} \,dz. $$

The set \(AC^{n} ( [ a,b ] ,\mathbb{R} ) \) corresponds to the space of functions \(u: [ a,b ] \rightarrow \mathbb{R} \) such that \(u\in C^{n-1} ( [ a,b ] ,\mathbb{R} ) \) and \(u^{ ( n-1 ) }\in AC^{n} ( [ a,b ] ,\mathbb{R} ) \). Here, as usual, \(C^{n-1} ( [ a,b ] ,\mathbb{R} ) \) denotes the set of mappings that are \(( n-1 ) \) times continuously differentiable on \([ a,b ] \). In particular, \(AC ( [ a,b ] ,\mathbb{R} ) :=AC^{1} ( [ a,b ] ,\mathbb{R} ) \).

Proposition 1

([30])

The following property of fractional integration holds:

$$ \int _{a}^{b} \bigl[ _{a}D_{t}^{-\alpha }u ( t ) \bigr] v ( t ) \,dt= \int _{a}^{b} \bigl[ _{t}D_{b}^{- \alpha }v ( t ) \bigr] u ( t ) \,dt, \quad \alpha >0, $$

provided that\(u\in L^{p} ( [ a,b ] ,\mathbb{R} ) \), \(v\in L^{q} ( [ a,b ] ,\mathbb{R} ) \)and\(p\geq 1\), \(q\geq 1\), \(\frac{1}{p}+\frac{1}{q}\leq 1+\alpha \)or\(p\neq 1\), \(q\neq 1\), \(\frac{1}{p}+\frac{1}{q}=1+\alpha \).

Proposition 2

([31])

If\(u ( a ) =u ( b ) =0\), \(u\in L^{\infty } ( [ a,b ] ,\mathbb{R} ^{N} ) \), \(v\in L^{1} ( [ a,b ] ,\mathbb{R} ) \), or\(v ( a ) =v ( b ) =0\), \(v\in L^{\infty } ( [ a,b ] ,\mathbb{R} ^{N} ) \), \(u\in L^{1} ( [ a,b ] ,\mathbb{R} ) \), then

$$ \int _{a}^{b} \bigl[ _{a}D_{t}^{\alpha }u ( t ) \bigr] v ( t ) \,dt= \int _{a}^{b} \bigl[ _{t}D_{b}^{ \alpha }v ( t ) \bigr] u ( t ) \,dt,\quad 0< \alpha \leq 1. $$

To establish a variational structure for the main problem, it is necessary to construct appropriate function spaces. Following [32], \(C_{0}^{\infty } ( [ 0,T ] ,\mathbb{R} ) \) denotes the set of all functions \(g\in C^{\infty } ( [ 0,T ] ,\mathbb{R} ) \) with \(g ( 0 ) =g ( T ) =0\).

Definition 2

([24])

For \(0<\alpha _{i}\leq 1\), and for \(1\leq i\leq n\), the fractional derivative space \(E_{0}^{\alpha i,p}\) is defined by

$$ E_{0}^{\alpha i,p}= \bigl\{ u\in L^{p}\bigl( [ 0,T ] ,\mathbb{R} \bigr): { }_{0}D_{t}^{\alpha _{i}}u_{i} \in L^{p}, u ( 0 ) =u ( T ) =0 \bigr\} $$

with the norm

$$ \Vert u_{i} \Vert _{\alpha _{i},p}= \bigl( \Vert u_{i} \Vert _{L^{p}}^{p}+\Vert _{0}D_{t}^{\alpha _{i}}u_{i} \Vert _{L^{p}}^{p} \bigr) ^{\frac{1}{p}},\quad \forall u_{i} \in E_{0}^{\alpha i,p} , $$
(2.1)

where

$$ \Vert u_{i} \Vert _{L^{p}}= \biggl( \int _{0}^{T} \vert u_{i} \vert ^{2}\,dt \biggr) ^{\frac{1}{p}} $$

is the norm of \(L^{p} ( [ 0,T ] ,\mathbb{R} ) \) for \(1\leq i\leq n\), and

$$ \Vert u_{i} \Vert _{\infty }=\max_{t\in [ 0,T ] } \bigl\vert u_{i} ( t ) \bigr\vert . $$
(2.2)

From [24, Lemma 3.1], one finds that, for \(0<\alpha _{i}\leq 1\) and \(1< p<+\infty \), the space \(E_{0}^{\alpha _{i},p}\) is a reflexive and separable Banach space.

Lemma 1

([24])

Let\(0<\alpha _{i}\leq 1\), for\(1\leq i\leq n\)and\(1< p<+\infty \), for all\(u_{i}\in E_{0}^{\alpha _{i},p}\)one has

$$\begin{aligned}& \Vert u_{i} \Vert _{L^{p}}\leq \frac{T^{\alpha _{i}}}{\varGamma ( \alpha _{i}+1 ) } \Vert _{0}D_{t}^{\alpha _{i}}u_{i}\Vert _{L^{p}} , \end{aligned}$$
(2.3)
$$\begin{aligned}& \Vert u_{i}\Vert _{\infty }\leq \frac{T^{\alpha _{i}-\frac{1}{p}}}{\varGamma ( \alpha _{i} ) ( ( \alpha _{i}-1 ) q+1 ) ^{\frac{1}{q}}}\Vert _{0}D_{t}^{\alpha _{i}}u_{i} \Vert _{L^{p}} . \end{aligned}$$
(2.4)

Hence, it is possible to consider \(E_{0}^{\alpha _{i},p}\) with respect to the norm

$$ \Vert u_{i} \Vert _{\alpha _{i},p}=\Vert _{0}D_{t}^{ \alpha _{i}}u_{i} \Vert _{L^{p}}= \biggl( \int _{0}^{T} \bigl\vert _{0}D_{t}^{\alpha _{i}}u_{i} \bigr\vert ^{p}\,dt \biggr) ^{ \frac{1}{p}},\quad \forall u_{i}\in E_{0}^{\alpha _{i},p} , $$
(2.5)

for \(1\leq i\leq n\), which is equivalent to (2.1).

