Research Article Advanced Discrete Halanay-Type Inequalities: Stability of Difference Equations

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1 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 29, Article ID , 11 pages doi:1.1155/29/ Research Article Advanced Discrete Halanay-Type Inequalities: Stability of Difference Equations Ravi P. Agarwal, 1 Young-Ho Kim, 2 and S. K. Sen 1 1 Department of Mathematical Sciences, Florida Institute of Technology, 15 West University Boulevard, Melbourne, FL , USA 2 Department of Applied Mathematics, Changwon National University, Changwon, Kyeong-Nam , South Korea Correspondence should be addressed to Ravi P. Agarwal, agarwal@fit.edu Received 7 December 28; Accepted 21 January 29 Recommended by Martin J. Bohner We derive new nonlinear discrete analogue of the continuous Halanay-type inequality. These inequalities can be used as basic tools in the study of the global asymptotic stability of the equilibrium of certain generalized difference equations. Copyright q 29 Ravi P. Agarwal et al. 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 The investigation of stability of nonlinear difference equations with delays has attracted a lot of attention from many researchers such as Agarwal et al. 1 3, Baĭnov and Simeonov 4, Bay and Phat 5, Cooke and Ivanov 6, Gopalsamy 7, Lizetal. 8 1, Niamsup et al. 11, 12, Mohamad and Gopalsamy 13, Pinto and Trofimchuk 14, and references sited therein. In 15, Halanay proved an asymptotic formula for the solutions of a differential inequality involving the maximum functional and applied it in the stability theory of linear systems with delay. Such an inequality was called Halanay inequality in several works. Some generalizations as well as new applications can be found, for instance, in Agarwal et al. 2, Gopalsamy 7, Lizetal. 8 1, Niamsup et al. 11, 12, Mohamad and Gopalsamy 13, and Pinto and Trofimchuk 14. In particular, in 2, 6, 1, 12, 13, the authors considered discrete Halanay-type inequalities to study some discrete version of functional differential equations. In the following results of Liz et al. 1, authors showed that some discrete versions of these maximum inequalities can be applied to study the global asymptotic stability of a family of difference equations.

2 2 Journal of Inequalities and Applications Theorem A. Assume that u, v satisfies the system of inequalities Δu n Au n Bũ n Cv n D v n, n, v n Eu n Fũ n, n, 1.1 where Δu n u n 1 u n, ũ n max{u n,...,u n r }, v n max{v n 1,...,v n r }, and r> is a natural number. If B, C, D, E, F, FD B>, E F> and B E F C D <A 1, 1.2 then there exist constants K 1, K 2, and λ, 1 such that u n K 1 λ n, v n K 2 λ n, n. 1.3 Moreover, λ can be chosen as the smallest root in the interval, 1 of equation h λ, where h λ λ 2r 1 1 A CE λ 2r B FC ED λ r FD. 1.4 By a simple use of Theorem A, authors also demonstrated the validity of the following statement, namely, Theorem B. Theorem B. Assume that f satisfies the following inequalities: f ( n, x n,...,x n r ( x n,...,x n r, ( x n,...,x n r R r 1, f ( n, x n,...,x n r xn r ( Δx n 1,...,Δx n r, ( x n,...,x n r R r If either a a 1 b, and <br<1, or b a<, and <br< a b a b 1 holds, then there exist K>and λ, 1 such that for every solution {x n } of Δx n ax n bf ( n, x n,x n 1,...,x n r, a >, 1.6 one has ( { } xn max xi λ n, n, 1.7 where λ can be calculated in the form established in Theorem A. As a consequence, the trivial solution of 1.6 is globally asymptotically stable. The main aim of the present paper is to establish some new nonlinear retarded Halanay-type inequalities, which extend Theorem A, along with the derivation of new global stability conditions for nonlinear difference equations.

