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Positive solutions of second-order linear difference equation with variable delays

Advances in Difference Equations20132013:82

https://doi.org/10.1186/1687-1847-2013-82

Received: 18 September 2012

Accepted: 8 March 2013

Published: 29 March 2013

Abstract

In this paper we consider the second-order linear difference equations with variable delays

Δ 2 a ( n ) + i = 1 m P i ( n ) a ( n k i ( n ) ) = 0 , n n 0 ,

where n 0 , n N , 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.

MSC:39A11, 39A12.

Keywords

  • second-order linear difference equation
  • variable delays
  • existence
  • positive solutions
  • non-oscillatory solutions
  • sufficient conditions

Introduction

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. [1], El-Sheik et al. [2], He [3], Huiqin and Zhen [4], Koplatadze and Kvinikadze [5], Krasznai et al. [6], Li et al. [7], Liu and Cheng [8], Medina and Pituk [9], Tang [10], Tang and Yu [11], Zhang [12], Zhang and Li [13], and the references therein. For comprehensive treatment, see the papers by Bastinec et al. [1417], Čermak [18, 19], Deng [20], Došly and Fišnarová [21], Hille [22], Jaroš and Stavroulakis [23], Thandapani et al. [24], Yang [25] 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 [30], by Bastinec et al. [31] and by Berezansky et al. [32]. Some sufficient conditions of oscillation and non-oscillation of second-order difference equations can be found in the papers by Lei [33], by Li and Jiang [34], by Zhou and Zhang [35], and the references therein.

Consider the second-order delay differential equation of the form
x ( t ) + i = 1 m p i ( t ) x ( t τ i ( t ) ) = 0
(1)

for t 0 t < T , where the following hypotheses are satisfied:

( H 1 ) p i C [ [ t 0 , T ) , R ] , i = 1 , 2 , , m ;

( H 2 ) τ i C [ [ t 0 , T ) , R + ] , i = 1 , 2 , , m .

As a special case we can formulate the following results given in [30] for half-linear delay differential equations.

Theorem A (see Theorem 2 in [30])

Assume that ( H 1 ) and ( H 2 ) hold and there exists a positive function μ ( t ) for t t 0 such that
t [ μ ( t ) 2 + i = 1 m | p i ( s ) | exp ( s τ i ( s ) s μ ( ξ ) d ξ ) ] d s μ ( t )

holds for t large enough. Then equation (1) has a positive solution.

Theorem B (see Corollary 2 in [30])

Assume that ( H 1 ) and ( H 2 ) hold and the functions τ i ( t ) , where i = 1 , 2 , , m , are bounded. If
lim sup t t t i = 1 m | p i ( s ) | d s 1 4 ,

then equation (1) has a positive solution.

Consider the second-order difference equation of the form
Δ 2 a ( n ) + P ( n ) a ( n ) = 0 , n n 0 ,
(2)

where n 0 N , N is the set of positive integers and { P ( n ) } is a sequence of real numbers.

The following result is a sufficient condition for the existence of non-oscillatory solutions of equation (2) and it is demonstrated in the paper [35].

Theorem C (see Lemma 4 in [35])

Assume that { P ( n ) } is a real sequence with P ( n ) 0 for all n n 0 . If
n i = n + 1 P i ( s ) 1 4

for all large n, then equation (2) has a non-oscillatory solution.

Consider now the second-order difference equation with variable delays
Δ 2 a ( n ) + i = 1 m P i ( n ) a ( n k i ( n ) ) = 0 , n n 0 ,
(3)

where n 0 N , N is the set of positive integers and the following hypotheses are satisfied:

( H 1 ) { P i ( n ) } is a sequence of real numbers for i = 1 , 2 , , m and n n 0 ;

( H 2 ) { k i ( n ) } is a sequence of positive integers such that k i ( n ) n for i = 1 , 2 , , m and n n 0 .

