AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 246723 10.1155/2013/246723 246723 Research Article On Some Symmetric Systems of Difference Equations Diblík Josef 1,2 Iričanin Bratislav 3 Stević Stevo 4,5 Šmarda Zdeněk 2 Yoshida Norio 1 Department of Mathematics and Descriptive Geometry Faculty of Civil Engineering Brno University of Technology 60200 Brno Czech Republic vutbr.cz 2 Department of Mathematics Faculty of Electrical Engineering and Communication Brno University of Technology 61600 Brno Czech Republic vutbr.cz 3 Faculty of Electrical Engineering University of Belgrade Bulevar Kralja Aleksandra 73 11000 Beograd Serbia bg.ac.rs 4 Mathematical Institute of the Serbian Academy of Sciences Knez Mihailova 36/III, 11000 Beograd Serbia sanu.ac.rs 5 Department of Mathematics King Abdulaziz University, Jeddah 21859 Saudi Arabia kau.edu.sa 2013 30 3 2013 2013 22 12 2012 12 02 2013 2013 Copyright © 2013 Josef Diblík et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Here we show that the main results in the papers by Yalcinkaya (2008), Yalcinkaya and Cinar (2010), and Yalcinkaya, Cinar, and Simsek (2008), as well as a conjecture from the last mentioned paper, follow from a slight modification of a result by G. Papaschinopoulos and C. J. Schinas. We also give some generalizations of these results.

1. Introduction

Studying difference equations and systems which possess some kind of symmetry attracted some attention recently (see, e.g.,  and the related references therein).

Paper  studied the following system of difference equations: (1)xn+1=ynxn-1+ayn+xn-1,yn+1=xnyn-1+axn+yn-1,n0.

In , authors claim that they study the system (2)xn+1=yn+xn-1ynxn-1+a,yn+1=xn+yn-1xnyn-1+a,n0, while in , the authors studied the system (3)xn+1=xnyn-1+axn+yn-1,yn+1=ynxn-1+ayn+xn-1,n0, where a>0.

Since a>0, it is clear that the change of variables (4)(xn,yn)(axnayn), reduces systems (1) and (3) to the case a=1. The authors of  claim that the change of variables (4) reduces (2) to the case a=1 too; however by using the change system (2) becomes (5)xn+1=yn+xn-1a(ynxn-1+1),yn+1=xn+yn-1a(xnyn-1+1),n0. Therefore, in fact,  studied only system (2) for the case a=1.

Based on this observation we may, and will, assume that a=1 in systems of difference equations (1)–(3).

In the main results in  it is proved that when a=1, the positive equilibrium point (x-,y-)=(1,1) of systems (1)–(3) is globally asymptotically stable.

The authors of  finish their paper by the statement that they believe that the results therein can be conveniently extended to the following higher order system of difference equations: (6)xn+1=xnyn-l+axn+yn-l,yn+1=ynxn-l+ayn+xn-l,n0, when l{1}.

Here, among others, we show that all the results and conjectures mentioned above follow from a slight modification of a result in the literature published before papers . For related systems see also [2, 510, 12, 1720].

2. Main Results

Let +=(0,+) and +n be the set of all positive n-dimensional vectors. The following theorem was proved in .

Theorem A.

Let (M,d) be a complete metric space, where d denotes a metric and M is an open subset of n, and let T:MM be a continuous mapping with the unique equilibrium x*M. Suppose that for the discrete dynamic system (7)xn+1=Txn,n0, there is a k such that for the kth iterate of T, the following inequality holds: (8)d(Tkx,x*)<d(x,x*), for all xx*. Then x* is globally asymptotically stable with respect to metric d.

The part-metric (see ) is a metric defined on +n by (9)p  (X,Y)=-logmin1in{xiyi,yixi}, for arbitrary vectors X=(x1,x2,,xn)T+n and Y=(y1,y2,,yn)T+n.

It is known that the part-metric p is a continuous metric on +n, (+n,p) is a complete metric space, and that the distances induced by the part-metric and by the Euclidean norm are equivalent on +n (see, e.g., ).

Based on these properties and Theorem A, the following corollary follows.

Corollary 1.