Hereafter, let X be the Cartesian product of the n spaces \(E_{0}^{\alpha _{i},p}\), i.e., \(X=E_{0}^{\alpha _{1},p}\times E_{0}^{\alpha _{2},p}\times \cdots\times E_{0}^{ \alpha _{n},p}\) equipped with the norm

$$ \Vert u \Vert =\sum_{i=1}^{n} \Vert u_{i} \Vert _{\alpha _{i},p},\quad u= ( u_{1},u_{2},\ldots,u_{n} ) , $$

where \(\Vert u_{i} \Vert _{\alpha _{i},p}\) is defined in (2.5). It is evident that X is compactly embedded in \(C ( [ 0,T ] ,\mathbb{R} ) ^{n}\).

Definition 3

A weak solution of system (1.4) consists of any function \(u= ( u_{1},u_{2},\ldots,u_{n} ) \in X\), such that, for all \(v= ( v_{1},v_{2},\ldots,v_{n} ) \in X\), one finds that

$$\begin{aligned}& \int _{0}^{T}\sum_{i=1}^{n} \bigl\vert _{0}D_{t}^{\alpha _{i}}u_{i} ( t ) \bigr\vert ^{p-2}{ }_{0}D_{t}^{\alpha _{i}}u_{i} ( t ) _{0}D_{t}^{\alpha _{i}}v_{i} ( t ) \,dt \\& \quad = \lambda \int _{0}^{T}\sum_{i=1}^{n}F_{u_{i}} \bigl( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) \bigr) v_{i} ( t ) \,dt \\& \qquad {}+\mu \int _{0}^{T}\sum_{i=1}^{n}G_{u_{i}} \bigl( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) \bigr) v_{i} ( t ) \,dt + \int _{0}^{T}\sum_{i=1}^{n}h_{i} \bigl( u_{i} ( t ) \bigr) v_{i} ( t ) \,dt. \end{aligned}$$

Remember the following result of [14, Theorem 1], with easy manipulations that are provided in the sequel.

Theorem 1

(Ricceri [14])

LetXbe a reflexive real Banach space; \(\varPhi :X\rightarrow \mathbb{R} \)be a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on\(X^{\ast }\), bounded on bounded subsets ofX, \(\varPsi :X\rightarrow \mathbb{R} \)a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that

$$ \varPhi ( 0 ) =\varPsi ( 0 ) =0. $$

Assume that there exists\(r>0\)and\(\overline{x}\in X\), with\(r<\varPhi ( \overline{x} ) \), such that

(\(a_{1}\)):

\(\frac{\sup_{\varPhi ( x ) \leq r}\varPsi ( x ) }{r}< \frac{\varPhi ( \overline{x} ) }{\varPsi ( \overline{x} ) }\);

(\(a_{2}\)):

for each\(\lambda \in \varLambda _{r};= ] \frac{\varPhi ( \overline{x} ) }{\varPsi ( \overline{x} ) }, \frac{r}{\sup_{\varPhi ( x ) \leq r}\varPsi ( x ) } [ \), the functional\(\varPhi -\lambda \varPsi \)is coercive.

Then, for each compact interval\([ a,b ] \subseteq \varLambda _{r}\), there exists\(\rho >0\)with the following property: for every\(\lambda \in [ a,b ] \)and every\(C^{1}\)functional\(\digamma :X\rightarrow \mathbb{R} \)with compact derivative, there exists\(\delta >0\)such that, for each\(\mu \in [ 0,\delta ] \), the equation

$$ \varPhi ^{\prime } ( x ) -\lambda \varPsi ^{\prime } ( x ) -\mu \digamma ^{\prime } ( x ) =0 $$

has at least three solutions inXwhose norms are less thanρ.

The main results

In the present section, the existence of multiple solutions for system (1.1) is discussed. For any \(\varsigma >0\), \(K ( \varsigma ) \) denotes

$$ \Biggl\{ ( x_{1},x_{2},\ldots,x_{n} ) \in \mathbb{R} ^{n}:\frac{1}{p}\sum _{i=1}^{n} \vert x_{i} \vert ^{p}\leq \varsigma \Biggr\} . $$

This set is one of the cornerstones of the given hypotheses for appropriate choices of ς. For \(u= ( u_{1},u_{2},\ldots,u_{n} ) \in X\) one has

$$ \varUpsilon ( u ) :=\sum_{i=1}^{n}\varUpsilon _{i} ( u_{i} ), $$

where

$$ \varUpsilon _{i} ( x ) := \int _{0}^{T}H_{i} \bigl( x ( s ) \bigr) \,ds\quad \text{and}\quad H_{i} ( x ) := \int _{0}^{x}h_{i} ( z ) \,dz, \quad 1\leq i\leq n, $$

for every \(t\in [ 0,T ] \) and \(x\in \mathbb{R} \). Moreover, let

$$\begin{aligned}& c :=\max_{1\leq i\leq n} \biggl\{ \frac{T^{\alpha _{i}-\frac{1}{p}}}{\varGamma ( \alpha _{i} ) ( ( \alpha _{i}-1 ) q+1 ) ^{\frac{1}{q}}} \biggr\} , \\& k :=\min_{1\leq i\leq n} \biggl\{ 1- \frac{L_{i}T^{\alpha _{i}p}}{ ( \varGamma ( \alpha _{i}+1 ) ) ^{p}} \biggr\} , \\& \tau :=\max_{1\leq i\leq n} \biggl\{ 1+ \frac{L_{i}T^{p\alpha _{i}}}{ ( \varGamma ( \alpha _{i}+1 ) ) ^{p}} \biggr\} . \end{aligned}$$