Let M n 0 be an arbitrary natural number and set
N M = { n | n N , n M } .

By a solution of equation (3) we mean a sequence of real numbers { a ( n ) } defined for n N M , which satisfies equation (3) for all n N M . A nontrivial solution { a ( n ) } of equation (3) is said to be oscillatory if for every ν > n 0 , there exists an n ν such that a ( n ) a ( n + 1 ) 0 . Otherwise, it is non-oscillatory. Thus, a non-oscillatory solution is either eventually positive or eventually negative.

The basic idea in the paper is to use the Riccati transformation technique by the substitution
λ ( n ) = Δ a ( n ) a ( n ) ,

where { a ( n ) } 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.

Let R : = { { ξ ( n ) } n = 1 : ξ ( n ) R , n = 1 , 2 , } and
: = { { ξ ( n ) } R : | ξ ( n ) | μ ( n ) , n N M } ,

where { μ ( n ) } is a fixed positive bounded sequence.

Let { x p ( n ) } denote the sequence of sequences defined for all natural numbers n [ n 0 , ) and p = 1 , 2 , 3 ,  .

Theorem D (Schauder-Tychonoff, see [3638])

Let F = , and let { μ ( n ) } be a fixed positive bounded sequence. Let S be a mapping of F into itself with the properties:
  1. (i)

    S is continuous in the sense that if x p ( n ) F for all natural number n n 0 , p = 1 , 2 ,  , and x p ( n ) x ( n ) , p , uniformly on every compact subinterval of [ n 0 , ) , then S x p ( n ) S x ( n ) , p , uniformly on every compact subinterval of [ n 0 , ) ;

     
  2. (ii)

    the sequences in the image set SF are bounded at every point of [ n 0 , ) .

     

Then the mapping S has at least one fixed point in F.

Main results

First, we apply the Riccati transformation and show the relationship between equation (3) and the Riccati-type equation (4).

Lemma 1 Assume that conditions ( H 1 ) and ( H 2 ) hold. Then the following statements are equivalent.
  1. (a)

    Equation (3) has an eventually positive solution.

     
  2. (b)
    There is a sequence { λ ( n ) } , n N M , for some M n 0 such that λ ( n ) + 1 > 0 for all n N M and { λ ( n ) } satisfies the Riccati-type equation
    Δ λ ( n ) + λ ( n + 1 ) λ ( n ) + i = 1 m P i ( n ) j = n k i ( n ) n 1 1 λ ( j ) + 1 = 0 for n N M .
    (4)
     
Proof (a) (b): Let { a ( n ) } be the solution of difference equation (3) and suppose, according to the hypotheses, that a ( n ) > 0 for n M n 0 . It will be shown that the sequence { λ ( n ) } defined by
λ ( n ) = Δ a ( n ) a ( n ) , n N M ,
(5)
is a solution of the Riccati-type equation (4) for n N M . Expressing Δ a ( n ) from (5) and applying the difference operator to the transformed equality, we get that
Δ a ( n ) = a ( n ) λ ( n ) and Δ [ Δ a ( n ) ] = a ( n ) λ ( n + 1 ) λ ( n ) + a ( n ) Δ λ ( n ) .
(6)
It follows from the first equality of (6) that
a ( n + 1 ) = a ( n ) ( λ ( n ) + 1 ) and a ( n ) = a ( M ) j = M n 1 ( λ ( j ) + 1 ) for  n N M ,
and hence the following equalities are valid:
a ( n k i ( n ) ) a ( n ) = a ( M ) j = M n k i ( n ) 1 ( λ ( j ) + 1 ) a ( M ) j = M n 1 ( λ ( j ) + 1 ) = j = n k i ( n ) n 1 1 λ ( j ) + 1 .
(7)
By dividing both sides of difference equation (3) by a ( n ) , we obtain that
Δ [ Δ a ( n ) ] a ( n ) + i = 1 m P i ( n ) a ( n k i ( n ) ) a ( n ) = 0 .
(8)
In virtue of the equalities (6) and (7), we obtain that
a ( n ) λ ( n + 1 ) λ ( n ) + a ( n ) Δ λ ( n ) a ( n ) + i = 1 m P i ( n ) j = n k i ( n ) n 1 1 λ ( j ) + 1 = 0 .
After reducing the first term, we get that
λ ( n + 1 ) λ ( n ) + Δ λ ( n ) + i = 1 m P i ( n ) j = n k i ( n ) n 1 1 λ ( j ) + 1 = 0 ,
and we conclude that the sequence { λ ( n ) } satisfies the Riccati-type equation (4), and the first part of the proof is complete.
  1. (b)
    (a): Let now { λ ( n ) } be a solution sequence of Riccati-type equation (4) for n N M such that λ ( n ) + 1 > 0 for all n N M . We show by direct substitution that the sequence defined by
    a ( n ) = j = M n 1 ( λ ( j ) + 1 ) for  n N M
     
