- Open Access
Positive solutions of second-order linear difference equation with variable delays
© Peics; licensee Springer. 2013
Received: 18 September 2012
Accepted: 8 March 2013
Published: 29 March 2013
In this paper we consider the second-order linear difference equations with variable delays
where , N is the set of positive integers. Using the method of Riccati transform and the generalized characteristic equations, we give sufficient conditions for the existence of positive solutions.
In the past few years, oscillation and non-oscillation theory of second-order linear and nonlinear difference equations has attracted considerable attention, and we refer the reader to the papers of Diblik et al. , El-Sheik et al. , He , Huiqin and Zhen , Koplatadze and Kvinikadze , Krasznai et al. , Li et al. , Liu and Cheng , Medina and Pituk , Tang , Tang and Yu , Zhang , Zhang and Li , and the references therein. For comprehensive treatment, see the papers by Bastinec et al. [14–17], Čermak [18, 19], Deng , Došly and Fišnarová , Hille , Jaroš and Stavroulakis , Thandapani et al. , Yang  and the monographs [26, 27]. Some sharp conditions of Hille-type criterion on the existence of oscillatory and non-oscillatory solutions of second-order differential equations are given by Kusano et al. [28, 29], by Péics and Karsai , by Bastinec et al.  and by Berezansky et al. . Some sufficient conditions of oscillation and non-oscillation of second-order difference equations can be found in the papers by Lei , by Li and Jiang , by Zhou and Zhang , and the references therein.
for , where the following hypotheses are satisfied:
() , ;
() , .
As a special case we can formulate the following results given in  for half-linear delay differential equations.
Theorem A (see Theorem 2 in )
holds for t large enough. Then equation (1) has a positive solution.
Theorem B (see Corollary 2 in )
then equation (1) has a positive solution.
where , N is the set of positive integers and is a sequence of real numbers.
Theorem C (see Lemma 4 in )
for all large n, then equation (2) has a non-oscillatory solution.
where , N is the set of positive integers and the following hypotheses are satisfied:
() is a sequence of real numbers for and ;
() is a sequence of positive integers such that for and .
By a solution of equation (3) we mean a sequence of real numbers defined for , which satisfies equation (3) for all . A nontrivial solution of equation (3) is said to be oscillatory if for every , there exists an such that . Otherwise, it is non-oscillatory. Thus, a non-oscillatory solution is either eventually positive or eventually negative.
where is a solution sequence of equation (3). This transformation leads to the generalization of the Riccati-type equation associated with equation (3). The aim of this paper is to prove theorems for the existence of positive solutions of equation (3), using the Riccati-type equation associated with equation (3). The obtained results are discrete analogues of results given for some differential equations and generalize results given for second-order linear difference equations with constant delays or without delays.
where is a fixed positive bounded sequence.
Let denote the sequence of sequences defined for all natural numbers and .
S is continuous in the sense that if for all natural number , , and , , uniformly on every compact subinterval of , then , , uniformly on every compact subinterval of ;
the sequences in the image set SF are bounded at every point of .
Then the mapping S has at least one fixed point in F.
Equation (3) has an eventually positive solution.
- (b)There is a sequence , , for some such that for all and satisfies the Riccati-type equation(4)
- (b)⇒ (a): Let now be a solution sequence of Riccati-type equation (4) for such that for all . We show by direct substitution that the sequence defined by
for and the proof of Lemma 1 is complete. □
Now, we introduce an antidifference equation associated with the Riccati-type equation (4), and we show in the main results that it is very useful to study the antidifference equation instead of equation (4).
- (a)There is a solution sequence , , of the Riccati-type equation (4) for some such that for all and(9)
- (b)There is a sequence , , for some such that for all and(10)
⇒ (a): Assume that there is a sequence which satisfies equation (10) for , where is an arbitrary natural number such that for all . Applying the difference operator to both sides of equation (10), we show that the sequence , defined by formula for , is a solution of the Riccati-type equation (4) for , and it satisfies assumption (9). The proof of the theorem is complete. □
Now we can formulate the main theorem.
- (a)There exists a natural number and there exist the real sequences and such that , for ,(13)
There exists a real solution sequence of equation (10) which satisfies the inequality for .
Proof (a) ⇒ (b): We have to show that equation (10) has a solution sequence for . To this end, using Theorem D, we prove that operator S, defined by (15), has a fixed point , which is a solution sequence of equation (10), and obviously satisfies the estimate for .
It follows from assumptions (13) and (14) that the operator S, defined for , satisfies the inequality and maps F to F. It follows immediately from assumption (14) that the sequences in the image set SF are uniformly bounded on any subset of .
Let the sequence of sequences tend to the sequence , , uniformly on any finite interval of , which means this convergence is uniform for all .
for , if p is sufficiently large. Thus, , , uniformly on any finite subset .
⇒ (a): If is a solution sequence of (10), then taking for , the conditions of Theorem 1 are satisfied because of the fact that . The proof is complete. □
Existence of positive solutions
Let the sequence be such that , and set and in Theorem 1. Now we can formulate the conditions for the existence of positive solutions of equation (3).
holds for n large enough. Then equation (3) has a positive solution for .
Proof Let the sequence be given such that the conditions of the theorem hold. Since , hence and . We show that the conditions of Theorem 1 are satisfied with and for n large enough.
Therefore, in virtue of Theorem 1, equation (3) has a positive solution and the proof is complete. □
Remark 1 The result of Theorem 2 is the discrete analogue of the result presented in Theorem A and generalizes the result given in  for first-order linear difference equations with variable delays.
Choosing A such that the function takes the maximum value, we obtain and we can formulate the following corollary of Theorem 2.
holds for n large enough. Then equation (3) has an eventually positive solution.
Hence, we obtain the following non-oscillation criterion.
holds, then equation (3) has an eventually positive solution.
Remark 2 The result of Corollary 2 is the discrete analogue of the result presented in Theorem B and at the same time generalizes the result given in Theorem C for second-order linear difference equations with variable delays.
The research is supported by the Serbian Ministry of Science, Technology and Development for Scientific Research Grant no. III44006. The author thanks the referees for the valuable comments.
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