Theorem 2

Suppose that\(k>0\)and the conditions\(( F1 ) \), \(( F2 ) \), \(( G ) \)and\(( H ) \)are satisfied. Furthermore, assume that there exist a positive constantrand a function\(\omega = ( \omega _{1},\omega _{2},\ldots,\omega _{n} ) \in X\)such that

  1. (i)
    $$\frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{ \alpha _{i}}^{p}>\frac{r}{k}; $$
  2. (ii)
    $$\frac{r\int _{0}^{T}F ( t,\omega _{1},\omega _{2},\ldots,\omega _{n} ) \,dt}{\frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{\alpha _{i}}^{p}-\varUpsilon ( \omega _{1},\omega _{2},\ldots,\omega _{n} ) }- \int _{0}^{T}\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( \frac{cr}{k} ) }F ( t,x_{1},x_{2},\ldots,x_{n} ) \,dt>0; $$
  3. (iii)
    $$ \lim_{ ( \vert x_{1} \vert , \vert x_{2} \vert ,\ldots, \vert x_{n} \vert ) \rightarrow ( +\infty ,+\infty ,\ldots,+\infty ) }\sup \frac{\sup_{t\in [ 0,T ] }F ( t,x_{1},x_{2},\ldots,x_{n} ) }{\frac{1}{p}\sum_{i=1}^{n} \vert x_{i} \vert ^{p}}\leq 0. $$

Then, setting

$$ \varLambda _{r}:= \biggl] \frac{\frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{\alpha _{i}}^{p}-\varUpsilon ( \omega _{1},\omega _{2},\ldots,\omega _{n} ) }{\int _{0}^{T}F ( t,\omega _{1},\omega _{2},\ldots,\omega _{n} ) \,dt}, \frac{r}{\int _{0}^{T}\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( \frac{cr}{k} ) }F ( t,x_{1},x_{2},\ldots,x_{n} ) \,dt} \biggr[ , $$

for each compact interval\([ a,b ] \subseteq \varLambda _{r}\), there exists\(\rho >0\)with the following property: for every\(\lambda \in [ a,b ] \)there exists\(\delta >0\)such that, for each\(\mu \in [ 0,\delta ] \), system (1.4) admits at least three solutions inXwhose norms are less thanρ.

Proof

For each \(u= ( u_{1},u_{2},\ldots,u_{n} ) \in X\), define Φ, \(\varPsi :X\rightarrow \mathbb{R} \) as

$$ \varPhi ( u ) :=\frac{1}{p}\sum_{i=1}^{n} \Vert u_{i} \Vert _{\alpha _{i},p}^{p}-\varUpsilon ( u ) $$

and

$$ \varPsi ( u ) := \int _{0}^{T}F \bigl( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) \bigr) \,dt. $$

Clearly, Φ and Ψ are continuously Gâteaux differentiable functionals whose Gâteaux derivatives at the point \(u\in X\) are given by

$$\begin{aligned}& \varPhi ^{\prime } ( u ) ( v ) := \int _{0}^{T} \sum _{i=1}^{n} \bigl\vert _{0}D_{t}^{\alpha _{i}}u_{i} ( t ) \bigr\vert ^{p-2}{ }_{0}D_{t}^{\alpha _{i}}u_{i} ( t ) _{0}D_{t}^{\alpha _{i}}v_{i} ( t ) \,dt- \int _{0}^{T} \sum _{i=1}^{n}h_{i} \bigl( u_{i} ( t ) \bigr) v_{i} ( t ) \,dt, \\& \varPsi ^{\prime } ( u ) ( v ) = \int _{0}^{T} \sum _{i=1}^{n}F_{u_{i}} \bigl( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) \bigr) v_{i} ( t ) \,dt, \end{aligned}$$

for every \(v= ( v_{1},v_{2},\ldots,v_{n} ) \in X\). Hence, \(\varPhi -\lambda \varPsi \in C^{1} ( X,\mathbb{R} ) \). Moreover, \(\varPsi ^{\prime }:X\rightarrow X^{\ast }\) is a compact operator (see the proof of [21, Theorem 3.1]). Furthermore, similar to the proof of [22, Theorem 3.1], we can show that Φ is sequentially weakly lower semicontinuous. As concerns functional Φ, it is easy to show that Φ is bounded on each bounded subset of X and its derivative admits a continuous inverse on \(X^{\ast }\). Moreover, we have \(\varPhi ( 0 ) =\varPsi ( 0 ) =0\).

