Abstract
In this paper, we study the persistence, boundedness, convergence, invariance and global asymptotic behavior of the positive solutions of the second-order difference system \(\begin{aligned} x_{n+1}&= \alpha _1 + a e ^{-x_{n-1}} + b y_{n} e ^{-y_{n-1}},\\ y_{n+1}&= \alpha _2 +c e ^{-y_{n-1}}+ d x_{n} e ^{-x_{n-1}} \quad n=0,1,2,\ldots \end{aligned}\) where \(\alpha _1, \alpha _2, a, b , c,d\) are positive real numbers and the initial conditions \(x_{-1},x_0, y_{-1}, y_0\) are arbitrary nonnegative numbers.
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Introduction
The theory of discrete dynamical system has many applications in applied sciences. Mathematical modeling of a physical, biological or ecological problem mostly leads to a nonlinear difference system. (See [1,2,3,4,5,6,7,8,9,10].)
In [4], Papachinopoulos et al. proposed a system of equation with exponents as
where a, b, c, d and the initial conditions \(f_{-1},f_0, g_{-1}, g_0\) are positive real values. They studied the existence, boundedness and asymptotic behavior of the positive solutions of (1).
In [5], G.Papaschinopoulos and C.J.Schinas together modified the system as
and put forward conditions for the positive solutions to be asymptotic.
In [11], authors multiplied \(f_n\) and \(g_n\) with a and c, respectively, in (2) and formed a new system of difference equations
and described the existence of a unique positive equilibrium, the boundedness, persistence and global attractivity of the positive solutions.
Parallelly in [12], the authors worked on the asymptotic behavior of the positive solutions of a similar difference system
N.Psarros and G.Papaschinopoulos in [13] proposed a new first-order model
and studied the asymptotic behavior of the positive solutions of the system.
Motivated by the above research articles, we propose a new second order difference system
where \(\alpha _1, \alpha _2, a, b , c,d\) are positive real numbers and the initial conditions \(x_{-1},x_0, y_{-1}, y_0\) are arbitrary nonnegative numbers, and investigate the persistence, boundedness, convergence, invariance, and global asymptotic behavior of the positive solutions of the system.
Methods
We use Theorem 1.16 of [14] to prove the lemma which we use to derive a condition for the existence, uniqueness of equilibrium solutions and the convergence of positive solutions to the equilibrium solution. We also use Remark 1.3.1 of [15] to obtain conditions for global asymptotic stability of the unique equilibrium point.
Results and discussion
The following theorem proposes conditions for persistence and boundedness for the positive solution \((x_n,y_n)\) of (3).
Theorem 1
Every positive solution \((x_n,y_n)\) of (3) is bounded and persists whenever \(bde^{-\alpha _1-\alpha _2}<1\).
Proof
\(x_n \ge \alpha _1, y_n \ge \alpha _2\), \(n=3,4,\ldots .\)
Hence, \((x_n,y_n)\) of system (3) persists.
Also, (3) becomes
where \(A= \alpha _1 + a e ^{-\alpha _1} + b \alpha _2e^{-\alpha _2}+ bce^{-\alpha _2-\alpha _2}\).
Similarly,
where \(C= \alpha _2 + c e ^{-\alpha _2} + d \alpha _1e^{-\alpha _1}+ ade^{-\alpha _1-\alpha _1}\).
Now, consider the difference equations
where \(B=D=bde^{-\alpha _1-\alpha _2}<1\). Therefore, an arbitrary solution \((z_n, v_n)\) of (6) can be written as
where \(r_1\), \(r_2\) rely on the initial conditions \(z_{-1}\), \(z_0\) and \(s_1\), \(s_2\) rely on the initial conditions \(v_{-1}\), \(v_0\). Hence, \((z_n, v_n)\) is bounded.
Let us examine the solution \((z_n, v_n)\) such that \(z_{-1}=x_{-1}, z_0=x_0,v_{-1}=y_{-1}, v_0=y_0.\)
Hence by induction, \(x_n \le z_n\) and \(y_n \le v_n, n=0,1,2,\ldots\).
Therefore, we get \((x_n, y_n)\) is bounded. \(\square\)
The following two theorems confirm the existence of invariant boxes of (3).