is the positive solution of difference equation (3). Completing the following transformations, we get
Δ [ Δ a ( n ) ] = a ( n ) ( λ ( n + 1 ) λ ( n ) + Δ λ ( n ) ) = a ( n ) ( i = 1 m P i ( n ) j = n k i ( n ) n 1 1 λ ( j ) + 1 ) = a ( n ) i = 1 m P i ( n ) a ( n k i ( n ) ) a ( n ) = i = 1 m P i ( n ) a ( n k i ( n ) )

for n N M 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).

Lemma 2 Assume that conditions ( H 1 ) and ( H 2 ) hold. Then the following statements are equivalent.
  1. (a)
    There is a solution sequence { λ ( n ) } , n N M , of the Riccati-type equation (4) for some M n 0 such that λ ( n ) + 1 > 0 for all n N M and
    = n ( i = 1 m P i ( ) j = k i ( ) 1 1 λ ( j ) + 1 ) < .
    (9)
     
  2. (b)
    There is a sequence { ω ( n ) } , n N M , for some M n 0 such that ω ( n ) + 1 > 0 for all n N M and
    ω ( n ) = = n ω ( + 1 ) ω ( ) + = n ( i = 1 m P i ( ) j = k i ( ) 1 1 ω ( j ) + 1 ) .
    (10)
     
Proof (a) (b): Let the sequence { λ ( n ) } , defined by formula λ ( n ) = ω ( n ) for n N M , be a solution of the Riccati-type equation (4) for n N M with the property (9). Let n M be a natural number fixed arbitrarily, and let us sum up both sides of the Riccati-type equation (4) from n to M 1 1 , where M 1 > n . Then we get the equality
ω ( M 1 ) ω ( n ) + = n M 1 1 ω ( + 1 ) ω ( ) + = n M 1 1 ( i = 1 m P i ( ) j = k i ( ) 1 1 ω ( j ) + 1 ) = 0 .
(11)
We claim that
= n ω ( + 1 ) ω ( ) < .
(12)
To this end, assume the contrary statement that = n ω ( + 1 ) ω ( ) = . Now, in view of (11) and the statement = n ω ( + 1 ) ω ( ) = , there are natural numbers M 1 and M large enough such that M 1 > M and
ω ( M 1 ) + = M M 1 1 ω ( + 1 ) ω ( ) = ω ( n ) = n M 1 ω ( + 1 ) ω ( ) = n M 1 1 ( i = 1 m P i ( ) j = k i ( ) 1 1 ω ( j ) + 1 ) 1
for M 1 > M n , or, equivalently,
ω ( M 1 ) = M M 1 1 ω ( + 1 ) ω ( ) + 1 .
Because of the substitution (5), it follows that
a ( M 1 + 1 ) a ( M 1 ) = ω ( M 1 ) 1 = M M 1 1 ω ( + 1 ) ω ( ) .
Since M 1 > M is an arbitrary large number, then the inequality
= M M 1 1 ω ( + 1 ) ω ( ) < 0
holds. If M 1 , then the inequality = M ω ( + 1 ) ω ( ) < 0 must hold, which contradicts the assumption
= n ω ( + 1 ) ω ( ) = .
We now let M 1 in (11). Using inequalities (12) and (9), we find that ω ( M 1 ) tends to a finite limit ω . But ω must be zero because in the other case inequality (12) would fail to hold. These argumentations complete the first part of the proof.
  1. (b)