It is shown that the required hypothesis \(\varPhi ( \overline{x} ) >r \) follows from \(( i ) \) and the definition of Φ, by choosing \(\overline{x}=\omega \). Indeed, since (1.5) holds for every \(x_{1},x_{2}\in \mathbb{R} \) and \(h_{1} ( 0 ) =h_{2} ( 0 ) =\cdots=h_{n} ( 0 ) =0\), one has \(\vert h_{i} ( x ) \vert \leq L_{i} \vert x \vert ^{p-1}\), \(1\leq i\leq n\), for all \(x\in \mathbb{R} \). Besides, it follows from (2.3) that

$$\begin{aligned} \varPhi ( \omega ) \geq &\frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{\alpha _{i},p}^{p}- \Biggl\vert \int _{0}^{T}\sum_{i=1}^{n}H_{i} \bigl( \omega _{i} ( t ) \bigr) \,dt \Biggr\vert \\ \geq &\frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{ \alpha _{i},p}^{p}- \frac{1}{p}\sum_{i=1}^{n}L_{i} \int _{0}^{T} \vert \omega _{i} \vert ^{p_{i}}\,dt \\ \geq &\sum_{i=1}^{n} \biggl( \frac{1}{p}- \frac{L_{i}T^{\alpha _{i}p}}{p ( \varGamma ( \alpha _{i}+1 ) ) ^{p_{i}}} \biggr) \Vert \omega _{i} \Vert _{\alpha _{i},p}^{p} \\ \geq &\frac{k}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{ \alpha _{i},p}^{p}>r. \end{aligned}$$
(3.1)

From (2.2) and (2.4), for every \(u_{i}\in E_{0}^{\alpha _{i},p}\) one has

$$ \max_{t\in [ 0,T ] } \bigl\vert u_{i} ( t ) \bigr\vert ^{p}\leq c \Vert u_{i} \Vert _{\alpha _{i},p}^{p}, $$
(3.2)

for each \(u= ( u_{1},u_{2},\ldots,u_{n} ) \in X\). From (2.4), (3.1) and (3.2), for each \(r>0\) one obtains

$$\begin{aligned}& \varPhi ^{-1} ( (-\infty ;r] ) \\& \quad = \bigl\{ u= ( u_{1},u_{2}, \ldots,u_{n} ) \in X:\varPhi ( u ) \leq r \bigr\} \\& \quad \subseteq \Biggl\{ u= ( u_{1},u_{2}, \ldots,u_{n} ) \in X: \frac{1}{p}\sum _{i=1}^{n} \Vert u_{i} \Vert _{\alpha _{i},p}^{p} \leq \frac{r}{k} \Biggr\} \\& \quad \subseteq \Biggl\{ u= ( u_{1},u_{2}, \ldots,u_{n} ) \in X: \frac{1}{p}\sum _{i=1}^{n} \frac{ ( \varGamma ( \alpha _{i} ) ) ^{p} ( ( ( \alpha _{i}-1 ) q+1 ) ) ^{\frac{p}{q}}}{T^{\alpha _{i}p-1}} \Vert u_{i} \Vert _{\infty }^{p} \leq \frac{r}{k} \Biggr\} \\& \quad \subseteq \Biggl\{ u= ( u_{1},u_{2}, \ldots,u_{n} ) \in X: \frac{1}{p}\sum _{i=1}^{n} \vert u_{i} \vert ^{p}\leq \frac{cr}{k}, \text{ for all }t\in [ 0,T ] \Biggr\} . \end{aligned}$$

Then

$$\begin{aligned} \sup_{u\in \varPhi ^{-1} ( (-\infty ;r] ) }\varPsi ( u ) =&\sup_{u\in \varPhi ^{-1} ( (-\infty ;r] ) } \int _{0}^{T}F \bigl( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) \bigr) \,dt \\ \leq & \int _{0}^{T}\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( \frac{cr}{k} ) }F ( t,x_{1},x_{2},\ldots,x_{n} ) \,dt. \end{aligned}$$

Therefore, from the condition (ii), one gets

$$\begin{aligned} \sup_{u\in \varPhi ^{-1} ( (-\infty ;r] ) }\varPsi ( u ) \leq & \int _{0}^{T}\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( \frac{cr}{k} ) }F ( t,x_{1},x_{2},\ldots,x_{n} ) \,dt \\ < &\frac{r\int _{0}^{T}F ( t,\omega _{1},\omega _{2},\ldots,\omega _{n} ) \,dt}{\frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{\alpha _{i},p}^{p}-\varUpsilon ( \omega _{1},\omega _{2},\ldots,\omega _{n} ) } \\ =&\frac{r\int _{0}^{T}F ( t,\omega _{1},\omega _{2},\ldots,\omega _{n} ) \,dt}{\frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{\alpha _{i},p}^{p}-\varUpsilon ( \omega _{1},\omega _{2},\ldots,\omega _{n} ) } \\ =&r\frac{\varPsi ( w ) }{\varPhi ( w ) }, \end{aligned}$$

from which assumption \((a_{1})\) of Theorem 1 follows. Fix \(0<\epsilon <\frac{1}{pTc\lambda }\); from (iii) there is a constant \(\tau _{\epsilon }\) such that

$$ F ( t,x_{1},x_{2},\ldots,x_{n} ) \leq \epsilon \sum_{i=1}^{n} \vert x_{i} \vert ^{p}+\tau _{\epsilon _{i}} $$
(3.3)

for every \(t\in [ 0,T ] \) and for every \(( x_{1},x_{2},\ldots,x_{n} ) \in \mathbb{R} ^{n}\). Taking (2.4) into account, from (3.3), it follows that, for each \(u\in X\),

