On a Max-Type Difference Inequality and Its Applications

(1)

Volume 2010, Article ID 975740,8pages doi:10.1155/2010/975740

Research Article

On a Max-Type Difference Inequality and Its Applications

Stevo Stevi ´c

¹

and Bratislav D. Iri ˇcanin

²

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia

2Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade 11120, Serbia

Correspondence should be addressed to Stevo Stevi´c,[email protected] Received 1 November 2009; Accepted 31 March 2010

Academic Editor: Leonid Berezansky

Copyrightq2010 S. Stevi´c and B. D. Iriˇcanin. 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.

We prove a useful max-type diﬀerence inequality which can be applied in studying of some max- type diﬀerence equations and give an application of it in a recent problem from the research area.

We also give a representation of solutions of the difference equationxnmax{xâ_n−1¹ , . . . , xâ_n−k^k }.

1. Introduction

The investigation of max-type diﬀerence equations attracted some attention recently; see, for example,1–20and the references therein. In the beginning of the study of these equations the following diﬀerence equation was investigated:

xnmax A¹n

xn−1,A²n

xn−2, . . . ,A^kn

xn−k

, n∈N0, 1.1

where k ∈ N,Aⁱn , i 1, . . . , k, are real sequences mostly constant or periodic, and the initial valuesx−1, . . . , x−kare diﬀerent from zerosee, e.g., monograph6or paper19and the references therein.

The study of the next diﬀerence equation

xnmax

Bn⁰, Bn¹

x_n−p^r¹ ₁ x^s_n−q¹ ₁, Bn²

x^r_n−p² ₂

x^s_n−q² ₂, . . . , B^kn

x^r_n−p^k _k x^s_n−q^k _k

, n∈N0, 1.2

(2)

wherepi, qiare natural numbers such thatp1 < p2 <· · ·< pk,q1 < q2 <· · ·< qk,ri, si ∈R 0,∞, i 1, . . . , k, andk ∈ N, was proposed by the first author in numerous talks; see, for example,11,13. For some results in this direction see1,4,5,7,8,12,14–18,20.

A particular case of the diﬀerence equation

ynmax

A

yn−1· · ·yn−m1, 1

yn−m−1· · ·yn−2m1

, n∈N0, 1.3

arises naturally in certain models in automatic control see 9. By the change xn ynyn−1· · ·yn−m1the equation is transformed into the equation

xnmax

A, xn−1

xn−m

, n∈N0, 1.4

which is a special case of1.2and which is a natural prototype for the equation.

The following result, which extends the main result from the study in18was proved by the first author in17 see also16.

Theorem A. Every positive solution to the diﬀerence equation

xnmax A1

x^α_n−1¹ , A2

x^α_n−2² , . . . , Ak

x^α_n−k^k

, n∈N0, 1.5

where−1< αi<1, Ai≥0, i1, . . . , k, converges to max_1≤i≤k{A^1/α_i ⁱ¹}.

Here we continue to study1.5by considering the cases when some ofαi’s are equal to one. We also give a representation of well-defined solutions of the difference equation xnmax{xâ_n−1¹ , . . . , xâ_n−k^k },whereai∈R,i1, . . . , k.

2. Main Results

In this section we prove the main results of this note. Before this we formulate the following very useful auxiliary result which can be found in10and give a definition.

Lemma A. Letan_n∈Nbe a sequence of positive numbers which satisfy the inequality

a_nk≤qmax{a_nk−1, a_nk−2, . . . , an}, forn∈N, 2.1

whereq >0 andk∈Nare fixed. Then there exists anM >0 such that

an≤M k

q_n

, n∈N. 2.2

(3)

Definition 2.1. For a sequencexn^∞_n−s,s∈N0, we say that it converges to zero geometrically if there is aq∈0,1andM >0 such that

|xn| ≤Mqⁿ, 2.3

forn−s, . . . ,−1,0,1, . . . .

Now we are in a position to formulate and prove the main results of this note.