    (a): Assume that there is a sequence { ω ( n ) } which satisfies equation (10) for n N M , where M n 0 is an arbitrary natural number such that ω ( n ) + 1 > 0 for all n N M . Applying the difference operator to both sides of equation (10), we show that the sequence { λ ( n ) } , defined by formula λ ( n ) = ω ( n ) for n N M , is a solution of the Riccati-type equation (4) for n N M , and it satisfies assumption (9). The proof of the theorem is complete. □

     

Now we can formulate the main theorem.

Theorem 1 Assume that conditions ( H 1 ) and ( H 2 ) hold. Then the following statements are equivalent.
  1. (a)
    There exists a natural number M n 0 and there exist the real sequences { β ( n ) } and { γ ( n ) } such that 1 < β ( n ) γ ( n ) , sup n γ ( n ) < for n N M ,
    = n ( i = 1 m | P i ( ) | j = k i ( ) 1 1 β ( j ) + 1 ) <
    (13)
     
and such that
β ( n ) ξ ( n ) γ ( n ) implies that β ( n ) S ξ ( n ) γ ( n )
(14)
for n N M and for real sequences { ξ ( n ) } , where
S ξ ( n ) = = n ξ ( + 1 ) ξ ( ) + = n ( i = 1 m P i ( ) j = k i ( ) 1 1 ξ ( j ) + 1 ) .
(15)
  1. (b)

    There exists a real solution sequence { ω ( n ) } of equation (10) which satisfies the inequality β ( n ) ω ( n ) γ ( n ) for n N M .

     

Proof (a) (b): We have to show that equation (10) has a solution sequence { ω ( n ) } for n N M . To this end, using Theorem D, we prove that operator S, defined by (15), has a fixed point { ω ( n ) } , which is a solution sequence of equation (10), and obviously satisfies the estimate β ( n ) ω ( n ) γ ( n ) for n N M .

Let now M 1 and M 2 be natural numbers such that M M 1 < M 2 < . Then the set
N 1 , 2 = { n | n N , M 1 n M 2 }
is an arbitrary compact subset of the set N M . Set
L : = max M n M 2 i = 1 m | P i ( n ) | , G : = sup n M γ ( n ) , B : = min M n M 2 β ( n ) , K : = max i = 1 m max M n M 2 k i ( n ) , K B : = { max i = 1 m max M n M 2 k i ( n ) , 1 < B 0 , min i = 1 m min M n M 2 k i ( n ) , B > 0 , K G : = { max i = 1 m max M n M 2 k i ( n ) , G > 1 , min i = 1 m min M n M 2 k i ( n ) , G < 1 .
Let
F : = { { ξ ( n ) } R : | ξ ( n ) | μ ( n ) , n N M } , where  μ ( n ) = max n n 0 { | γ ( n ) | , 1 } .

It follows from assumptions (13) and (14) that the operator S, defined for { ξ ( n ) } F , satisfies the inequality = n ξ ( + 1 ) ξ ( ) < 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 N M .

Let the sequence of sequences { ξ p ( n ) } F tend to the sequence { ξ ( n ) } , p , uniformly on any finite interval of N M , which means this convergence is uniform for all n N 1 , 2 .

