Selçuk J. Appl. Math. Selçuk Journal of Vol. 14. No. 1. pp. 3-10, 2013 Applied Mathematics
Global Behavior of a Recursive Sequence ·
Ibrahim Yalcinkaya1, Alaa E. Hamza2, Cengiz Cinar3
1Department of Mathematics, Faculty of Education, Necmettin Erbakan University,
42090, Meram Yeni Yol, Konya, Turkiye.
e-mail:iyalcinkaya42@ gm ail.com
2Department of Mathematics, Faculty of Science, Cairo University, Giza, 12211, Egypt.
e-mail:ham zaaeg2003@ yaho o.com
3Department of Mathematics, Faculty of Education, Gazi University, 06500, Be¸sevler,
Ankara, Turkiye.
e-mail:ccinar2525@ gm ail.com
Received Date: March 05, 2012 Accepted Date: February 28, 2013
Abstract. In this paper, we investigate the global behavior of the di¤erence equation xn+1= xn m + k Q j=0 xn ij ; n = 0; 1; :::
where the parameters ; ; and initial conditions are non-negative real num-bers, fi0< i1< ::: < ikg is a set of non-negative even integers and m is an odd positive integer.
Key words: Di¤erence equation; Global asymptotic stability; Oscillation; Pe-riod (m+1) solutions; Semicycles.
AMS Classi…cation: 39A10. 1. Introduction
Di¤erence equations appear naturally as discrete analogues and as numerical solutions of di¤erential and delay di¤erential equations having applications in biology, ecology, physics, etc. [21] Rational di¤erence equations is an important class of di¤erence equations where they have many applications in real life for example the di¤erence equation xn+1 = (a + bxn)=(c + dxn) which is known Riccati Di¤erence Equation has an applications in Optics and Mathematical Biology (see [19]). Recenlty there has been an increasing interest in the study of global behavior of rational di¤erence equations. Although di¤erence equations’
forms very simple, it is extremely di¢ cult to understand thoroughly the global behaviors of their solutions. For example see Refs. [1-32].
In [13], Hamza and Khalaf-Allah studied the di¤erence equaion xn+1= Axn 1 B + C k Q i=l xn 2i ; n = 0; 1; :::
where non-negative parameters.
In [14], Hamza and Khalaf-Allah studied the global asymptotic stability of the di¤erence equaion xn+1= A k Q i=l xn 2i 1 B + C kQ1 i=l xn 2i ; n = 0; 1; :::
where non-negative parameters and l; k are nonnegative integers for l < k: They discussed the existence of unbounded solutions under certain conditions for l = 0:
In [22], Zayed and El-Moneam studied the global asymptotic stability, the pe-riodicity nature and the boundedness character of the positive solutions of the di¤erence equation
xn+1=
+ xn k xn
; n = 0; 1; :::
where the parameters > 0; ; > 0 and the positive initial conditions. Also, Zayed and El-Moneam have many papers about the rational recursive sequence. See, Ref.[22-32]
In this paper, we investigate the global behavior of the di¤erence equation
(1) xn+1= xn m + k Q j=0 xn ij ; n = 0; 1; :::
where the parameters ; ; and initial conditions are non-negative real num-bers, fi0< i1< ::: < ikg is a set of non-negative even integers and m is an odd positive integer.
We need the following de…nitions and theorem [17]:
De…nition 1. Let I be an interval of real numbers and let f : Ik+1! I be a continuously di¤erentiable function. Consider the di¤erence equation
with x k; :::; x02 I . Let x be the equilibrium point of Eq.(2). The linearized equation of Eq.(2) about the equilibrium point x is (3) yn+1= c1yn+ c2yn 1+::: + c(k+1)yn k; n = 0; 1; ::: where c1= @f @xn (x; :::; x); c2= @f @xn 1 (x; :::; x); :::; c(k+1)= @f @xn k (x; :::; x) The characteristic equation of Eq.(3) is
(4) (k+1) c1 k c2 (k 1) ::: ck c(k+1)= 0:
De…nition 2. Let x be an equilibrium point of Eq.(2).
(a)The equilibrium x is called locally stable if for every " > 0, there exists > 0 such that if x0; : : : ; x k 2 I and jx0 xj + + jx k xj < , then jxn xj < " for all n k:
(b)The equilibrium x is called locally asymptotically stable if it is locally stable and if there exists > 0 such that if x0; : : : ; x k 2 I and jx0 xj + + jx k xj < , then lim
n!1xn = x:
(c)The equilibrium x is called global attractor if for every x0; : : : ; x k2 I we have lim
n!1xn= x:
(d)The equilibrium x is called globally asymptotically stable if it is locally stable and is a global attractor.
De…nition 3. A positive semicycle of a solution fxng1n= kof Eq.(2) consists of a "string" of terms fxl; xl+1; :::; xmg ; all greater than or equal to equilibrium x with l k and m 1 such that either l = k or l > k and xl 1< x and either m = 1 or m 1 and xm+1< x.
A negative semicycle of a solution fxng1n= kof Eq.(2) consists of a "string" of terms fxl; xl+1; :::; xmg all less than x with l k and m 1 such that either l = k or l > k and xl 1 x and either m = 1 or m 1 and xm+1 x. De…nition 4. A solution fxng1n= k of Eq.(2) is called nonoscillatory if there exists N k such that either xn > x f or 8n N or xn < x f or 8n N; and it is called oscillatory if it is not nonoscillatory.
