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Elementary variational approach to positive and negative solutions of a nonlinear algebraic system
Advances in Difference Equations volumeÂ 2017, ArticleÂ number:Â 322 (2017)
Abstract
By using variational methods and elementary analysis, the conditions for the existence of positive and negative solutions to a nonlinear system are given. Furthermore, some examples are offered to explain that the conditions are new and the results are sharp.
1 Introduction
In [1], the periodic solutions of difference equations, the solutions of boundary value problems and the steady state solutions of the majority of neural networks can be summed up in the solutions of the nonlinear algebraic system as follows:
where \(B=(b_{ij})_{n\times n}\) is a real symmetric n by n matrix and \(f(u)=(f_{1}(u_{1}),f_{2}(u_{2}),\ldots,f_{n}(u_{n}))^{T}\) for \(u=(u_{1},u_{2},\ldots,u_{n})^{T}\in R^{n}\), where \(f_{k}:R\rightarrow R\) for eachÂ k.
A column vector \(u=(u_{1},u_{2},\ldots,u_{n})^{T}\in R^{n}\) is said to be a solution corresponding to it if substitution of u into (1) renders it an identity. A vector \(u=(u_{1},u_{2},\ldots,u_{n})^{T}\) is said to be positive if \(u_{i}>0\) for \(i\in \{ 1,2,\ldots,n \} \), negative if \(u_{i}<0\) for \(i\in \{ 1,2,\ldots,n \} \), zerofree if \(u_{i}\neq 0\) for all \(i\in \{ 1,2,\ldots,n \} \), respectively.
It has come to our attention that: if B in (1) is a real positive definite matrix, the existence of zerofree solutions of (1) is studied in [2â€“4]; if B in (1) takes the form of a real symmetric matrix and has a positive eigenvalue, the existence of nontrivial solutions of (1) is discussed in [1, 5, 6]; if B in (1) represents a symmetric nonnegative matrix, the existence of positive and negative solutions of (1) is explored in [7].
In case B is a real symmetric matrix with some additional conditions attached to its elements, and by using variational approaches (see [8]), this paper presents the existence criteria for the positive and negative solutions of (1). Our proofs are elementary. Furthermore, we provide some examples to show that our conditions are new and our results are sharp.
2 Preliminaries
In this section, we give some notations and definitions.
Let
and
Consider the function \(I:\Omega^{+}\rightarrow R\) defined by
Since
a column vector \(u=(u_{1},u_{2},\ldots,u_{n})^{T}\) is a positive solution of (1) if and only if u is a critical point of function I in the interior of \(\Omega^{+}\).
Let \(B=(b_{ij})_{n\times n}\) be a real symmetric n by n matrix. For convenience, for some \(\{ i_{1},i_{2},\ldots, i_{k_{0}} \} \subset \{ 1,2,\ldots,n \} \), let
and let \(\lambda_{\max }\) be the maximum eigenvalue of B and \(\lambda_{\min } \) the minimum eigenvalue ofÂ B.
3 Superlinear case
In this section, we are concerned with \(f_{1},\ldots,f_{n}\) that are â€˜superlinearâ€™ near 0.
Theorem 3.1
Let \(B=(b_{ij})_{n\times n}\) be a real symmetric matrix with \(b_{ij}\geqslant 0\) for \(i,j\in \{ 1,2,\ldots,n \} \), while \(i\neq j\), \(\max \{ b_{11},b_{12} \} >0\), \(\max \{ b_{i+1,i},b_{i+1,i+1} \} >0\) for \(i\in \{ 1,2,\ldots,n1 \} \), and there exists some \(\{ i_{1},i_{2},\ldots,i _{k_{0}} \} \subset \{ 1,2,\ldots,n \} \) such that \(\overline{B}(i_{1},i_{2},\ldots,i_{k_{0}})>0\). Assume furthermore that \(f_{k}\in C ( [ 0,+\infty ) ,R ) \) for each k and
 (\(\mathrm{G}_{1}\)):

There exist constants \(a_{1}>\frac{1}{2}\lambda_{\max }\), \(a_{2}>0 \) and \(M>0\) such that
$$ \int_{0}^{z}f_{k}(s)\,ds\geq a_{1}z^{2}a_{2}\quad \textit{for } z\geq M \textit{ and } k\in \{ 1,2,\ldots,n \} . $$  (\(\mathrm{G}_{2}\)):

For each \(k\in \{ 1,2,\ldots,n \} \),
$$ \lim_{z\rightarrow 0^{+}}\frac{\int_{0}^{z}f_{k}(s)\,ds}{z ^{2}}=0. $$
Then system (1) has at least one positive real solution.