$$\begin{aligned} \varPhi ( u ) -\lambda \varPsi ( u ) =&\frac{1}{p}\sum _{i=1}^{n} \Vert u_{i} \Vert _{\alpha _{i},p}^{p}- \lambda \int _{0}^{T}F ( t,u_{1},u_{2}, \ldots,u_{n} ) \,dt \\ \geq &\frac{1}{p}\sum_{i=1}^{n} \Vert u_{i} \Vert _{ \alpha _{i},p}^{p}-T\lambda c \epsilon \sum_{i=1}^{n} \Vert u_{i} \Vert _{\alpha _{i},p}^{p}-\lambda \tau _{\epsilon } \\ \geq & \biggl( \frac{1}{p}-T\lambda c\epsilon \biggr) \sum _{i=1}^{n} \Vert u_{i} \Vert _{\alpha _{i},p}^{p}-\lambda \tau _{ \epsilon }, \end{aligned}$$

and thus

$$ \lim_{ \Vert u \Vert \rightarrow +\infty } \bigl( \varPhi ( u ) -\lambda \varPsi ( u ) \bigr) =+ \infty , $$

which means the functional \(\varPhi ( u ) -\lambda \varPsi ( u ) \) is coercive for every parameter λ, in particular, for every \(\lambda \in \varLambda \subset ] \frac{\varPhi ( \omega ) }{\varPsi ( \omega ) }, \frac{r}{\sup_{\varPhi ( u ) \leq r}\varPsi ( u ) } [ \). Then also condition \(( a_{2} ) \) holds.

In addition, since \(G: [ 0,T ] \times \mathbb{R} ^{n}\rightarrow \mathbb{R} \) is a measurable function with respect to \(t\in [ 0,T ] \) for every \(( x_{1},x_{2},\ldots,x_{n} ) \in \mathbb{R} ^{n}\) belonging to \(C^{1}\) with respect to \(( x_{1},x_{2},\ldots,x_{n} ) \in \mathbb{R} ^{n}\) for a.e. \(t\in [ 0,T ] \) satisfying condition \(( \mathbf{G} ) \), the functional

$$ \digamma ( u ) = \int _{0}^{T}G \bigl( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) \bigr) \,dt $$

is well defined and continuously Gâteaux differentiable on X with a compact derivative, and

$$ \digamma ^{\prime } ( u ) = \int _{0}^{T}\sum_{i=1}^{n}G_{u_{i}} \bigl( t,u_{1} ( t ) ,u_{2} ( t ) ,\ldots,u_{n} ( t ) \bigr) v_{i} ( t ) \,dt $$

for all \(( u_{1},u_{2},\ldots,u_{n} ) \), \(( v_{1},v_{2},\ldots,v_{n} ) \in X\). Thus, all the hypotheses of Theorem 1 are satisfied. Also note that the solutions of the equation

$$ \varPhi ^{\prime } ( x ) -\lambda \varPsi ^{\prime } ( x ) -\mu \digamma ^{\prime } ( x ) =0 $$

are exactly the solutions of (1.4) (see [21]). So, the conclusion follows from Theorem 1. □

Example 1

Consider the following fractional boundary value problem:

$$ \textstyle\begin{cases} {}_{t}D_{T}^{0,75}\phi _{3} ( _{0}D_{t}^{0,75}u_{1} ( t ) ) =\lambda F_{u_{1}} ( t,u_{1} ( t ) ,u_{2} ( t ) ) +\mu G_{u_{1}} ( t,u_{1} ( t ) ,u_{2} ( t ) ) +h_{1} ( u_{1} ( t ) ) \\\quad \text{a.e. }t\in [ 0,T ] , \\ {}_{t}D_{T}^{0,6}\phi _{3} ( _{0}D_{t}^{0,6}u_{2} ( t ) ) =\lambda F_{u_{2}} ( t,u_{1} ( t ) ,u_{2} ( t ) ) +\mu G_{u_{1}} ( t,u_{1} ( t ) ,u_{2} ( t ) ) +h_{2} ( u_{2} ( t ) )\\ \quad \text{a.e. }t\in [ 0,T ], \\ u_{1} ( 0 ) =u_{2} ( 0 ) =u_{1} ( 1 ) =u_{2} ( 1 ) =0,\end{cases} $$
(3.4)

where \(\alpha _{1}=0.75\), \(\alpha _{2}=0.6\), \(p=3\), \(T=1\), \(h_{1} ( u_{1} ) = ( \sin ( \frac{u_{1}}{2} ) ) ^{2}\), \(h_{2} ( u_{2} ) = ( \arctan ( \frac{u_{2}}{3} ) ) ^{2}\) and \(G : [ 0,1 ] \times \mathbb{R} ^{2}\rightarrow \mathbb{R} \) is an arbitrary function which is measurable with to respect to \(t\in [ 0,1 ] \) for every \(( x_{1},x_{2} ) \in \mathbb{R} ^{2}\) and is \(C^{1}\) with respect to \(( x_{1},x_{2} ) \in \mathbb{R} ^{2}\) for a.e. \(t\in [ 0,1 ] \), satisfying

$$ \sup_{ \vert ( x_{1},x_{2} ) \vert \leq M} \bigl\vert G_{u_{i}} ( t,x_{1},x_{2} ) \bigr\vert \in L^{1} \bigl( [ 0,T ] \bigr) , $$

for every \(M>0\) and \(i=1,2\). Moreover, for all \(( t,x_{1},x_{2} ) \in [ 0,1 ] \times \mathbb{R} ^{2} \), put \(F ( t,x_{1},x_{2} ) = ( 1+t^{2} ) H ( x_{1},x_{2} ) \), where

$$ H ( x_{1},x_{2} ) = \textstyle\begin{cases} ( x_{1}^{3}+x_{2}^{3} ) ^{2}, & x_{1}^{3}+x_{2}^{3}\leq 1, \\ 2\sqrt{x_{1}^{3}+x_{2}^{3}}- ( x_{1}^{3}+x_{2}^{3} ) , & x_{1}^{3}+x_{2}^{3}>1. \end{cases} $$