In virtue of the following transformations:
| ξ ( + 1 ) ξ ( ) ξ p ( + 1 ) ξ p ( ) | = | ξ ( + 1 ) ξ ( ) ξ p ( + 1 ) ξ ( ) + ξ p ( + 1 ) ξ ( ) ξ p ( + 1 ) ξ p ( ) | | ξ ( + 1 ) ξ p ( + 1 ) | | ξ ( ) | + | ξ ( ) ξ p ( ) | | ξ p ( + 1 ) | ,
we obtain that
= n M 2 | ξ ( + 1 ) ξ ( ) ξ p ( + 1 ) ξ p ( ) | 2 G = n M 2 + 1 | ξ ( ) ξ p ( ) | ,
and also we obtain that the inequalities
j = k i ( ) 1 1 ξ ( j ) + 1 j = k i ( ) 1 1 ξ p ( j ) + 1 = j = k i ( ) 1 ( ξ p ( j ) + 1 ) j = k i ( ) 1 ( ξ ( j ) + 1 ) j = k i ( ) 1 ( ξ ( j ) + 1 ) ( ξ p ( j ) + 1 ) ( ( K 1 ) G K G 1 + 1 ) j = k i ( ) 1 | ξ ( ) ξ p ( ) | j = k i ( ) 1 ( β ( j ) + 1 ) ( β p ( j ) + 1 ) ( K G K G + 1 ) j = K 1 | ξ ( ) ξ p ( ) | ( B + 1 ) 2 K B
are valid. Using the previous inequalities, we can get the following transformations:
| S ξ ( n ) S ξ p ( n ) | lim M 2 ( = n M 2 | ξ ( + 1 ) ξ ( ) ξ p ( + 1 ) ξ p ( ) | + = n M 2 i = 1 m | P i ( ) | ( j = k i ( ) 1 1 ξ ( j ) + 1 j = k i ( ) 1 1 ξ p ( j ) + 1 ) ) lim M 2 ( 2 G = n M 2 + 1 | ξ ( ) ξ p ( ) | + L ( K G K G + 1 ) ( B + 1 ) 2 K B = n M 2 j = K 1 | ξ ( ) ξ p ( ) | ) .
The uniform convergence of the sequence ξ p ( n ) ξ ( n ) , p , for all n N 1 , 2 implies that
| ξ ( n ) ξ p ( n ) | < δ for sufficiently large  p  and  n N 1 , 2 .
If we use the following form of δ:
δ = ε C ( M 2 + 2 ) , where  C = 2 G + L K ( K G K G + 1 ) ( B + 1 ) 2 K B ,
we obtain
| S ξ ( n ) S ξ p ( n ) | lim M 2 ( 2 G δ ( M 2 + 2 n ) + L ( K G K G + 1 ) ( B + 1 ) 2 K B K δ ( M 2 + 1 n ) ) < lim M 2 ( 2 G + L ( K G K G + 1 ) ( B + 1 ) 2 K B K ) δ ( M 2 + 2 ) = lim M 2 C ε C ( M 2 + 2 ) ( M 2 + 2 ) = lim M 2 ε = ε

for n N 1 , 2 , if p is sufficiently large. Thus, S ξ p ( n ) S ξ ( n ) , p , uniformly on any finite subset N M .

We obtained that the conditions of the Schauder-Tychonoff theorem are satisfied, and hence the mapping S has at least one fixed point { ω ( n ) } in F. Moreover, because of the equality ω ( n ) = S ω ( n ) for n N M , we conclude that { ω ( n ) } is the solution sequence of equation (10) with the property that β ( n ) ω ( n ) γ ( n ) for n N M . The first part of the proof is complete.
  1. (b)

    (a): If { ω ( n ) } is a solution sequence of (10), then taking β ( n ) = γ ( n ) = ω ( n ) for n N M , the conditions of Theorem 1 are satisfied because of the fact that ω ( n ) = S ω ( n ) . The proof is complete. □

     

Existence of positive solutions

Let the sequence { μ ( n ) } be such that 0 < μ ( n ) < 1 , and set β ( n ) = μ ( n ) and γ ( n ) = μ ( n ) in Theorem 1. Now we can formulate the conditions for the existence of positive solutions of equation (3).