Proof
First, we prove that I is bounded from above in \(\Omega^{+}\). According to (\(\mathrm{G}_{1}\)), if
then for any \(z\geq 0\) and \(k\in \{ 1,2,\ldots,n \} \),
For any \(u\in \Omega^{+}\),
Since \(a_{1}>\frac{1}{2}\lambda_{\max }\), for any \(u\in \Omega^{+}\),
That is, I is bounded from above in \(\Omega^{+}\).
Claim 1
Let \(c_{0}=\sup_{u\in \Omega ^{+}} I(u)\), then \(c_{0}>0\).
By (\(\mathrm{G}_{2}\)), there exist constants \(\delta >0\) and \(\beta \in ( 0,\frac{1}{2k_{0}}\overline{B}(i_{1},i_{2},\ldots,i_{k_{0}}) ) \) such that for any \(0\leq z\leq \delta \),
Choosing \(v=(v_{1},v_{2},\ldots,v_{n})^{T}\in \Omega^{+}\), where \(v_{i}= \delta \) if \(i\in \{ i_{1},i_{2},\ldots,i_{k_{0}} \} ,v_{i}=0\) if \(i\notin \{ i_{1},i_{2},\ldots, i_{k_{0}} \} \). By (2) and (4),
This shows that \(c_{0}=\sup_{u\in \Omega ^{+}}I(u)>0\).
By the definition of \(c_{0}\), there is a sequence \(\{ u^{ ( i ) } \} \subset \Omega^{+}\) such that \(\lim_{i\rightarrow \infty } I ( u^{ ( i ) } ) =c_{0}\). It is easy to see that there is a positive number p such that for \(i\geqslant 1\),
It follows that
Thus \(\{ u^{ ( i ) } \} \) is a bounded sequence in \(\Omega^{+}\). Consequently, it has a convergent subsequence \(\{ u^{ ( i_{j} ) } \} \). Let \(\{ u^{ ( i _{j} ) } \} \) tend to \(u^{ ( 0 ) }= ( u_{1} ^{ ( 0 ) },u_{2}^{ ( 0 ) },\ldots,u_{n}^{ ( 0 ) } ) ^{T}\) as \(j\rightarrow \infty \). Then \(u^{ ( 0 ) } \in \Omega^{+}\) and \(I ( u^{ ( 0 ) } ) =c_{0}\). Note that \(I ( 0 ) =0\) and \(c_{0}>0\). Clearly, there exists \(i_{0}\in \{ 1,2,\ldots,n \} \) such that \(u_{i_{0}}^{ ( 0 ) }>0\).
Claim 2
\(u^{ ( 0 ) }\) is a positive critical point of functionÂ I.