Obviously, \(F ( t,0,0 ) =0\) for all \(t\in [ 0,1 ] \), and a direct calculation shows that

$$ c\approx 1.0727,\qquad k\approx 0.3559. $$

By choosing, for instance,

$$ \omega _{1} ( t ) =\varGamma ( 1,25 ) t ( 1-t ) , \qquad \omega _{2} ( t ) =\varGamma ( 1,4 ) t ( 1-t ) , $$

and \(r=\frac{1}{10^{3}}\) all assumptions of Theorem 2 are satisfied. In fact, \(\omega _{i} ( 0 ) =\omega _{i} ( 1 ) =0\), \(i=1,2\), and

$$ {}_{0}D_{t}^{0,75}\omega _{1} ( t ) =t^{0,25}- \frac{2\varGamma ( 1,25 ) }{\varGamma ( 2,25 ) }t^{1,25},\qquad {}_{0}D_{t}^{0,6}\omega _{2} ( t ) =t^{0,4}- \frac{2\varGamma ( 1,4 ) }{\varGamma ( 2,4 ) }t^{1,4}. $$

Then one has

$$ \Vert \omega _{1} \Vert _{0.75}^{3} \approx 0.0498, \qquad \Vert \omega _{2} \Vert _{0.6}^{3}\approx 0.0233, $$

which implies that the condition (i) holds, and

$$\begin{aligned} \frac{\int _{0}^{1}\max \limits _{ ( x_{1},x_{2} ) \in \pi ( \frac{cr}{k} ) }F ( t,x_{1},x_{2} ) \,dt}{r} =&\frac{12c^{2}r}{k^{2}}\approx 0.1090 \\ < &\frac{\int _{0}^{1}F ( t,\omega _{1},\omega _{2} ) \,dt}{\frac{1}{3}\sum_{i=1}^{2} \Vert \omega _{i} \Vert _{\alpha _{i}}^{3}-\varUpsilon ( \omega _{1},\omega _{2} ) } \approx 0.5548 \end{aligned}$$

and

$$ \lim_{ ( \vert x_{1} \vert , \vert x_{2} \vert ) \rightarrow ( +\infty ,+\infty ) } \sup \frac{\sup_{t\in [ 0,1 ] }F ( t,u_{1},u_{2} ) }{\frac{1}{3}\sum_{i=1}^{2} \vert u_{i} \vert ^{3}}=0. $$

Thus, conditions (ii) and (iii) are satisfied. Then, in view of Theorem 2 for each \(\lambda \in ] 1.8025,9.1743 [ \), system \(( 3.4 ) \) has at least three weak solutions in \(X=E_{0}^{.0,75,3}\times E_{0}^{.0,6,3}\).

Next, it is desirable to give a verifiable consequence of Theorem 2 for a fixed text function ω. For a given constant \(\gamma \in ( 0,\frac{1}{2} ) \) and for all \(1\leq i\leq n\), set

$$\begin{aligned}& \begin{aligned} C_{i} ( \alpha _{i},\gamma ) ={}& \frac{1}{p ( \gamma T ) ^{p}} \biggl\{ \int _{0}^{\gamma T}t^{p ( 1-\alpha _{i} ) }\,dt+ \int _{\gamma T}^{ ( 1-\gamma ) T} \bigl( t^{1-\alpha _{i}}- ( t- \gamma T ) ^{1-\alpha _{i}} \bigr) ^{p}\,dt \\ &{}+ \int _{ ( 1-\gamma ) T}^{T} \bigl( t^{1- \alpha _{i}}- ( t- \gamma T ) ^{1-\alpha _{i}} \bigr) - \bigl( 1- \bigl( ( 1-\gamma ) T \bigr) ^{1-\alpha _{i}} \bigr) ^{p}\biggr\} , \end{aligned} \\& \triangle =\min_{1\leq i\leq n} \Biggl\{ \sum _{i=1}^{n}C_{i} ( \alpha _{i},\gamma ) \Biggr\} , \\& \triangle ^{\prime } =\max \Biggl\{ \sum_{i=1}^{n}C_{i} ( \alpha _{i},\gamma ) \Biggr\} . \end{aligned}$$

Corollary 1

Let assumption (iii) in Theorem 2hold. Assume that there exist positive constantsdandηsuch that\(\frac{d}{\triangle ckn}\geq \eta ^{p}\), and also

  1. (j)

    \(F ( t,x_{1},x_{2},\ldots,x_{n} ) \geq 0\), for each\(( t,x_{1},x_{2},\ldots,x_{n} ) \in [ 0,T ] \times {}[ 0;+\infty )\times \cdots\times {}[ 0;+\infty )\);

  2. (jj)

    \(\frac{\int _{0}^{T}\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( d ) }F ( t,x_{1},x_{2},\ldots x_{n} ) \,dt}{kd} < \frac{\int _{\gamma T}^{ ( 1-\gamma ) T}F ( t,\varGamma ( 2-\alpha _{1} ) \eta ,\ldots,\varGamma ( 2-\alpha _{n} ) \eta ) \,dt}{nc\tau \triangle ^{\prime }\eta ^{p}}\).