Theorem 2
Let \(bde^{-\alpha _1-\alpha _2}<1\). Let \((x_n, y_n)\) denote a positive solution of (3). Then \(\displaystyle [\alpha _1,\frac{\alpha _1 + a e ^{-\alpha _1} + b \alpha _2e^{-\alpha _2}+ bce^{-\alpha _2-\alpha _2}}{(1-bde^{-\alpha _1-\alpha _2})}]\) \(\displaystyle \times [\alpha _2,\frac{\alpha _2 + c e ^{-\alpha _2} + d \alpha _1e^{-\alpha _1}+ ade^{-\alpha _1-\alpha _1}}{(1-bde^{-\alpha _1-\alpha _2})}]\) is an invariant set for (3).
Proof
Let \(I_1=\displaystyle [\alpha _1,\frac{\alpha _1 + a e ^{-\alpha _1} + b \alpha _2e^{-\alpha _2}+ bce^{-\alpha _2-\alpha _2}}{(1-bde^{-\alpha _1-\alpha _2})}]\) and \(\displaystyle I_2=[\alpha _2,\frac{\alpha _2 + c e ^{-\alpha _2} + d \alpha _1e^{-\alpha _1}+ ade^{-\alpha _1-\alpha _1}}{(1-bde^{-\alpha _1-\alpha _2})}]\).
Let \(x_{-1}, x_{0} \in I_1\) and \(y_{-1}, y_{0} \in I_2.\)
Then
Hence, we get \(x_1 \le \displaystyle \frac{\alpha _1 + a e ^{-\alpha _1} + b \alpha _2e^{-\alpha _2}+ bce^{-\alpha _2-\alpha _2}}{1-bde^{-\alpha _1-\alpha _2}}.\), i.e., \(x_{1} \in I_1\). Similarly, we get \(y_1 \in I_2.\)
Hence, the proof follows by applying the method of induction. \(\square\)
Theorem 3
Let \(bde^{-\alpha _1-\alpha _2}<1\). Consider the intervals
and
where \(\epsilon\) is an arbitrary positive number. If \((x_n,y_n)\) is any arbitrary solution of (3), then there exists an \(N \in {\mathbb {N}}\) such that \(x_n \in I_3\) and \(y_n \in I_4, n \ge N\).
Proof
Let \((x_n,y_n)\) denote an arbitrary solution of (3).
Then by Theorem 1, \(\limsup _{n \rightarrow \infty }{x_n}=M< \infty\) and \(\limsup _{n \rightarrow \infty }{y_n}=L< \infty\).
Hence from Theorem 1, \(x_{n+1} \le A + bdx_{n-1}e^{-\alpha _1-\alpha _2}\) and \(y_{n+1}\le C + bdy_{n-1}e^{-\alpha _1-\alpha _2}\)
Hence \(\displaystyle M \le \frac{A}{1-bde^{-\alpha _1-\alpha _2}}\), and \(\displaystyle L \le \frac{C}{1-bde^{-\alpha _1-\alpha _2}}\).
Hence, there exists an \(N \in {\mathbb {N}}\) such that the theorem holds. \(\square\)
Now we prove a lemma which is an alteration of Theorem 1.16 of [14].
Lemma 4
Let [a, b] and [c, d] denote intervals of real numbers. Let \(f:[a,b]\times [c,d]\times [c,d] \rightarrow [a,b]\) and \(g:[a,b]\times [a,b]\times [c,d] \rightarrow [c,d]\) be continuous functions. Consider the difference system
such that the initial values \(x_{-1},x_0 \in [a,b]\) and \(y_{-1}, y_0 \in [c,d]\). (or \(x_{n_0},x_{n_0+1} \in [a,b],\) \(y_{n_0},y_{n_0+1} \in [c,d], n_0 \in {\mathbb {N}}\)). Suppose the following are true.
-
1.
If f(x, y, z) is nonincreasing in x, f(x, y, z) is nondecreasing in y and f(x, y, z) is nonincreasing in z.
-
2.
If g(x, y, z) is nondecreasing in x, g(x, y, z) is nonincreasing in y and g(x, y, z) is nonincreasing in z.