Let \(i_{0}\in \{ 3,4,\ldots,n1 \} \). First, we prove that \(u_{i_{0}+1}^{ ( 0 ) },u_{i_{0}+2}^{ ( 0 ) },\ldots,u _{n}^{ ( 0 ) }\)are greater than zero. Assume that \(u_{i_{0}+1} ^{ ( 0 ) }=0\), then
For \(r\geq 0\), let
We assert that there exists a positive constant \(r_{1}\) such that \(I_{1} ( r_{1} ) >0\). Since \(\max \{ b_{i_{0}+1,i_{0}}, b _{i_{0}+1,i_{0}+1} \} >0\), the problem can be discussed on two scenarios: if \(b_{i_{0}+1,i_{0}}>0\), then
By (7), (8) and (\(\mathrm{G}_{2}\)), the assertion holds; if \(b_{i_{0}+1,i_{0}+1}>0\), note that \(b_{ij}\geqslant 0\) for \(i,j\in \{ 1,2,\ldots,n \} \) while \(i\neq j\), then
By (9) and (\(\mathrm{G}_{2}\)), the assertion holds. In conclusion, there exists a positive constant \(r_{1}\) such that \(I_{1} ( r_{1} ) >0\).
Choosing
then \(\overline{u}^{ ( 0 ) }\in \Omega^{+}\) and
which is contrary to the definition of \(c_{0}\). Thus \(u_{i_{0}+1}^{ ( 0 ) }>0\). Similarly, \(u_{i_{0}+2}^{ ( 0 ) },\ldots,u _{n}^{ ( 0 ) }\)are greater than zero.
It can be deduced that \(u_{i_{0}1}^{ ( 0 ) }>0\) by changing \(i_{0}+1\) as \(i_{0}1\) in the above mentioned procedure. Similarly, \(u_{2}^{ ( 0 ) },\ldots,u_{i_{0}2}^{ ( 0 ) }\) are greater than zero.
Finally, we prove that \(u_{1}^{ ( 0 ) }>0\). Assume that \(u_{1}^{ ( 0 ) }=0\). Since \(u_{2}^{ ( 0 ) }>0\), we see that
For \(r\geq 0\), let
We assert that there exists a positive constant \(r_{2}\) such that \(I_{2} ( r_{2} ) >0\). Since \(\max \{ b_{11},b_{12} \} >0\), the problem can be discussed on two scenarios: if \(b_{12}>0\), then
By (11), (12) and (\(\mathrm{G}_{2}\)), the assertion holds; if \(b_{11}>0\), note that \(b_{ij}\geqslant 0\) for \(i,j\in \{ 1,2,\ldots,n \} \) while \(i\neq j\), then
By (\(\mathrm{G}_{2}\)), the assertion holds. In conclusion, there exists a positive constant \(r_{2}\) such that \(I_{2} ( r_{2} ) >0\).
Choosing \(\overline{u}^{ ( 0 ) }= ( r_{2},u_{2}^{ ( 0 ) },\ldots,u_{n}^{ ( 0 ) } ) ^{T}\), we have \(\overline{u}^{ ( 0 ) }\in \Omega^{+}\) and
which is contrary to the definition of \(c_{0}\). Thus \(u_{1}^{ ( 0 ) }>0\).
All cases are exhausted, then \(u^{ ( 0 ) }\) belongs to the interior of \(\Omega^{+}\)and \(I ( u^{ ( 0 ) } ) =\sup_{u\in \Omega ^{+}} I(u)\). It follows that \(u^{ ( 0 ) }\) is a positive critical point of I, that is, \(u^{ ( 0 ) }= ( u_{1}^{ ( 0 ) },u_{2}^{ ( 0 ) },\ldots,u_{n} ^{ ( 0 ) } ) ^{T}\) is a positive solution of (1).
The proof is completed.â€ƒâ–¡
Theorem 3.2
Let \(B=(b_{ij})_{n\times n}\) be a real symmetric matrix which satisfies the conditions in TheoremÂ 3.1. Assume furthermore that \(f_{k}\in C( ( \infty,0 ] ,R ) \) for each k and
 (\(\mathrm{G}_{1}^{\prime }\)):

There exist constants \(a_{1}>\frac{1}{2} \lambda_{\max }\), \(a_{2}>0\) and \(M>0\) such that
$$ \int_{0}^{z}f_{k}(s)\,ds\geq a_{1}z^{2}a_{2}\quad \textit{for } z\leq M\textit{ and } k\in \{ 1,2,\ldots,n \} . $$  (\(\mathrm{G}_{2}^{\prime }\)):

For each \(k\in \{ 1,2,\ldots,n \} \),
$$ \lim_{z\rightarrow 0^{}}\frac{\int_{0}^{z}f_{k}(s)\,ds}{z^{2}}=0. $$
Then system (1) has at least one negative real solution.
