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
-
(i)
$$\frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{ \alpha _{i}}^{p}>\frac{r}{k}; $$
-
(ii)
$$\frac{r\int _{0}^{T}F ( t,\omega _{1},\omega _{2},\ldots,\omega _{n} ) \,dt}{\frac{1}{p}\sum_{i=1}^{n} \Vert \omega _{i} \Vert _{\alpha _{i}}^{p}-\varUpsilon ( \omega _{1},\omega _{2},\ldots,\omega _{n} ) }- \int _{0}^{T}\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( \frac{cr}{k} ) }F ( t,x_{1},x_{2},\ldots,x_{n} ) \,dt>0; $$
-
(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
-
(j)
\(F ( t,x_{1},x_{2},\ldots,x_{n} ) \geq 0\), for each\(( t,x_{1},x_{2},\ldots,x_{n} ) \in [ 0,T ] \times {}[ 0;+\infty )\times \cdots\times {}[ 0;+\infty )\);
-
(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
-
(H)
\(F ( x_{1},\ldots,x_{n} ) \geq 0\), for each\(( x_{1},\ldots,x_{n} ) \in {}[ 0;+\infty )\times \cdots \times {}[ 0;+\infty )\);
-
(HH)
\(\frac{\max_{ ( x_{1},x_{2},\ldots,x_{n} ) \in K ( d ) }F ( x_{1},x_{2},\ldots x_{n} ) }{kd}\)\(< \frac{ ( 1-2\gamma ) F ( \varGamma ( 2-\alpha _{1} ) \eta ,\ldots,\varGamma ( 2-\alpha _{n} ) \eta ) }{nc\tau \triangle ^{\prime }\eta ^{p}}\);
-
(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.