Proposition 2.2. Assume thatan^∞_n−k is a sequence of nonnegative numbers satisfying the diﬀerence inequality

an≤max{α1a_n−1−d1, . . . , αka_n−k−dk}, n∈N0, 2.4

wherek ∈ N,αi ∈ 0,1,di ∈ R,i∈ {1, . . . , k}, and if, for somei,αi 1, thendi >0. Then the sequenceanconverges geometrically to zero asn → ∞.

Proof. Letβ∈0,1be chosen such that

max{a_−k, . . . , a₋₁} ≤ cm

1−β, 2.5

where

cmmin

1,min

i:αi1{di}

. 2.6

Then from2.4and using the fact thatanare nonnegative numbers, we have that

an≤max

A max

i:αi∈0,1{an−i},max

j:αj1 an−j−cm

, 2.7

whereAmax_i:α_i_∈0,1{αi}.

From2.7,2.5and since 0<max{A, β}<1, we have that

a0≤max

A max

i:αi∈0,1{a−i},max

j:αj1 a−j−cm

≤max Acm

1−β, cm

1−β−cm,

max Acm

1−β, βcm

1−β

< cm

1−β.

2.8

(4)

Now assume thatan≤cm/1−β, for 0≤n≤n0−1.Then from2.4we get

an0≤max

A max

i:αi∈0,1{an0−i},max

j:αj1 an0−j−cm

≤max Acm

1−β, cm

1−β−cm,

max Acm

1−β, βcm

1−β

< cm

1−β.

2.9

Inequalities2.8and2.9along with the method of induction show that

0≤an≤ cm

1−β, forn∈ {−s, . . . ,−1} ∪N0. 2.10

Now note that from2.10we have that

an−cm≤βan, forn∈ {−s, . . . ,−1} ∪N0. 2.11

From2.7,2.11and the choice ofcm, it follows that forn∈N0

an≤max

A max

i:αi∈0,1{an−i}, βmax

j:αj1 an−j

≤max A, β

max1≤i≤k{an−i}.

2.12

Applying Lemma A in inequality2.12withqmax{A, β}, the result follows.

Remark 2.3. Note that the constant β in the proof of Proposition 2.2 depends on initial conditions of solutions to diﬀerence equation2.4, so that this is not a uniform constant.

Lemma 2.4. Consider the diﬀerence equation

znmin{C1−α1z_n−1, C2−α2z_n−2, . . . , Ck−αkz_n−k}, n∈N0, 2.13

wherek∈N,Ci∈R, αj∈R,i1, . . . , k, and there isi0∈ {1, . . . , k}such thatCi0 0.Then

|zn| ≤max{|α1||z_n−1| −C₁,|α2||z_n−2| −C2, . . . ,|αk||z_n−k| −Ck}, n∈N0. 2.14

Proof. If all terms in the right-hand side of 2.13 are nonnegative then clearly 0 ≤ zn ≤

−αi0zn−i0, so that

|zn| ≤ |αi0||zn−i0||αi0||zn−i0| −Ci0. 2.15

(5)

Otherwise, the setS ⊆ {1, . . . , k} of all indices for which the terms in2.13are negative is nonempty, so thatznmin_i∈S{Ci−αiz_n−i}<0.

From this and since for suchi∈S,αizn−imust be positive, it follows that

|zn|max

i∈S {αiz_n−i−Ci}max

i∈S {|αi||z_n−i| −Ci}. 2.16

From2.15and2.16inequality2.14easily follows.

ByProposition 2.2andLemma 2.4we obtain the following theorem.

Theorem 2.5. Consider the diﬀerence equation xnmax

A1

x^α_n−1¹ , A2

x^α_n−2² , . . . , Ak

x^α_n−k^k

, n∈N0, 2.17

wherek ∈N, 0≤Ai≤1,−1≤αi ≤1,−1< αiAi<1 for eachi∈ {1, . . . , k}, andAi 1 for at least onei∈ {1, . . . , k}.Then every positive solution of 2.17converges to one.