Consider the function \(\overline{I}:\Omega^{}\rightarrow R\) defined by
Let \(u=v\), then finding a critical point of IÌ… in the interior of \(\Omega^{}\) is equivalent to finding a critical point of function (14) in the interior of \(\Omega^{+}\),
Similar to the proof of TheoremÂ 3.1, it can be proved that there exists a negative critical point of J in the interior of \(\Omega^{+}\), so the conclusion of TheoremÂ 3.2 is true.
According to Theorems 3.1 and 3.2, the following can be directly obtained.
Corollary 3.1
Let \(B=(b_{ij})_{n\times n}\) be a real symmetric matrix which satisfies conditions in TheoremÂ 3.1. Assume furthermore that \(f_{k}\in C ( R,R ) \) for each k and
 (\(\mathrm{G}_{1}^{\prime\prime }\)):

There exist constants \(a_{1}> \frac{1}{2}\lambda_{\max }\), \(a_{2}>0\) and \(M>0\) such that
$$ \int_{0}^{z}f_{k}(s)\,ds\geq a_{1}z^{2}a_{2}\quad \textit{for }\vert z\vert \geq M\textit{ and }k\in \{ 1,2,\ldots,n \} . $$  (\(\mathrm{G}_{2}^{\prime\prime }\)):

For each \(k\in \{ 1,2,\ldots,n \} \),
$$ \lim_{z\rightarrow 0} \frac{\int_{0}^{z}f_{k}(s)\,ds}{z^{2}}=0. $$
Then system (1) has at least one positive real solution and one negative real solution.
Example 3.1
The system
is one of the form (1), where
and \(f_{k}(s)=s^{r}\), \(k=1,2,3\). B is neither a nonnegative matrix nor a positive definite matrix because the main diagonal elements of B are negative, but a real symmetric matrix with \(b_{ij}>0\) for \(i,j=1,2,3\) while \(i\neq j\), \(\max \{ b_{11},b_{12} \} =\max \{ b_{21},b_{22} \} =\max \{ b_{32},b_{33} \} =1\), and the sum of all its elements is 3. The eigenvalues of B are 1, âˆ’2 and âˆ’2. Let \(a_{1}=1\), \(a_{2}=1\) and \(M=(r+1)^{\frac{1}{r1}}\), then for any \(z\geq M\) and \(k\in \{ 1,2,3 \} \), we have
For each \(k\in \{ 1,2,3 \} \),
That is, (15) satisfies all the conditions of TheoremÂ 3.1. Thus, (15) has at least one positive solution.
Indeed, \((1,1,1)^{T}\) is the unique positive solution to (15). Let \((x_{1},x_{2},x_{3})^{T}\) be the positive solution to (15) such that
we assert that \(x_{1}=x_{2}=x_{3}\). By the first and second equations of (16) and differential mean value theorem,
where \(\theta_{1}\) is a certain positive number between \(x_{1}\) and \(x_{2}\). If \(x_{1}\neq x_{2}\), by (17), then \(r\theta_{1}^{r1}=2\), which is contrary to the condition that r and \(\theta_{1}\) are positive. Similarly, by the second and third equations of (16) and differential mean value theorem, it is easy to find \(x_{2}=x_{3}\). In conclusion, \(x_{1}=x_{2}=x_{3}\). Let \(x_{1}=x_{2}=x _{3}=t\). By (15), \(t=1\). Thus \((1,1,1)^{T}\) is the unique positive solution of (15) and TheoremÂ 3.1 is sharp.
