On the Dynamics of the Recursive Sequence x n +1 = [alpha] + x n -k p / xnq

Abstract

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation [subscript] x n +1[/subscript] = α +[superscript] x n -k p[/superscript] /[superscript] x n q[/superscript] , n =0,1 , ... , where k ∈ {2,3 ... } , α , p , q ∈ (0 , ∞ ) and the initial conditions [subscript] x -k[/subscript] , ... ,[subscript] x 0[/subscript] are arbitrary positive numbers. We investigate the boundedness character for p ∈ (0 , ∞ ) . Also, we investigate the existence of a prime two periodic solution for k is odd. Moreover, when k is even, we prove that there are no prime two periodic solutions of the equation above.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Mehmet Gümüs 1 and Özkan Öcalan 2 and Nilüfer B. Felah 2

Recommended by M. De la Sen

1, Department of Mathematics, Faculty of Science and Arts, Bülent Ecevit University, 67100 Zonguldak, Turkey
2, Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey

Received 5 July 2012; Accepted 13 September 2012

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.

1. Introduction

Our aim in this paper is to study the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation [figure omitted; refer to PDF] where k ∈ {2,3 , ... } , α is a positive, p ,q ∈ (0 , ∞ ) and the initial conditions _{x -k} , ... , _{x 0} are arbitrary positive numbers. Equation ( 1.1) was studied by many authors for different cases of k ,p ,q .

In [ 1] the authors studied the global stability, the boundedness character, and the periodic nature of the positive solutions of the difference equation [figure omitted; refer to PDF] where α is positive, and the initial values _{x -1} , _{x 0} are positive numbers (see also [ 2- 4] for more results on this equation).

In [ 5] the authors studied the boundedness, the global attractivity, the oscillatory behaviour, and the periodicity of the positive solutions of the difference equation [figure omitted; refer to PDF] where α , p are positive, and the initial values _{x -1} , _{x 0} are positive numbers (see also [ 6- 8] for more results on this equation).

In [ 9] the authors studied general properties, the boundedness, the global stability, and the periodic character of the solutions of the difference equation [figure omitted; refer to PDF] where α ,p are positive, k ∈ {2,3 , ... } , and the initial values _{x -k} , _{x -k +1} , ... , _{x -1} , _{x 0} are positive numbers.

In [ 10, 11] the authors studied the boundedness, the persistence, the attractivity, the stability, and the periodic character of the positive solutions of the difference equation [figure omitted; refer to PDF] where α , p , q are positive, and the initial values _{x -1} , _{x 0} are positive numbers.

Finally in [ 12, 13] the authors studied the oscillatory, the behaviour of semicycle, and the periodic character of the positive solution of the difference equation [figure omitted; refer to PDF] where k ∈ {2,3 , ... } and α , p >0 under the initial conditions _{x -k} , _{x -k +1} , ... , _{x -1} , _{x 0} are positive numbers.

There exist many other papers related with ( 1.1) and on its extensions (see [ 14- 16]).

Motivated by the above papers, we study of the boundedness character, the oscillatory, and the periodic character of positive solutions of ( 1.1).

In this paper, also we investigate the case p =1 , k =1 and q ∈ (0 , ∞ ) of ( 1.1) and we give a correction for [ 2, Theorem 2.5].

We say that the equilibrium point x ¯ of the equation [figure omitted; refer to PDF] is the point that satisfies the condition [figure omitted; refer to PDF] A solution { _{x n} ^{} n = -k ∞} of ( 1.1) is called nonoscillatory if there exists N ...5; -k such that either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] A solution { _{x n} ^{} n = -k ∞} of ( 1.1) is called oscillatory if it is not nonoscillatory. We say that a solution { _{x n} ^{} n = -k ∞} of ( 1.1) is bounded and persists if there exist positive constant P and Q such that P ...4; _{x n} ...4;Q for n = -k , -k +1 , ... .

The linearized equation for ( 1.1) about the positive equilibrium x ¯ is [figure omitted; refer to PDF]

2. Semicycle Analysis

A positive semicycle of a solution { _{x n} ^{} n = -k ∞} of ( 1.1) consists of a "string" of terms { _{x l} , _{x l +1} , ... , _{x m} } all greater than or equal to x ¯ , with l ...5; -k and m ...4; ∞ , such that [figure omitted; refer to PDF] A negative semicycle of a solution { _{x n} ^{} n = -k ∞} of ( 1.1) consists of a "string" of terms { _{x l} , _{x l +1} , ... , _{x m} } all less than x ¯ , with l ...5; -k and m ...4; ∞ , such that [figure omitted; refer to PDF]

Lemma 2.1.