Proof. Taking the logarithm of2.17and using the changeyn−lnxn, we obtain that

yn min

i:Ai/0

ln 1

Ai −αiyn−i

, n∈N0. 2.18

Now note that ln1/Ai ≥ 0 for thoseisuch that Ai/0, since Ai ∈ 0,1, and there is an S1 ⊂ {1, . . . , k}such that ln1/Ai 0 wheni ∈ S1.ByLemma 2.4we have that for every n∈N0

yn≤ max

i:Ai/0

|αi|yn−i−ln 1 Ai

. 2.19

From 2.19, noticing that if |αi| 1 and Ai/0, thenAi ∈ 0,1so that ln1/Ai > 0 and by applyingProposition 2.2we obtain that|yn| → 0 asn → ∞, from which it follows that xne^−yⁿ → 1 asn → ∞, as desired.

Remark 2.6. Recently Gelis¸ken and C¸ inar in the paper: “On the global attractivity of a max- type diﬀerence equation,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 812674, 5 pages, 2009, have studied the asymptotic behavior to positive solutions of the diﬀerence equation

xnmax A

xn−1, 1 x_n−3^α

, n∈N0, 2.20

whereα ∈ 0,1 andA > 0. They claim that ifA ∈ 0,1, then every positive solution to 2.20converges to one. However the proof given there cannot be regarded as complete one.

Namely, they first formulated the following lemma.

(6)

Lemma 2.7. Letynbe a solution to the diﬀerence equation

ynmax 1−yn−1,−αy_n−3

, n∈N0. 2.21

Then for alln∈N0, the following inequality holds:

yn≤max yn−1−1, αyn−3. 2.22 Then they tried to show thatyn → 0 asn → ∞. Note that2.21is obtained by the changexn A^yⁿ from2.20, so that if it is proved thatyn → 0 asn → ∞then xn → 1 asn → ∞,from which the claim follows. In the beginning of the proof of the theorem they choose a numberβsuch that 0 < |yn−1| −1 ≤β|yn|,but do not say if these inequalities hold for allnor not, which is a bit confusing. Note that for diﬀerentnthe chosen numberβcan be diﬀerent, which means that in this caseβmight be a function ofn. Hence it is important that these inequalities hold for everyn∈N0∪ {−2,−1},which was not proved. This motivated us to proveProposition 2.2which, among others, removes the gap.

Now we present a representation of solutions of a particular case of1.5. The first author would like to express his sincere thanks to Professor L. Berg for a nice communication regarding this2.

Theorem 2.8. Consider the equation

xnmax x^a_n−1¹ , . . . , x^a_n−k^k

, n∈N0, 2.23

where k ∈ N,ai ∈ R, i 1, . . . , k.Then every well-defined solution of equation 2.23 has the following form:

xnd

k j1aⁱ

jn j

n , 2.24

where

nk k

≤i¹n · · ·i^kn ≤n1, n∈N0, 2.25

i^jn ≥0,j 1, . . . , k,and wherednis equal to one of the initial valuesx−k, . . . , x−1. Moreover, if−1< ai <1,i1, . . . , k,thenxn → 1 asn → ∞.

Proof. The casek1 is well known and simple. Just note thatxnx^a₋₁ⁿ¹¹ . Hence assume that k≥2.We prove the result by induction. Forn0 we have

x0max x^a₋₁¹, . . . , x^a_−k^k

. 2.26

(7)

Note thatx0can be equal to one of the numbersx^a₋₁¹, . . . , x^a_−k^k and that

x^a_−iⁱ x^aⁱ

i /ja⁰_j

−i , i1, . . . , k, 2.27

which is nothing but formula2.24in this case. From this we also have that 0k

k

1i¹₁ · · ·i^k₁ 110, 2.28

which is2.25in this case.

Now assume that we have proved2.24and2.25forl≤n−1.Then

xnmax x^a_n−1¹ , . . . , x^a_n−k^k

max

⎧⎨

⎩d

k j1aⁱ

jn−1δj 1 j

n−1 , . . . , d

k j1aⁱ

jn−kδj k j

n−k

⎫⎬

⎭, 2.29

whereδ_i^jis the Kronecker symbol andlk/k≤i¹_l · · ·i^k_l ≤l,forl≤n−1.Thus

i¹_n−s· · ·i^k_n−sδs^j≤n−s11≤n1, 2.30

fors1, . . . , kand

i¹_n−s· · ·i^k_n−sδ^js≥

n−sk k

1

nk−sk k

≥ nk

k

, 2.31

s1, . . . , k.Hence the first statement follows by induction.