Example 3.2
Consider the scenario that \(r=3\) for the nonlinear algebra system (15). It is easy to verify that (15) satisfies all the conditions of CorollaryÂ 3.1. Thus by CorollaryÂ 3.1, the system has one positive solution and one negative solution. From the discussion in ExampleÂ 3.1, \((1,1,1)^{T}\) is the unique positive solution. It is easy to find that \((1,1,1)^{T}\) is the unique negative solution because \(r=3\). This example shows that CorollaryÂ 3.1 is sharp.
Example 3.3
The system
is one of the form (1), where
\(f_{1}(s)=f_{4}(s)=s^{3}\), and \(f_{2}(s)=f_{3}(s)=\frac{1}{2}s^{3}\). B is neither a nonnegative matrix nor a positive definite matrix because the main diagonal elements of B are negative, but a real symmetric matrix with \(b_{ij}>0\) for \(i,j=1,2,3,4\) while \(i\neq j\), \(\max \{ b_{11},b_{12} \} =\max \{ b_{21},b_{22} \} =\max \{ b_{43},b_{44} \} = \frac{3}{2}\), \(\max \{ b_{32},b_{33} \} =\frac{13}{4}\), and the sum of all its elements is \(\frac{9}{2}\). The eigenvalues of B are \(\frac{3}{8}+\frac{1}{8}\sqrt{313}\), \(\frac{3}{8}\frac{1}{8} \sqrt{313}\), \(\frac{29}{8}+\frac{1}{8}\sqrt{313}\) and \( \frac{29}{8}+\frac{1}{8}\sqrt{313}\). Let \(a_{1}=1\), \(a_{2}=3\) and \(M=3\), it is easy to verify that (18) satisfies all the conditions of CorollaryÂ 3.1. Thus, (18) has at least one positive solution and one negative solution. Indeed, all the solutions of (18) can be found and given by \((0,0,0,0)^{T}\), \((1,2,2,1)^{T}\) and \((1,2,2,1)^{T}\). This example shows that CorollaryÂ 3.1 is sharp.
4 Sublinear case
In this section, we are concerned with \(f_{1},\ldots,f_{n}\) that are â€˜sublinearâ€™ near 0.
Theorem 4.1
Let \(B=(b_{ij})_{n\times n}\) be a real symmetric matrix with \(b_{ij}\leq 0\) for \(i,j\in \{ 1,2,\ldots,n \} \) while \(i\neq j\). Assume furthermore that \(f_{k}\in C ( [ 0,+\infty ) ,R ) \) for each k and
 (\(\mathrm{G}_{3}\)):

There exist constants \(a_{1}<\frac{1}{2}\lambda_{\min }\), \(a_{2}>0 \) and \(M>0\) such that
$$ \int_{0}^{z}f_{k}(s)\,ds\leq a_{1}z^{2}+a_{2}\quad \textit{for } z>M \textit{ and } k\in \{ 1,2,\ldots,n \} . $$  (\(\mathrm{G}_{4}\)):

For each \(k\in \{ 1,2,\ldots,n \} \),
$$ \lim_{z\rightarrow 0^{+}}\frac{\int_{0}^{z}f_{k}(s)\,ds}{z ^{2}}=+\infty . $$
Then system (1) has one positive real solution.
Proof
Let I be defined by (2). First, we prove that I is bounded from below in \(\Omega^{+}\). By (\(\mathrm{G}_{3}\)), if
then for any \(z\geq 0\) and \(k\in \{ 1,2,\ldots,n \} \),
For any \(u\in \Omega^{+}\),
Since \(a_{1}<\frac{1}{2}\lambda_{\min }\), for any \(u\in \Omega^{+}\),
That is, I is bounded from below in \(\Omega^{+}\).
Claim 3
Let \(c_{0}=\inf_{u\in \Omega ^{+}}I(u)\), then \(c_{0}<0\).