Let T:+n+n be a continuous mapping with a unique equilibrium x*+n. Suppose that for the discrete dynamic system (7), there is some k such that for the part-metric p inequality (10)p  (Tkx,x*)<p  (x,x*) holds for all xx*. Then x* is globally asymptotically stable.

Some applications of various part-metric-related inequalities and some asymptotic methods in studying difference equations related to symmetric ones can be found, for example, in [1, 35, 10, 11, 1316, 22] (see also the related references therein).

In Lemma  2.3 in , Papaschinopoulos and Schinas formulated a variant of the following result, without giving a proof. However, the part concerning the equality in inequality (12) below, is not mentioned, but it is crucial in applying Corollary 1 (see inequality (10)). For this reason, the completeness and the benefit of the reader we will give a complete proof of it.

Proposition 2.

Let f:+2m+, g:+2m+ be continuous functions. We suppose that the system of two difference equations, (11)un+m=f(un,un+1,,un+m-1,vn,vn+1,,vn+m-1),vn+m=g(un,un+1,,un+m-1,vn,vn+1,,vn+m-1), has a unique positive equilibrium (w,w). Suppose also that there is an r such that for any positive solution (un,vn)n0 of system (11), the following inequalities: (12)(un-un+r)(w2un-un+r)0,(vn-vn+r)(w2vn-vn+r)0,n0, hold, with the equalities if and only if un=w, for every n0, and vn=w, for every n0, respectively. Then the equilibrium (w,w) is globally asymptotically stable.

Proof.

First, we prove that for every n0(13)min{un+rw,vn+rw,wun+r,wvn+r}>min{unw,vnw,wun,wvn}, if and only if (un,vn)(w,w).

To prove (13), it is enough to prove that (14)min{un+rw,wun+r}>min{unw,wun}, if and only if unw, and (15)min{vn+rw,wvn+r}>min{vnw,wvn}, if and only if vnw.

The proofs of inequalities (14) and (15) are the same (up to the interchanging letters u and v) so it is enough to prove (14).

Now note that if the equality holds in the first inequality in (12), then we have that (16)un=un+rorwun=un+rw, from which, in both cases, it easily follows that (17)min{un+rw,wun+r}=min{unw,wun}. On the other hand, if (17) holds, then we easily obtain that one of the equalities in (16) holds, and consequently it follows that the equality holds in the first inequality in (12). Hence, by one of the assumptions, we have that (17) holds if and only if un=w for every n0.

Now suppose that the first inequality in (12), is strict. Then, if un>un+r, directly follows that w/un+r>w/un, while from the first inequality in (12) it follows that un+r/w>w/un. Hence (18)min{un+rw,wun+r}>wun, from which inequality (14) easily follows.

If un<un+r, then un+r/w>un/w, while from the first inequality in (12), it follows that w/un+r>un/w. From these two inequalities, we have that (19)min{un+rw,wun+r}>unw, and consequently (14).

If (14) and (15) hold then if unw and vnw, inequality (13) immediately follows by using the following elementary implication: if a>b and c>d, then min{a,c}>min{b,d}.

If unw and vn=w, then from the second inequality in (12), we have that vn+r=vn=w. Hence (20)min{vn+rw,wvn+r}=min{vnw,wvn}=1>min{unw,wun}, which along with (14) implies (13). The case un=w and vnw directly follows from the case unw and vn=w, by the symmetry.

Finally, note that if un=vn=w, then from (12), we have that un+r=un=w and vn+r=vn=w, so that the first equality in (20) holds and (21)min{un+rw,wun+r}=min{unw,wun}=1, from which it follows that both minima in (13) are equal, finishing the proof of the claim.

Now we define the map T:+2m+2m as follows: (22)T(x1,x2,,xm-1,xm,y1,y2,,ym-1,ym)=(x2,,xm,f(x1,,xm,y1,,ym),y2,,ym,g(x1,,xm,y1,,ym)).