Then, setting

$$\begin{aligned} \varLambda _{1} :=& \biggl( \frac{n\tau \triangle ^{\prime }\eta ^{p}}{\int _{\gamma T}^{ ( 1-\gamma ) T}F ( t,\varGamma ( 2-\alpha _{1} ) \eta ,\ldots,\varGamma ( 2-\alpha _{n} ) \eta ) \,dt}, \\ &\frac{kd}{\int _{0}^{T}\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( d ) }F ( t,x_{1},x_{2},\ldots x_{n} ) \,dt} \biggr) \end{aligned}$$

for each compact interval \([ a,b ] \subseteq \varLambda _{1}\), there exists \(\rho >0\) with the following property: for every \(\lambda \in ] a,b [ \), there exists δ such that, for each \(\mu \in [ 0,\delta ] \), system (1.4) admits at least three solutions in X whose norms are less than ρ.

Proof

For \(\gamma \in ( 0,\frac{1}{2} ) \) choose \(\omega ( t ) = ( \omega _{1} ( t ) ,\ldots, \omega _{n} ( t ) ) \) for every \(t\in [ 0,T ] \) with

$$ \omega _{i} ( t ) = \textstyle\begin{cases} \frac{\varGamma ( 2-\alpha _{i} ) \eta }{\gamma T}t, & t\in {}[ 0;\gamma T), \\ \varGamma ( 2-\alpha _{i} ) \eta ,& t\in {}[ \gamma T;(1-\gamma )T], \\ \frac{\varGamma ( 2-\alpha _{i} ) \eta }{\gamma T} ( t-T ) , & t\in ((1-\gamma )T;T], \end{cases} $$

for \(1\leq i\leq n\), Clearly \(\omega _{i} ( 0 ) =\omega _{i} ( T ) =0\) and \(\omega _{i}\in L^{2} ( [ 0,T ] ,\mathbb{R} ) \) for \(1\leq i\leq n\),A direct calculation shows that

$$ _{0}D_{t}^{\alpha _{i}}\omega _{i} ( t ) = \textstyle\begin{cases} \frac{\eta }{\gamma T}t^{1-\alpha _{i}},& t\in {}[ 0; \gamma T), \\ \frac{\eta }{\gamma T} ( t^{1-\alpha _{i}}- ( t-\gamma T ) ^{1-\alpha _{i}} ) ,& t\in {}[ \gamma T;(1- \gamma )T],\\ \frac{\eta }{\gamma T} ( t^{1-\alpha _{i}}- ( t- \gamma T ) ^{1-\alpha _{i}}- ( t- ( 1-\gamma ) T ) ^{1-\alpha _{i}} ) ,& t\in ((1-\gamma )T;T], \end{cases} $$

for \(1\leq i\leq n\). Furthermore,

$$\begin{aligned} \int _{0}^{T} \bigl\vert _{0}D_{t}^{\alpha _{i}} \omega _{i} ( t ) \bigr\vert ^{p}\,dt =& \biggl( \frac{\eta }{\gamma T} \biggr) ^{p} \biggl\{ \int _{0}^{ \gamma T}t^{ ( 1-\alpha _{i} ) p}\,dt+ \int _{hT}^{ ( 1-\gamma ) T} \bigl( t^{1-\alpha _{i}}- ( t- \gamma T ) ^{1-\alpha _{i}} \bigr) ^{p}\,dt \\ & + \int _{ ( 1-h ) T}^{T} \bigl( t^{1- \alpha _{i}}- ( t- \gamma T ) ^{1-\alpha _{i}}- \bigl( t- ( 1-\gamma ) T \bigr) ^{1-\alpha _{i}} \bigr) ^{p}\,dt \biggr\} \\ =&p\eta ^{p}C_{i} ( \alpha _{i},h ) , \end{aligned}$$

for \(1\leq i\leq n\). Thus, \(\omega \in X\), and

$$ \Vert \omega _{i} \Vert _{\alpha _{i},p}^{p}=p \eta ^{p}C_{i} ( \alpha _{i},h ) , $$

with \(1\leq i\leq n\). This and (3.1) imply that

$$ \begin{aligned}[b] \varPhi ( \omega ) =\varPhi ( \omega _{1}, \ldots,\omega _{n} ) &=\frac{1}{p}\sum _{i=1}^{n} \Vert \omega _{i} \Vert _{ \alpha _{i},p}^{p}-\varUpsilon ( \omega _{i} ) \\ &\geq \frac{k}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{ \alpha _{i},p}^{p} \\ &\geq k\eta ^{p}\sum_{i=1}^{n}C_{i} ( \alpha _{i},h ) \\ &\geq nk\triangle \eta ^{p}. \end{aligned} $$
(3.5)

Similarly to (3.1) and (3.5) one has

$$ \varPhi ( \omega ) \leq n\tau \triangle ^{\prime }\eta ^{p}. $$

Let \(r=\frac{kd}{c}\). From \(\frac{d}{\triangle ckn}<\eta ^{p}\), it is found as a result that

$$ \frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{ \alpha _{i},p}^{p}\geq \varPhi ( \omega ) \geq nk \triangle \eta ^{p}\geq nk\triangle \times \frac{d}{\triangle ckn}= \frac{r}{k}, $$

which is assumption (i) of Theorem 2.