-
3.
If \((m_1,M_1,m_2,M_2) \in [a,b]^2\times [c,d]^2\) satisfies the systems \(m_1=f(M_1,m_2,M_2),\) \(M_1=f(m_1,M_2,m_2)\) and \(m_2=g(m_1,M_1,M_2), M_2=g(M_1,m_1,m_2)\) then \(M_1=m_1\) and \(M_2=m_2\),
then there exists a unique equilibrium solution \(({\bar{x}},{\bar{y}})\) of (9) with \({\bar{x}} \in [a,b]\), \({\bar{y}} \in [c,d]\). Also every solution of (9) converges to \(({\bar{x}},{\bar{y}})\).
Proof
Set \(m_1^{-1}=a, m_1^{0}=a,m_2^{-1}=c, m_2^{0}=c.\)
For each \(i \ge 0\), let \(m_1^{i+1}=f(M_1^{i-1},m_2^i,M_2^{i-1}), M_1^{i+1}=f(m_1^{i-1},M_2^i,m_2^{i-1})\) and
Hence \(m_1^1= f(M_1^{-1},m_2^0,M_2^{-1}) \le f(m_1^{-1},M_2^0,m_2^{-1})= M_1^{1} ,\) and
Therefore,
Also \(m_1^{0}=a \le x_n \le b =M_1^0, n\ge 0\) and \(m_2^{0}=c \le y_n \le d =M_2^0, n\ge 0\).
For all \(n\ge 0\), we have
Hence \(m_1^{1} \le x_n \le M_1^1, n\ge 1\) and \(m_2^{1} \le y_n \le M_2^1, n\ge 1\).
We then obtain by induction that for \(i \ge 0\), the following are true.
-
1.
\(a=m_1^{-1}\le m_1^0 \le m_1^1 \ldots \le m_1^{i-1}\le m_1^{i}\le M_1^{i}\ldots \le M_1^1 \le M_1^0 \le M_1^{-1}=b\).
-
2.
\(c=m_2^{-1}\le m_2^0 \le m_2^1 \ldots \le m_2^{i-1}\le m_2^{i}\le M_2^{i}\ldots \le M_2^1 \le M_2^0 \le M_2^{-1}=d\).
-
3.
\(m_1^{i} \le x_n \le M_1^i, n\ge 1\) and \(m_2^{i} \le y_n \le M_2^i, n\ge 1\).
Set \(m_1= \lim _{i \rightarrow \infty } m_1^{i} , m_2= \lim _{i \rightarrow \infty } m_2^{i}\) and \(M_1= \lim _{i \rightarrow \infty } M_1^{i} , M_2= \lim _{i \rightarrow \infty } M_2^{i}\).
Since f and g are continuous, we get \(m_1=f(M_1,m_2,M_2), M_1=f(m_1,M_2,m_2)\) and \(m_2=g(m_1,M_1,M_2), M_2=g(M_1,m_1,m_2).\)
Hence \(M_1=m_1={\bar{x}}\) and \(M_2=m_2={\bar{y}}\), from which we get the proof. \(\square\)
The following theorem proposes conditions for the convergence of the equilibrium solution of (3).
Theorem 5
Suppose
and
Then (3) has a unique positive equilibrium \(E({\bar{x}},{\bar{y}})\). Also, every solution of (3) converges to \(E({\bar{x}},{\bar{y}})\).
Proof
Let \(f: {\mathbb {R}}^+ \times {\mathbb {R}}^+ \times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}^+,g: {\mathbb {R}}^+ \times {\mathbb {R}}^+ \times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}^+\) be continuous functions such that \(f(x,y,z)= \alpha _1 + ae^{-x} +bye^{-z}\), \(g(x,y,z)=\alpha _2 + ce^{-z} +dxe^{-y}\).