Theorem 1. (i)If all roots of Eq.(4) have absolute values less than one, then the equilibrium point xof Eq.(2) is locally asymptotically stable.
(ii)If at least one of the roots of Eq.(4) has absolute value greater than one, then the equilibrium point xof Eq.(2) is unstable.
The following cases can be obtained for Eq.(1) when one or more of the para-meters are zero. If = 0; then Eq.(1) is trivial. If = 0; then Eq.(1) can be
reduced to a linear di¤erence equation by the change of variables xn = eyn: If = 0; then Eq.(1) is linear.
2. Main Results
In this section, we investigate the dynamics of Eq.(1) under the assumptions that all parameters in the equation are positive and the initial conditions are non-negative.
The change of variables xn = ( ) 1
(k + 1) yn reduces Eq.(1) to the di¤erence equation (5) yn+1= ryn m 1 + k Q j=0 yn ij , n = 0; 1; :::;
where r = > 0: Note that y1 = 0 is always an equilibrium point of Eq.(5). When r > 1, Eq.(5) also possesses the unique positive equilibrium y2= (r 1)
1 (k+1):
Theorem 2. The following statements are true:
(i)If r < 1, then the equilibrium point y1= 0 of Eq.(5) is locally asymp-totically stable,
(ii)If r > 1, then the equilibrium point y1= 0 of Eq.(5) is unstable, (iii)If r > 1, then the positive equilibrium point y2= (r 1)
1
(k+1) of Eq.(5)
is unstable.
Proof. The linearized equation of Eq.(5) about the equilibrium point y1= 0 is zn+1= rzn m for n = 0; 1; :::;
so the proof of (i) and (ii) follows from Theorem 1.
For (iii), we assume that r > 1, then the linearized equation of Eq.(5) about the equilibrium point y2= (r 1)
1
(k+1) has the form
zn+1= zn m (r 1) r k P j=0 zn ij ! for n = 0; 1; :::; so it is clear that y2= (r 1) 1
(k+1) is an unstable equilibrium point from
Theo-rem 1. This completes the proof.
Theorem 3. Assume that r > 1 and let fyng1n= maxfm;ikg be a solution of
Eq.(5) such that
or
(7) y0; y 2; :::; y maxfm 1;ikg< y2 and y 1; y 3; :::; y maxfm;ik 1g y2:
Then, fyng1n= maxfm;ikg oscillates about y2 with semicycles of length one.
Proof. Assume that (6) holds. Then, y1 = ry m 1+ k Q j=0 y ij < y2 and y2 = ry (m 1) 1+ k Q j=0 y (ij 1)
> y2, the proof follows by induction. The case where (7) holds is similar and will be omitted.
Theorem 4. Assume that r < 1, then the equilibrium point y1= 0 of Eq.(5) is globally asymptotically stable.
Proof. We know by Theorem 2 that the equilibrium point y1= 0 of Eq.(5) is locally asymptotically stable. So, let fyng1n= maxfm;ikg be a solution of Eq.(5).
It su¢ ces to show that lim
n!1yn= 0: Since 0 yn+1= ryn m 1+ k Q j=0 yn ij ryn m; we
obtain yn+1 ryn m: Then, we can write,
yt(m+1)+1 r(t+1)y m; yt(m+1)+2 r(t+1)y (m 1) ::: yt(m+1)+(m+1) r(t+1)y0 for t = 0; 1; ::: If r < 1, then lim n!1r (t+1)= 0:So lim
n!1yn = 0: This completes the proof.
Theorem 5. Assume that m > ik: A necessary and su¢ cent condition for Eq.(5) to have a prime period two solution is that r = 1: In this case the prime period two solution is of the form :::; 0; ; 0; ; ::: where > 0: Furthermore every solution converges to a period (m + 1) solution.
Proof. Su¢ ciency: Let r = 1; then for every > 0; we have :::; 0; ; 0; ; ::: is a prime period two solution.
Necessity: Assume that Eq.(5) has a prime period two solution :::; ; ; ; ; :::; then
= r
1 + k+1 and = r 1 + k+1:
Then, + k+1= r and + k+1= r : This implies that 1 r = k k : Hence r 1: Also, r 1 = k+ k + :
Therefore r = 1. Also from last equation, = 0: Now let fyng1n= maxfm;ikg be a solution of Eq.(5) (with r = 1). Then
< yn+m+3< yn+m+2 < yn+1< yn m;
and in this way we obtain m+1 decreasing sequences yn+(m+i) ; i = 1; :::; m+ 1; so it decreases to i: We can check that
1; 2; :::; m+1; 1; 2; :::; m+1; ::: is a period m + 1 solution.
Theorem 6. Assume that r > 1; then Eq.(5) possesses an unbounded solution. Proof. From Theorem 3, we can assume that (5) without loss of generality that the solution fyng1n= maxfm;ikg of Eq.(5) is such that
y2n+1< y2and y2n+2> y2 for n> 0: Then, y2n+1= ry2n m 1 + k Q j=0 y2n ij < ry2n m 1 + (r 1) = y2n m and y2n+2= ry2n (m 1) 1 + k Q j=0 y2n (ij 1) > ry2n (m 1) 1 + (r 1) = y2n (m 1)
which it follows that lim
n!1y2n= 1 and limn!1y2n+1= 0: Then, the proof is complete.
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