On the other hand, by using assumption (j), one can infer

$$\begin{aligned} \varPsi ( \omega ) :=& \int _{0}^{T}F \bigl( t,\omega _{1} ( t ) ,\omega _{2} ( t ) ,\ldots,\omega _{n} ( t ) \bigr) \,dt \\ \geq & \int _{\gamma T}^{ ( 1-\gamma ) T}F \bigl( t, \varGamma ( 2-\alpha _{1} ) \eta ,\varGamma ( 2-\alpha _{2} ) \eta ,\ldots, \varGamma ( 2-\alpha _{n} ) \eta \bigr) \,dt. \end{aligned}$$

Moreover, by condition (jj) one gets

$$\begin{aligned}& \frac{\int _{0}^{T}\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( \frac{cr}{k} ) }F ( t,x_{1},x_{2},\ldots x_{n} ) \,dt}{r} \\& \quad = \frac{c\int _{0}^{T}\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( d ) }F ( t,x_{1},x_{2},\ldots x_{n} ) \,dt}{kd} \\& \quad < \frac{\int _{\gamma T}^{ ( 1-\gamma ) T}F ( t,\varGamma ( 2-\alpha _{1} ) \eta ,\ldots,\varGamma ( 2-\alpha _{n} ) \eta ) \,dt}{n\tau \triangle ^{\prime }\eta ^{p}} \\& \quad \leq \frac{\int _{\gamma T}^{ ( 1-\gamma ) T}F ( t,\varGamma ( 2-\alpha _{1} ) \eta ,\ldots,\varGamma ( 2-\alpha _{n} ) \eta ) \,dt}{\varPhi ( \omega ) } \\& \quad \leq \frac{p\int _{0}^{T}F ( t,\omega _{1},\omega _{2},\ldots,\omega _{n} ) \,dt}{\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{\alpha _{i},p}^{p}-p\varUpsilon ( \omega _{1},\omega _{2},\ldots,\omega _{n} ) }, \end{aligned}$$

which implies that (ii) is satisfied. Thus, all the assumptions of Theorem 2 are satisfied and the proof is complete. □

Corollary 2

Let\(F:\mathbb{R} ^{n}\rightarrow \mathbb{R} ^{n}\)be a\(C^{1}\)-function and\(F ( 0,\ldots,0 ) =0\). Assume that there exist positive constantsdandηsuch that\(\frac{d}{\triangle ckn}<\eta ^{p}\), and also

  1. (H)

    \(F ( x_{1},\ldots,x_{n} ) \geq 0\), for each\(( x_{1},\ldots,x_{n} ) \in {}[ 0;+\infty )\times \cdots \times {}[ 0;+\infty )\);

  2. (HH)

    \(\frac{\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( d ) }F ( x_{1},x_{2},\ldots x_{n} ) }{kd}\)\(< \frac{ ( 1-2\gamma ) F ( \varGamma ( 2-\alpha _{1} ) \eta ,\ldots,\varGamma ( 2-\alpha _{n} ) \eta ) }{nc\tau \triangle ^{\prime }\eta ^{p}}\);

  3. (HHH)

    \(\lim_{ ( \vert x_{1} \vert , \vert x_{2} \vert ,\ldots, \vert x_{n} \vert ) \rightarrow ( +\infty ,+\infty ,\ldots,+\infty ) }\sup \frac{F ( x_{1},x_{2},\ldots,x_{n} ) }{\frac{1}{p}\sum_{i=1}^{n} \vert x_{i} \vert ^{p}} \leq 0\).

Then, setting

$$\begin{aligned} \varLambda _{2} :=& \biggl( \frac{n\tau \triangle ^{\prime }\eta ^{p}}{T ( 1-2\gamma ) F ( \varGamma ( 2-\alpha _{1} ) \eta ,\ldots,\varGamma ( 2-\alpha _{n} ) \eta ) }, \\ &{}\frac{kd}{cT\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( d ) }F ( x_{1},x_{2},\ldots x_{n} ) } \biggr) , \end{aligned}$$

for each compact interval \([ a,b ] \subseteq \varLambda _{2}\), there exists \(\rho >0\) with the following property: for every \(\lambda \in ] a,b [ \), there exists \(\delta >0\) such that, for each \(\mu \in [ 0,\delta ] \), system (1.4) admits at least three solutions in X whose norms are less than ρ.

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Acknowledgements

We authors acknowledge to Prof. Aldo Jonathan Muñoz-Vázquez for a first revision and kind comments on this work. The authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions.

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Affiliations

  1. Department of Mathematics, Faculty of Exact Sciences, Larbi Tebessi University, Tebessa, Algeria

    Rafik Guefaifia & Fares Kamache

  2. Department of Mathematics, College of Sciences and Arts, Qassim University, Al-Rass, Kingdom of Saudi Arabia

    Salah Boulaaras

  3. Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Algeria

    Salah Boulaaras

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  1. Rafik Guefaifia
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  2. Salah Boulaaras
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  3. Fares Kamache
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The authors contributed equally in this article. They have all read and approved the final manuscript.

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Correspondence to Salah Boulaaras.

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On the occasion of the 80th birthday of the second author’s mother, Mrs. Fatma Bint Al-Tayeb Zeghdoud.

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Guefaifia, R., Boulaaras, S. & Kamache, F. On the existence of three solutions of Dirichlet fractional systems involving the p-Laplacian with Lipschitz nonlinearity. Bound Value Probl 2020, 131 (2020). https://doi.org/10.1186/s13661-020-01429-x

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MSC

  • 35J60
  • 35B30
  • 35B40
  • 35J15
  • 35J25

Keywords

  • Fractional differential equations
  • Riemann–Liouville fractional derivatives
  • Variational methods
  • Three solutions
  • p-Laplacian