Then we get (23)T(un,un+1,,un+m-2,un+m-1,vn,vn+1,,vn+m-2,vn+m-1)=(un+1,,un+m-1,un+m,vn+1,,vn+m-1,vn+m), and by induction (24)Ts(un,un+1,,un+m-2,un+m-1,vn,vn+1,,vn+m-2,vn+m-1)=(un+s,,un+m-2+s,un+m-1+s,vn+s,,vn+m-2+s,vn+m-1+s), for every s.

By using inequality (13) and the fact that the inequalities 1ai>bi,iI{1,,m}, I, along with equalities ai=bi=1, i{1,,m}I, imply the inequality min1imai>min1imbi, we have that for each vector x+2m such that x(w,w,,w)=:w+2m, (25)p(Tr(x),w)=-logmin{un+rw,wun+r,,un+r+m-1w,wun+r+m-1,vn+rw,wvn+r,,vn+r+m-1w,wvn+r+m-1}<-logmin{unw,wun,,un+m-1w,wun+m-1,vnw,wvn,,vn+m-1w,wvn+m-1}=p(x,w), from which the proof follows by Corollary 1.

It is not difficult to see that the following extension of Proposition 2 can be proved by slight modifications of the proof of Proposition 2.

Proposition 3.

Let fi:+lm+,i=1,,l, be continuous functions. Suppose that the system of difference equations (26)un+m(1)=f1(un(1),un+1(1),,un+m-1(1),,un(l),un+1(l),,un+m-1(l)),un+m(i)=fi(un(1),un+1(1),,un+m-1(1),,un(l),un+1(l),,un+m-1(l)),un+m(l)=fl(un(1),un+1(1),,un+m-1(1),,un(l),un+1(l),,un+m-1(l)) has a unique positive equilibrium (w,,w)+l, and that there is an r such that for any solution (un(1),,un(l))n0+l of system (26), the following inequalities: (27)(un(i)-un+r(i))(w2un(i)-un+r(i))0,n0,i=1,,l, hold, with the equalities if and only if un(i)=w, for every n0, and i=1,,l. Then the equilibrium (w,,w) is globally asymptotically stable.

Now we use Proposition 2 in proving the results in papers .

Corollary 4.

Let k,l0,kl. Consider the system (28)xn+1=xn-kyn-l+1xn-k+yn-l,yn+1=yn-kxn-l+1yn-k+xn-l,n0. Then the positive equilibrium point (x-,y-)=(1,1) of system (28) is globally asymptotically stable with respect to the set +m×+m, where m=max{k,l}.

Proof.

We may assume that m=k. From system (28), we have that (29)xn+1-xn-k=1-xn-k2yn-l+xn-k,xn+1-1xn-k=yn-l(xn-k2-1)xn-k(yn-l+xn-k),(30)yn+1-yn-k=1-yn-k2xn-l+yn-k,yn+1-1yn-k=xn-l(yn-k2-1)yn-k(xn-l+yn-k) from which it follows that (31)(xn+1-xn-k)(xn+1-1xn-k)0,(yn+1-yn-k)(yn+1-1yn-k)0 so that condition (12) in Proposition 2 is fulfilled with r=k+1.

Clearly if (32)xn=1=ynforeveryn-max{l,k}, then in (31) equalities follow. On the other hand, if equality holds in the first inequality in (31), we have that (33)xn+1=xn-korxn+1=1xn-k. If xn+1=xn-k, then from the first equality in (29) we have that xn-k=1, while if xn+1=1/xn-k, then from the second equality in (29), we have that xn-k=1.

By symmetry (see (30)), we have that if equality holds in the second inequality in (31), then yn-k=1. Therefore, equalities in (31) hold if and only if (xn-k,yn-k)=(1,1). Hence all the conditions of Proposition 2 are fulfilled from which it follows that the positive equilibrium (1,1) is globally asymptotically stable with respect to the set +m×+m.

Remark 5.

Corollary 4 extends and gives a very short proof of the main result in , which is obtained for k=1 and l=0. Further, it also extends and gives a very short proof of the main result in , which is obtained for k=0 and l=1. Moreover, it confirms the conjecture in , which is obtained for k=0 and l{1}.

Corollary 6.

Let k,l0,kl. Consider the system (34)xn+1=xn-k+yn-lxn-kyn-l+1,yn+1=yn-k+xn-lyn-kxn-l+1,n0. Then the positive equilibrium point (x-,y-)=(1,1) of system (34) is globally asymptotically stable with respect to the set +m×+m, where m=max{k,l}.