Let { _{x n} ^{} n = -k ∞} be a solution of ( 1.1). Then either { _{x n} ^{} n = -k ∞} consists of a single semicycle or { _{x n} ^{} n = -k ∞} oscillates about equilibrium x ¯ with semicycles having at most k terms.

Proof.

Suppose that { _{x n} ^{} n = -k ∞} has at least two semicycles. Then there exists N ...5; -k such that either _{x N} < x ¯ ...4; _{x N +1} or _{x N +1} < x ¯ ...4; _{x N} . We assume that the former case holds. The latter case is similar and will be omitted. Now suppose that the positive semicycle beginning with the term _{x N +1} has k terms. Then _{x N} < x ¯ ...4; _{x N +k} and so the case [figure omitted; refer to PDF] holds for every p , q ∈ (0 , ∞ ) , from which the result follows.

3. Boundedness and Global Stability of ( 1.1)

In this section, we consider the case p ∈ (0,1 ) with no restriction on other parameters and we consider the case p >1 with some specified conditions. For these cases, we have the following results which give a complete picture as regards the boundedness character of positive solutions of ( 1.1).

Theorem 3.1.

Suppose that [figure omitted; refer to PDF] then every positive solution of ( 1.1) is bounded.

Proof.

We have [figure omitted; refer to PDF] Hence we will prove that { _{x N} } is bounded. If [figure omitted; refer to PDF] then we have ^{f [variant prime]} ( x ) >0 , ^{f [variant prime][variant prime]} ( x ) <0 . Hence, the function f is increasing and concave. Thus, we get that there is a unique fixed point x ¯ of the equation f (x ) =x . Also the function f satisfies the condition [figure omitted; refer to PDF] Using [ 15, 2.6.2] we obtain that x ¯ is a global attractor of all positive solutions of ( 1.1) and so { _{x N} } is bounded, from which the result follows.

Now we study the boundedness of ( 1.1) for the case p >1 . We give better result than Theorem 3.1for the boundedness of ( 1.1) and we prove that in this case, there exist unbounded solutions of ( 1.1).

Theorem 3.2.

Consider ( 1.1) and assume that p >1 , p ... ∞ , α >1 and q [arrow right] ∞ . Then every positive solution of ( 1.1) is bounded and _{lim n [arrow right] ∞ x n} = α .

Proof.

Let [figure omitted; refer to PDF] Suppose on the contrary that every positive solution of ( 1.1) is unbounded. Then, from ( 1.1), we obtain _{x n} ...5; α >1 for n ...5;1 . Therefore we get [figure omitted; refer to PDF] Thus the proof is complete. We note that in here f (x ) is a continuous function for x ,p ,q ∈ (0 , ∞ ) .

Theorem 3.3.

Consider ( 1.1) when the case k is odd and suppose that [figure omitted; refer to PDF] then there exists unbounded solutions of ( 1.1).

Proof.

Let { _{x n} ^{} n = -k ∞} be a solution of ( 1.1) with initial values _{x -k} , _{x -k +1} , ... , _{x -1} , _{x 0} such that [figure omitted; refer to PDF] Then from ( 1.1), ( 3.7), and ( 3.8) we have [figure omitted; refer to PDF] Also, from ( 3.8) and ( 3.10)-( 3.14), it is clear that [figure omitted; refer to PDF] Moreover from ( 1.1) and ( 3.8)-( 3.14) and arguing as above we get [figure omitted; refer to PDF] Therefore working inductively we can prove that for n =0,1 ,2 , ... , [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] So { _{x n} ^{} n = -k ∞} is unbounded. From which the result follows.

Now, in the next theorem, we will provide an alternative proof for the theorem above when 1 ...4;p < ∞ and k is odd, whose proof can be used for some practical applications.

Theorem 3.4.

Consider ( 1.1) when the case k is odd and suppose that 1 ...4;p < ∞ . If 0 ...4; α <1 , then there exists solutions of ( 1.1) that are unbounded.

Proof.