Let \(M_1,m_1,M_2,m_2\) be positive real numbers satisfying
and
Therefore, \(M_1-m_1=a[e^{-m_1}-e^{-M_1}] + b[M_2e^{-m_2}-m_2e^{-M_2}].\)
Also, there exists a \(\zeta\) , \(m_2 \le \zeta \le M_2\) satisfying
Now, \(a[e^{-m_1}-e^{-M_1}] = ae^{-m_1-M_1}[e^{M_1}-e^{m_1}].\)
Also there exists a \(\lambda\), \(m_1 \le \lambda \le M_1\) satisfying
Since \(M_1,m_1 \ge \alpha _1\) and \(\lambda \le M_1,\)
Thus, from (15) and (17) we get,
Since \(M_2,m_2 \ge \alpha _2\) and \(\zeta \le M_2\), (18) becomes
, i.e.,
Also, (12) can be written as
Since \(\zeta \le M_2\) we get,
Therefore, (20) becomes
Similarly, we get
Therefore from (11) and (26), we get \(M_1=m_1\) and \(M_2=m_2\).
Therefore by applying Lemma 4, the result is obtained. \(\square\)
In the next theorem, we derive conditions for the global asymptotic stability of the equilibrium solution of (3).
Theorem 6
-
1.
Let \((a+ac+c)<1\). If \((1+{\bar{x}})(1+{\bar{y}})< \displaystyle \frac{1-(a+ac+c)}{bd}\), then the unique equilibrium \(E({\bar{x}},{\bar{y}})\) is globally asymptotically stable.
-
2.
If \((a+c+ac+bd)+ bd[\frac{A}{1-B}+\frac{C}{1-B}+\frac{AC}{(1-B)^2}]<1\), where A, B and C are defined as in (4) and (5), then the unique equilibrium \(E({\bar{x}},{\bar{y}})\) is globally asymptotically stable.
Proof
First we show that \(E({\bar{x}},{\bar{y}})\) is locally asymptotically stable in both the cases. The Jacobian \(JF({\bar{x}},{\bar{y}})\) about the equilibrium point \(E({\bar{x}},{\bar{y}})\) is given by
Hence the characteristic equation of the Jacobian \(JF({\bar{x}},{\bar{y}})\) about the equilibrium point \(E({\bar{x}},{\bar{y}})\) is given by
Then
is satisfied whenever
-
1.
From (27), we get
$$\begin{aligned} (1+{\bar{x}})(1+{\bar{y}})< \displaystyle \frac{1-(a+ac +c)}{bd}. \end{aligned}$$(28) -
2.
Since \(E({\bar{x}},{\bar{y}})\) is the equilibrium point of (3), we get
$$\begin{aligned} {\bar{x}} \le \alpha _1 + a e ^{-\alpha _1} + b e^{-\alpha _2} [\alpha _2 +d{\bar{x}}e^{-\alpha _1}+ce^{-\alpha _2}]. \end{aligned}$$, i.e.,
$$\begin{aligned} {\bar{x}} \le \frac{A}{(1-bde^{-\alpha _1-\alpha _2})}. \end{aligned}$$(29)Similarly
$$\begin{aligned} {\bar{y}} \le \frac{C}{(1-bde^{-\alpha _1-\alpha _2})}. \end{aligned}$$(30)Substituting (29), (30) in (27), we get
$$\begin{aligned} (a+c+ac+bd)+ bd\left[ \frac{A}{1-B}+\frac{C}{1-B}+\frac{AC}{(1-B)^2}\right] <1. \end{aligned}$$Hence by Remark 1.3.1 of [15], we get the result.
Therefore by using Theorem 5, we obtain the conditions for global asymptotic stability. \(\square\)
Conclusions
In this paper, we analyzed the persistence, boundedness, convergence, invariance and global asymptotic behavior of the positive solutions of a second-order difference system. Here we expressed all the conditions in terms of the parameters occurring in the system. We also obtained two conditions for the occurrence of global stability where in the first one the condition was given in terms of the equilibrium point and in the second one the condition was given in terms of parameters of the system.
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The authors would like to thank the referees for their valuable suggestions.
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DSD wrote the title, abstract, introduction and references. SMM wrote the main results. Both authors read and approved the final manuscript.
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Dilip, D.S., Mathew, S.M. Dynamics of a second-order nonlinear difference system with exponents. J Egypt Math Soc 29, 10 (2021). https://doi.org/10.1186/s42787-021-00119-6
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DOI: https://doi.org/10.1186/s42787-021-00119-6