Proof.

We may assume that m=k. From system (34), we have that (35)xn+1-xn-k=yn-l(1-xn-k2)xn-kyn-l+1,xn+1-1xn-k=xn-k2-1xn-k(xn-kyn-l+1),yn+1-yn-k=xn-l(1-yn-k2)xn-lyn-k+1,yn+1-1yn-k=yn-k2-1yn-k(xn-lyn-k+1) from which it follows that (36)(xn+1-xn-k)(xn+1-1xn-k)0,(yn+1-yn-k)(yn+1-1yn-k)0. Hence condition (12) in Proposition 2 is fulfilled with r=k+1. On the other hand, similarly as in the proof of Corollary 4 it is proved that equalities in (36) hold if and only if (xn-k,yn-k)=(1,1). Hence all the conditions of Proposition 2 are fulfilled from which it follows that the positive equilibrium (1,1) is globally asymptotically stable with respect to the set +m×+m.

Remark 7.

Corollary 6 extends and gives a very short proof of the main result in , which is obtained for k=1 and l=0.

Remark 8.

Corollary 6 is also a consequence of Corollary 4. Namely, by using the change of variables (xn,yn)=(1/un,1/vn), system (34) is transformed into the system (37)vn+1=vn-kun-l+1vn-k+un-l,un+1=un-kvn-l+1un-k+vn-l,n0, which is system (28). In particular, this shows that systems (1) and (2), for the case a=1, are equivalent and consequently the results in [23, 24].

Remark 9.

Similar type of issues appear in some literature on scalar difference equations (see, e.g., related results in papers [1, 5, 11, 13]).

It is of some interest to extend results in Corollaries 4 and 6 by using Proposition 2. The next result is of this kind and it extends a result in .

Corollary 10.

Let fC(+k,+) and gC(+l,+) with k,l,0r1<<rk and 0m1<<mlrk and satisfy the following two conditions:

[f(u1,u2,,uk)]*=f(u1*,u2*,,uk*),

f(u1*,u2*,,uk*)uk*,

where a*:=max{a,1/a}.

Then (x-,y-)=(1,1) is the unique positive equilibrium of the system of difference equations (38)xn=f(xn-r1-1,,xn-rk-1)g(yn-m1-1,,yn-ml-1)+1f(xn-r1-1,,xn-rk-1)+g(yn-m1-1,,yn-ml-1),n,yn=f(yn-r1-1,,yn-rk-1)g(xn-m1-1,,xn-ml-1)+1f(yn-r1-1,,yn-rk-1)+g(xn-m1-1,,xn-ml-1),n, and it is globally asymptotically stable.

Proof.

Let (39)fn=f(xn-r1-1,,xn-rk-1),gn=g(yn-m1-1,,yn-ml-1). We should determine the sign of the product of the following expressions: (40)Pnfngn+1fn+gn-xn-rk-1=1fn+gn(fngn(1-xn-rk-1fn)+1-xn-rk-1fn),(41)Qn:=fngn+1fn+gn-1xn-rk-1=1xn-rk-1(fn+gn)(1xn-r1-1(fn+gn)gn(xn-rk-1fn-1)+fn(xn-rk-1fn-1)1xn-r1-1(fn+gn)).

From (40) and (41), we see if we show that xn-rk-1fn-1 and (xn-rk-1/fn)-1 have the same sign for n, then PnQn will be nonpositive.

We consider four cases.

Case  1. x n - r k - 1 1 , fn1. Clearly in this case xn-rk-1fn-10. By (H1) and (H2), we have that (42)1fn=(fn)*=f(xn-r1-1*,,xn-rk-1*)xn-rk-1*=xn-rk-1.

Hence (xn-rk-1/fn)-10 and consequently (43)(xn-rk-1fn-1)(xn-rk-1fn-1)0.

Case  2. xn-rk-11, fn1. Since 1/fn1, we obtain (xn-rk-1/fn)-10. On the other hand, by (H1) and (H2), we have (44)1fn=(fn)*=f(xn-r1-1*,,xn-rk-1*)xn-rk-1*=xn-rk-1, so that xn-rk-1fn-10. Hence (43) follows in this case.