We assume that 0 < α <1 and choose the initial conditions such that [figure omitted; refer to PDF] So, [figure omitted; refer to PDF] Therefore, we obtain α < _{x k +1} <1 and _{x k +2} >2 α + ^{x -k p} . By induction, for i =1,2 , ... , we have α < _{x (k +1 )i} <1 and _{x (k +1 )i +1} > (i +1 ) α + ^{x -k p} . Thus, [figure omitted; refer to PDF] Now, we assume that α =0 and choose the initial conditions such that [figure omitted; refer to PDF] So, we have [figure omitted; refer to PDF] Further, we have [figure omitted; refer to PDF] By induction for i =1,2 , ... , we have 0 < _{x (k +1 )i} <1 and _{x (k +1 )i +1} > ( ^{x -k p ) (i +1 )} . Thus, [figure omitted; refer to PDF] from which the result follows.

The following result is essentially proved in [ 10, 11] for k =1 . The result is satisfied for k ∈ {2,3 , ... } and its proof is omitted.

Lemma 3.5.

If Either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] holds, then ( 1.1) has a unique equilibrium point x ¯ .

The following result is essentially proved in [ 10, 11] for k =1 . It is clear that the result is satisfied when k is odd and its proof is omitted.

Lemma 3.6.

Consider ( 1.1) when the case k is odd. Suppose that [figure omitted; refer to PDF] or [figure omitted; refer to PDF] holds. Then the unique positive equilibrium of ( 1.1) is globally asymptotically stable.

4. Periodicity of the Solutions of ( 1.1)

In this section, we investigate the existence of a prime two periodic solution for k is odd. Moreover, when k is even, we prove that there are no positive prime two periodic solutions and lastly, we give a correction for Theorem 2.5 which was given in [ 2].

The following result is given when the case k =1 in [ 10]. If k is odd, the result is still satisfied and its proof is omitted.

Lemma 4.1.

Assume that k is odd. Then, ( 1.1) has prime two periodic solutions if and only if [figure omitted; refer to PDF] and there exists a sufficient small positive number _{[straight epsilon] 1} , such that [figure omitted; refer to PDF]

Now, let consider the case where k is even.

Theorem 4.2.

Consider ( 1.1) when the case k is even and the following conditions are satisfied separately: [figure omitted; refer to PDF] Then, there are no positive prime two periodic solutions of ( 1.1).

Proof.

Firstly, we consider the case 0 <q <p <1 and k is even of ( 1.1) and suppose that [figure omitted; refer to PDF] where x , y ∈ ( α , ∞ ) is a prime two periodic solution of ( 1.1). Then it must be [figure omitted; refer to PDF] Substituting ( 4.6) into ( 4.5), it follows that [figure omitted; refer to PDF] Taking logarithm on both sides of ( 4.7), we obtain that [figure omitted; refer to PDF] So from ( 4.8) [figure omitted; refer to PDF] From x > α , thus ^{F [variant prime]} ( x ) >0 , which implies that x ¯ is a unique solution of ( 1.1).

We consider the case 0 <p <q <1 and k is even of ( 1.1). The proof of this case is similar to the first case's proof and will be omitted.

Now, suppose that 1 <p , q <p +1 and k is even of ( 1.1). In this case we have that [figure omitted; refer to PDF] Considering the numerator on the right hand side in ( 4.10) let [figure omitted; refer to PDF] From x > α and 1 <p , q <p +1 [figure omitted; refer to PDF] which implies that x ¯ is a unique solution of ( 1.1).

Suppose that 1 <q , p <q +1 and k is even of ( 1.1). The proof of this case is similar to the third case's proof and will be omitted.

The following result was given in [ 2, Theorem 2.5] for ( 1.1) when the case p =1 , k =1 and q ∈ (0 , ∞ ) . But the authors make some mistakes in this theorem. Now, we give a correction and a conjecture for this result.

Theorem A.

Consider ( 1.1). Let be p =1 , k =1 , q ∈ ( 0 , ∞ ) , q ... ^{0 +} and q ... ∞ . Suppose that [figure omitted; refer to PDF] hold. Then the unique positive equilibrium x ¯ of ( 1.1) is globally asymptotically stable.

Correction B

Consider ( 1.1). Let be p =1 , k =1 , q ∈ (1 , ∞ ) , and q ... ∞ . Suppose that [figure omitted; refer to PDF] holds. Then the unique positive equilibrium x ¯ of ( 1.1) is globally asymptotically stable.

Proof.

It is easy to see the proof from Theorem 2.5 in [ 2].

Conjecture 4.3.

Consider ( 1.1). Let be p =1 , k =1 , q ∈ (0,1 ) , and q ... ^{0 +} . Suppose that [figure omitted; refer to PDF] holds. Then the unique positive equilibrium x ¯ of ( 1.1) is globally asymptotically stable.

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