By (\(\mathrm{G}_{4}\)), there exist constants \(\delta >0\) and \(\beta > \frac{1}{2n}\sum_{i,j=1}^{n} b_{ij}\) such that for any \(0\leq z\leq \delta \),
Choosing \(v=(\delta,\delta,\ldots,\delta )^{T}\in \Omega^{+}\), by (2) and (20),
This shows that \(c_{0}=\inf_{u\in \Omega ^{+}}I(u)<0\).
By the definition of \(c_{0}\), there is a sequence \(\{ u^{ ( i ) } \} \subset \Omega^{+}\) such that \(\lim_{i\rightarrow \infty } I ( u^{ ( i ) } ) =c_{0}\). It is easy to see that there is a positive number p such that for any \(i\geq 1\),
It follows that
Thus \(\{ u^{ ( i ) } \} \) is a bounded sequence in \(\Omega^{+}\). Consequently, it has a convergent subsequence \(\{ u^{ ( i_{j} ) } \} \). Let \(\{ u^{ ( i _{j} ) } \} \) tend to \(u^{ ( 0 ) }= ( u_{1} ^{ ( 0 ) },u_{2}^{ ( 0 ) },\ldots,u_{n}^{ ( 0 ) } ) ^{T}\) as \(j\rightarrow \infty \). Then \(u^{ ( 0 ) } \in \Omega^{+}\) and \(I ( u^{ ( 0 ) } ) =c_{0}\). Note that \(I ( 0 ) =0\) and \(c_{0}<0\), therefore \(u^{ ( 0 ) }\neq 0\).
Claim 4
\(u^{ ( 0 ) }\) is a zerofree critical point of functionalÂ I.
Assume that there exists \(i_{0}\in \{ 1,2,\ldots,n \} \) such that \(u_{i_{0}}^{ ( 0 ) }=0\),
For \(r\geq 0\), let
Since \(b_{i_{0},j}\leq 0\) for \(j\in \{ 1,2,\ldots,n \} \) while \(j\neq i_{0}\), by (\(\mathrm{G}_{4}\)),
then there exists a positive constant \(r_{3}\) such that \(I_{3} ( r _{3} ) <0\).
Choosing \(\overline{u}^{ ( 0 ) }= ( u_{1}^{ ( 0 ) },\ldots,u_{i_{0}1}^{ ( 0 ) },r_{3},u_{i_{0}+1}^{ ( 0 ) },\ldots,u_{n}^{ ( 0 ) } ) ^{T}\), we have \(\overline{u} ^{ ( 0 ) }\in \Omega^{+}\)and
which is contrary to the definition of \(c_{0}\). Thus \(u^{ ( 0 ) }\) belongs to the interior of \(\Omega^{+}\) and \(I ( u^{ ( 0 ) } ) =\inf_{u\in \Omega ^{+}}I(u)\). It follows that \(u^{ ( 0 ) }\)is a positive critical point of I, that is, \(u^{ ( 0 ) }= ( u_{1}^{ ( 0 ) },u_{2}^{ ( 0 ) },\ldots,u_{n}^{ ( 0 ) } ) ^{T}\) is a positive solution of (1).
The proof is completed.â€ƒâ–¡
Theorem 4.2
Let \(B=(b_{ij})_{n\times n}\) be a real symmetric matrix with \(b_{ij}\leq 0\) for \(i,j\in \{ 1,2,\ldots,n \} \) while \(i\neq j\). Assume furthermore that \(f_{k}\in C( (\infty,0 ],R ) \) for each k and
 (\(\mathrm{G}_{3}^{\prime }\)):

There exist constants \(a_{1}<\frac{1}{2} \lambda_{\min }\), \(a_{2}>0\) and \(M>0\) such that
$$ \int_{0}^{z}f_{k}(s)\,ds\leq a_{1}z^{2}+a_{2}\quad \textit{for } z< M \textit{ and } k\in \{ 1,2,\ldots,n \} . $$  (\(\mathrm{G}^{\prime}_{4}\)):

For each \(k\in \{ 1,2,\ldots,n \} \),
$$ \lim_{z\rightarrow 0^{}}\frac{\int_{0}^{z}f_{k}(s)\,ds}{z ^{2}}=+\infty . $$
Then system (1) has one negative real solution.