Case  3. Case xn-rk-11, fn1. Then we have that 1/fn1 and consequently (xn-rk-1/fn)-10. On the other hand, we have (45)fn=(fn)*=f(xn-r1-1*,,xn-rk-1*)xn-rk-1*=1xn-rk-1, so that xn-rk-1fn-10. Hence (43) follows in this case too.

Case  4. Case xn-rk-11, fn1. Then xn-rk-1fn-10. On the other hand, we have (46)1fn=(fn)*=f(xn-r1-1*,,xn-rk-1*)xn-rk-1*=1xn-rk-1, so that (xn-rk-1/fn)-10. Hence (43) also holds in this case. Thus PnQn0, for every n.

Assume that PnQn=0, then Pn=0 or Qn=0. Using (40) or (41) along with (43) in any of these two cases, we have that (47)fn=1xn-rk-1=xn-rk-1,n. Hence xn-rk-1=1,n.

Let (48)f^n=f(yn-r1-1,,yn-rk-1),g^n=g(xn-m1-1,,xn-ml-1). Using the following expressions: (49)P^nf^ng^n+1f^n+g^n-yn-rk-1=1f^n+g^n(f^ng^n(1-yn-rk-1f^n)+1-yn-rk-1f^n(1-yn-r1-1f^n)),Q^n:=f^ng^n+1f^n+g^n-1yn-rk-1=1yn-rk-1(f^n+g^n)((yn-r1-1f^n-1)g^n(yn-rk-1f^n-1)+f^n(yn-rk-1f^n-1)), it can be proved similarly that P^nQ^n0, for every n, and that P^nQ^n=0, if and only if yn-rk-1=1,n.

Finally, let (x*,y*) be a solution of the system (50)x*=f(xk*)g(yl*)+1f(xk*)+g(yl*),y*=f(yk*)g(xl*)+1f(yk*)+g(xl*).

Then we have that (51)0=f(xk*)g(yl*)+1f(xk*)+g(yl*)-x*=1f(xk*)+g(yl*)(f(xk*)g(yl*)(1-x*f(xk*))+1-x*f(xk*)(1-x*f(xk*))), where zj*=(z*,,z*) denotes the vector consisting of j copies of z*. Then similar to the considerations in Cases (i)–(iv), it follows that f(xk*)=x*=1/x*, so that x*=1, and similarly it is obtained that y*=1. Hence (x*,y*)=(1,1) is a unique positive equilibrium of system (26).

From all above mentioned and by Proposition 2, we get the result.

Acknowledgments

S. Stević would like to express his sincere thanks to Professors G. Papaschinopoulos and C. J. Schinas for useful conversations and their help during writing this paper. The first author is supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague). The fourth author is supported by the grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is also supported by the Serbian Ministry of Science Projects III 41025, III 44006, and OI 171007.