Now assume that max_1≤j≤k{|aj|}<1. From this and2.25we have

k

j1

aⁱ_j^jⁿ ≤

max1≤j≤k aj_nk/k

. 2.32

Inequality2.32, the assumption max1≤j≤k{|aj|} < 1, and2.24imply thatxn tends to 1 as n → ∞,finishing the proof of the theorem.

Remark 2.9. Note that formula2.24holds for each value of parametersaj, j 1, . . . , k, and for all solutions whose initial values are diﬀerent from zero if one of these exponents is negative.

Remark 2.10. The second statement inTheorem 2.8follows easily also from Lemma A.

(8)

References

1 K. S. Berenhaut, J. D. Foley, and S. Stević, “Boundedness character of positive solutions of a max difference equation,” Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1193–1199, 2006.

2 L. Berg, “Email Communication,” October 2009.

3 C. Ç inar, S. Stević, and I. Yalçinkaya, “On positive solutions of a reciprocal difference equation with minimum,” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 307–314, 2005.

4 E. M. Elsayed and S. Stevi´c, “On the max-type equationxn1 {A/xn, xn−2},” Nonlinear Analysis:

Theory, Methods & Applications, vol. 71, no. 3-4, pp. 910–922, 2009.

5 E. M. Elsayed, B. Iriˇcanin, and S. Stevi´c, “ On the max-type equationxn1max{An/xn, xn−1},” Ars Combinatoria, vol. 95, pp. 187–192, 2010.

6 E. A. Grove and G. Ladas, Periodicities in Nonlinear Diﬀerence Equations, vol. 4 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC Prss, Boca Raton, Fla, USA, 2005.

7 B. D. Iriˇcanin and E. M. Elsayed, “On a max-type diﬀerence equationxn1{A/xn, xn−3},” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 675413, 2010.

8 B. Iriˇcanin and S. Stevi´c, “On a class of third-order nonlinear diﬀerence equations,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 479–483, 2009.

9 E. P. Popov, Automatic Regulation and Control, Nauka, Moscow, Russia, 1966.

10 S. Stevi´c, “Behavior of the positive solutions of the generalized Beddington-Holt equation,”

Panamerican Mathematical Journal, vol. 10, no. 4, pp. 77–85, 2000.

11 S. Stevi´c, “Boundedness character of a max-type diﬀerence equation,” in Proceedings of the Honour of Allan Peterson Conference, p. 28, Novacella, Italy, July- August 2007, Book of Abstracts.

12 S. Stevi´c, “On the recursive sequencexn1Ax^pn/x^r_n−1,” Discrete Dynamics in Nature and Society, vol.

2007, Article ID 40963, 9 pages, 2007.

13 S. Stevi´c, “On behavior of a class of diﬀerence equations with maximum,” in Proceedings of the Mathematical Models in Engineering, Biology and Medicine. Conference on Boundary Value, p. 35, Santiago de Compostela, Spain, September 2008, Book of abstracts.

14 S. Stevi´c, “On the recursive sequencexn1max{c, x^pn/x_n−1^p },” Applied Mathematics Letters, vol. 21, no.

8, pp. 791–796, 2008.

15 S. Stevi´c, “Boundedness character of a class of diﬀerence equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 839–848, 2009.

16 S. Stevi´c, “Global stability of a diﬀerence equation with maximum,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 525–529, 2009.

17 S. Stevi´c, “Global stability of a max-type diﬀerence equation,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 354–356, 2010.

18 F. Sun, “On the asymptotic behavior of a diﬀerence equation with maximum,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 243291, 6 pages, 2008.

19 H. D. Voulov, “On a difference equation with periodic coefficients,” Journal of Difference Equations and Applications, vol. 13, no. 5, pp. 443–452, 2007.

20 X. Yang, X. Liao, and Chuandong Li, “On a diﬀerence equation with maximum,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 1–5, 2006.