Let J be defined by (14). Similar to the proof of TheoremÂ 4.1, it can be proved that there exists a critical point of J in the interior of \(\Omega^{+}\), so the conclusion of TheoremÂ 4.2 is true.
According to Theorems 4.1 and 4.2, the following can be directly obtained.
Corollary 4.1
Let \(B=(b_{ij})_{n\times n}\) be a real symmetric matrix with \(b_{ij}\leq 0\) for \(i,j\in \{ 1,2,\ldots,n \} \) while \(i\neq j\). Assume furthermore that \(f_{k}\in C ( R,R ) \) for each k and
 (\(\mathrm{G}^{\prime\prime}_{3}\)):

There exist constants \(a_{1}< \frac{1}{2}\lambda_{\min }\), \(a_{2}>0\) and \(M>0\) such that
$$ \int_{0}^{z}f_{k}(s)\,ds\leq a_{1}z^{2}+a_{2}\quad \textit{for }\vert z\vert >M \textit{ and } k\in \{ 1,2,\ldots,n \}. $$  (\(\mathrm{G}^{\prime\prime}_{4}\)):

For each \(k\in \{ 1,2,\ldots,n \}\),
$$ \lim_{z\rightarrow 0} \frac{\int_{0}^{z}f_{k}(s)\,ds}{z^{2}}=+\infty. $$
Then system (1) has one positive real solution and one negative real solution.
Example 4.1
The system
where
(24) is one of the form (1), where
B is not a positive definite matrix but a positive semidefinite matrix. The eigenvalues of B are \(0, 3\) and 3, therefore \(\lambda_{\min }=0\). Let \(a_{1}=\frac{1}{4}\), \(a_{2}=4\) and \(M=8\), it is easy to verify that (24) satisfies all conditions of CorollaryÂ 4.1. Thus, (24) has one positive solution and one negative solution.
Indeed, let \((x_{1},x_{2},x_{3})^{T}\) be a positive solution, we assert \(\min \{ x_{1},x_{2},x_{3} \} \geq 1\).
Let \(\min \{ x_{1},x_{2},x_{3} \} =x_{1}\). If \(0<\min \{ x_{1},x_{2},x_{3} \} <1\), by (24),
It follows that \(x_{1}>2x_{1}x_{1}^{\frac{1}{3}}=x_{2}+x_{3}\), which is contrary to \(\min \{ x_{1},x_{2},x_{3} \} =x_{1}\). This shows \(\min \{ x_{1},x_{2},x_{3} \} \geq 1\).
The positive solution of (24) is obtained by
It follows that
The solution of (26) can be found and given by \((2,2,2)^{T}\). Thus \((2,2,2)^{T}\) is the unique positive solution of (24). Similarly, \((2,2,2)^{T}\) is the unique negative solution of (24). This example shows that CorollaryÂ 4.1 is sharp.
Example 4.2
The system
where
(27) is one of the form (1), where
B is a negative definite matrix. The eigenvalues of B are \(4,1\) and âˆ’1. Let \(a_{1}=\frac{17}{4}\), \(a_{2}>0\) and \(M=2\), it is easy to verify that system (27) satisfies all the conditions of CorollaryÂ 4.1. Thus, (27) has one positive solution and one negative solution. Indeed, all the solutions of (27) can be found and given by \(( 0,0,0 ) ^{T}\), \((1,1,1)^{T}\) and \((1,1,1)^{T}\). This example shows that Corollary 4.1Â is sharp.
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Acknowledgements
The authors would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by GDSFC project (No. 9151008002000012).
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Zhang, JL., Wang, GQ. Elementary variational approach to positive and negative solutions of a nonlinear algebraic system. Adv Differ Equ 2017, 322 (2017). https://doi.org/10.1186/s136620171359z
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DOI: https://doi.org/10.1186/s136620171359z