Berg L. Stević S. On the asymptotics of some difference equations Journal of Difference Equations and Applications 2012 18 5 785 797 10.1080/10236198.2010.512918 MR2929114 ZBL1252.39007 Iričanin B. D. Stević S. Some systems of nonlinear difference equations of higher order with periodic solutions Dynamics of Continuous, Discrete & Impulsive Systems A 2006 13 3-4 499 507 MR2220850 ZBL1098.39003 Iričanin B. Stević S. Eventually constant solutions of a rational difference equation Applied Mathematics and Computation 2009 215 2 854 856 10.1016/j.amc.2009.05.044 MR2561544 ZBL1178.39012 Kruse N. Nesemann T. Global asymptotic stability in some discrete dynamical systems Journal of Mathematical Analysis and Applications 1999 235 1 151 158 10.1006/jmaa.1999.6384 MR1758674 ZBL0933.37016 Liu W. Yang X. Stević S. Iričanin B. Part metric and its applications to cyclic discrete dynamical systems Abstract and Applied Analysis 2011 2011 16 534974 10.1155/2011/534974 Papaschinopoulos G. Schinas C. J. On a system of two nonlinear difference equations Journal of Mathematical Analysis and Applications 1998 219 2 415 426 10.1006/jmaa.1997.5829 MR1606350 ZBL1110.39301 Papaschinopoulos G. Schinas C. J. On the behavior of the solutions of a system of two nonlinear difference equations Communications on Applied Nonlinear Analysis 1998 5 2 47 59 MR1621223 ZBL1110.39301 Papaschinopoulos G. Schinas C. J. Invariants for systems of two nonlinear difference equations Differential Equations and Dynamical Systems 1999 7 2 181 196 MR1860787 ZBL0978.39014 Papaschinopoulos G. Schinas C. J. Invariants and oscillation for systems of two nonlinear difference equations Nonlinear Analysis: Theory, Methods & Applications A 2001 46 7 967 978 10.1016/S0362-546X(00)00146-2 MR1866733 ZBL1003.39007 Papaschinopoulos G. Schinas C. J. Oscillation and asymptotic stability of two systems of difference equations of rational form Journal of Difference Equations and Applications 2001 7 4 601 617 10.1080/10236190108808290 MR1922592 ZBL1009.39006 Papaschinopoulos G. Schinas C. J. Global asymptotic stability and oscillation of a family of difference equations Journal of Mathematical Analysis and Applications 2004 294 2 614 620 10.1016/j.jmaa.2004.02.039 MR2061346 ZBL1055.39017 Stefanidou G. Papaschinopoulos G. Schinas C. J. On a system of two exponential type difference equations Communications on Applied Nonlinear Analysis 2010 17 2 1 13 MR2669014 ZBL1198.39028 Stević S. Global stability and asymptotics of some classes of rational difference equations Journal of Mathematical Analysis and Applications 2006 316 1 60 68 10.1016/j.jmaa.2005.04.077 MR2201749 ZBL1090.39009 Stević S. On positive solutions of a (k+1)-th order difference equation Applied Mathematics Letters 2006 19 5 427 431 10.1016/j.aml.2005.05.014 MR2213143 ZBL1095.39010 Stević S. Existence of nontrivial solutions of a rational difference equation Applied Mathematics Letters 2007 20 1 28 31 10.1016/j.aml.2006.03.002 MR2273123 ZBL1131.39009 Stević S. Nontrivial solutions of a higher-order rational difference equation Mathematical Notes 2008 84 5-6 718 724 10.1134/S0001434608110138 Stević S. On a system of difference equations Applied Mathematics and Computation 2011 218 7 3372 3378 10.1016/j.amc.2011.08.079 MR2851439 ZBL1242.39017 Stević S. On a solvable rational system of difference equations Applied Mathematics and Computation 2012 219 6 2896 2908 10.1016/j.amc.2012.09.012 MR2991990 Stević S. On a third-order system of difference equations Applied Mathematics and Computation 2012 218 14 7649 7654 10.1016/j.amc.2012.01.034 MR2892731 ZBL1243.39011 Stević S. On some solvable systems of difference equations Applied Mathematics and Computation 2012 218 9 5010 5018 10.1016/j.amc.2011.10.068 MR2870025 ZBL1253.39001 Thompson A. C. On certain contraction mappings in a partially ordered vector space Proceedings of the American Mathematical Society 1963 14 438 443 MR0149237 ZBL0147.34903 Yang X. Yang M. Liu H. A part-metric-related inequality chain and application to the stability analysis of difference equation Journal of Inequalities and Applications 2007 2007 9 19618 10.1155/2007/19618 MR2291654 ZBL1133.26302 Yalcinkaya I. On the global asymptotic stability of a second-order system of difference equations Discrete Dynamics in Nature and Society 2008 2008 12 860152 10.1155/2008/860152 MR2457146 ZBL1161.39013 Yalcinkaya N. Çinar C. Global asymptotic stability of a system of two nonlinear difference equations Fasciculi Mathematici 2010 43 171 180 MR2666115 ZBL1207.39024 Yalcinkaya I. Cinar C. Simsek D. Global asymptotic stability of a system of difference equations Applicable Analysis 2008 87 6 677 687 10.1080/00036810802140657 MR2440851 ZBL1161.39013