Theorem 2 Assume that conditions ( H 1 ) and ( H 2 ) hold and there exists a positive sequence { μ ( n ) } for n N M for some natural number M n 0 such that 0 < μ ( n ) < 1 for n N M and
= n ( μ ( + 1 ) μ ( ) + i = 1 m | P i ( ) | j = k i ( ) 1 1 1 μ ( j ) ) μ ( n )
(16)

holds for n large enough. Then equation (3) has a positive solution { a ( n ) } for n N M .

Proof Let the sequence { μ ( n ) } be given such that the conditions of the theorem hold. Since 0 < μ ( n ) < 1 , hence 1 < μ ( n ) < 0 and 0 < 1 μ ( n ) < 1 . We show that the conditions of Theorem 1 are satisfied with β ( n ) = μ ( n ) and γ ( n ) = μ ( n ) for n large enough.

Let { ξ ( n ) } be a real sequence such that | ξ ( n ) | μ ( n ) . Because of the property that μ ( n ) ξ ( n ) μ ( n ) , the inequality
1 μ ( n ) ξ ( n ) + 1 implies that 1 ξ ( n ) + 1 1 1 μ ( n ) for  n N M .
Because of assumption (16) and some transformations, it follows that
| S ξ ( n ) | = n ( | ξ ( + 1 ) | | ξ ( ) | + i = 1 m | P i ( ) | j = k i ( ) 1 1 ξ ( j ) + 1 ) = n ( μ + 1 μ ( ) + i = 1 m | P i ( ) | j = k i ( ) 1 1 1 μ ( j ) ) μ ( n ) .

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 [39] for first-order linear difference equations with variable delays.

Now we would like to find the sequence { μ ( n ) } in the form μ ( n ) = A n , where A is some constant. Since
= n μ ( + 1 ) μ ( ) = = n A 2 ( + 1 ) = lim M = n M A 2 ( + 1 ) = lim M = n M ( A 2 A 2 + 1 ) = lim M ( A 2 n A 2 n + 1 + A 2 n + 1 A 2 n + 2 + + A 2 M A 2 M + 1 ) = A 2 n
and
j = k i ( ) 1 1 1 μ ( j ) = j = k i ( ) 1 1 1 A j = j = k i ( ) 1 j j A ,
condition (16) takes the form
= n ( i = 1 m | P i ( ) | j = k i ( ) 1 j j A ) A n A 2 n .

Choosing A such that the function f ( A ) = A A 2 takes the maximum value, we obtain A = 1 2 and we can formulate the following corollary of Theorem 2.

Corollary 1 Assume that conditions ( H 1 ) and ( H 2 ) hold and
n = n ( i = 1 m | P i ( ) | j = k i ( ) 1 2 j 2 j 1 ) 1 4
(17)

holds for n large enough. Then equation (3) has an eventually positive solution.

In particular, if the sequences of delays { k i ( n ) } ( i = 1 , 2 , , m ) are bounded by K, we have
= n ( i = 1 m | P i ( ) | j = k i ( ) 1 2 j 2 j 1 ) ( 2 ( n K ) 2 ( n K ) 1 ) K = n ( i = 1 m | P i ( ) | )
and
lim sup n ( 2 ( n K ) 2 ( n K ) 1 ) K = 1 .

Hence, we obtain the following non-oscillation criterion.

Corollary 2 Assume that conditions ( H 1 ) and ( H 2 ) hold and the sequences of delays { k i ( n ) } ( i = 1 , 2 , , m ) are bounded. If
lim sup n n = n i = 1 m | P i ( ) | 1 4
(18)

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.

Declarations

Acknowledgements

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.

Authors’ Affiliations

(1)
Faculty of Civil Engineering, University of Novi Sad, Subotica, Serbia

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