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

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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 ,nN, 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.

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 ) ] dsμ(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)|ds 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|nN,nM}.

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 nν 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)Sx(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

    Δλ(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 nM 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 Δa(n) from (5) and applying the difference operator to the transformed equality, we get that

Δa(n)=a(n)λ(n)andΔ [ Δ a ( n ) ] =a(n)λ(n+1)λ(n)+a(n)Δλ(n).
(6)

It follows from the first equality of (6) that

a(n+1)=a(n) ( λ ( n ) + 1 ) anda(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)+Δλ(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 nM 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 thatβ(n)Sξ(n)γ(n)
(14)

for n N M and for real sequences {ξ(n)}, where

Sξ(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|nN, 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 ( ) | 2G = 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 ) | <δfor sufficiently large p and n N 1 , 2 .

If we use the following form of δ:

δ= ε C ( M 2 + 2 ) ,where C=2G+ 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)Sξ(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)=Sω(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)=Sω(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<1μ(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

1μ(n)ξ(n)+1implies 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.

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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.

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Correspondence to Hajnalka Peics.

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The author declares that she has no competing interests.

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Since there is one author, she completed all tasks necessary for the article.

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Keywords

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