3 Journal of Inequalities and Applications 3 2. Halanay-Type Discrete Inequalities Let R denote the set of all real numbers, R the set of positive real numbers, R the set of nonnegative real numbers, Z the set of integers, Z the set of positive integers, and Z r {z Z : z r}. Consider the following nonlinear difference equation: Δx n f ( n, x n,x n 1,...,x n r, n Z, 2.1 where Δx n x n 1 x n,andf : N R r 1 R. Equation 2.1 is a generalized difference equation see 3, Section 21 and 11. The initial value problem for this equation requires the knowledge of the initial data {x r,x r 1,...,x }. This vector is called the initial string in 6. For every initial string, there exists a unique solution {x n } n Z r of 2.1 that can be calculated using the explicit recurrence formula x n 1 x n f ( n, x n,x n 1,...,x n r, n Z. 2.2 In this section, we introduce new discrete inequalities which will be used to derive global stability conditions in the next section. Theorem 2.1. Let a i,b i,c i,d i,e i,f i R, r d ie>, h i Z i,...,r, h <h 1 < <h r ; h r Z, and b c d e f <a 1, 2.3 where a r a i,b r b i,c r c i, d r d i,e r e i, and f r f i. Also, let {u n,v n } n Z hr be a sequence of nonnegative real numbers satisfying the system of inequalities Δu n ( ai u n b i u p n h i c i v n d i v n hi, n Z, v n (e i u n f i u pn hi, n Z, 2.4 where p is a constant. Then there exist constants K 1, K 2, and λ, 1 such that u n K 1 λ n, v n K 2 λ n, n Z, 2.5 where K 1 max i r {u hi,α 1 v hi }, and K 2 αk 1 with α e r f iλ n n h i p. Moreover, λ can be chosen as the smallest root in the interval, 1 of equation g λ, where g λ λ 1 a ce d i eλ h i ( bi cf i λ n h i p n ( 2.6 d i f j λ n h j h i p n with n Z.

4 4 Journal of Inequalities and Applications Proof. Let {x n,y n } n Z hr inequalities be a sequence of nonnegative real numbers satisfying the system of x n 1 a n n 1 x 1 a n j 1 y n ( ai x j b i x p j h i c i y j d i y j hi, (e i x n f i x pn hi, 2.7 where n Z. Since 1 a, it is easy to prove by induction that if u n x n and v n y n for n h r,...,, then u n x n and v n y n for all n Z. On the other hand, the system 2.7 is equivalent to Δx n y n ( ai x n b i x p n h i c i y n d i y n hi, (e i x n f i x pn hi, 2.8 where n Z. Next we prove, under the assumptions of the theorem, that there exists a solution {x n,y n } n Z hr to system 2.8 in the form x n λ n,y n αλ n with α>, λ, 1. Indeed, such {x n,y n } n Z hr is a solution of 2.8 if and only if ( 1 a λ n λ n 1 ( αλ n b i λ n h i p c i αλ n d i e i λ n f iλ n h i p ( αλ n h i, n Z,, n Z. 2.9 This is equivalent to the existence of a solution λ, 1 of equation g λ, where g is the polynomial defined by 2.6. Now, g lim λ g λ < inviewof r d ie >. On the other hand, g 1 a b c d e f > inviewof 2.3. As a consequence, there exists λ, 1 such that g λ. Hence, λ,α is a solution of 2.9 with α e r f iλ n n h i p >. For this value of λ, the pair {Kλ n,kαλn } is a solution of 2.8 for every K. Thus, choosing K max i r {u hi,α 1 v hi }, we have that u n Kλ n x n,andv n Kαλ n y n for all n h r,...,. Hence, using the first part of the proof, we can conclude that u n x n,andv n y n for all n Z. By the similar argument used in Theorem 2.1, we obtain the following result.

5 Journal of Inequalities and Applications 5 Theorem 2.2. Let a, b, c, d, e, f R,h i Z,i,...,r, h <h 1 < <h r ; r 1, and b c e f d e f r 1 <a with ce >. Also, let {u n,v n } n Z hr of inequalities be a sequence of nonnegative real numbers satisfying the system Δu n au n bu n hi cv n dv n hi, n Z, v n eu n fu n hi, n Z Then there exist constants K 1, K 2, and λ, 1 such that u n K 1 λ n, v n K 2 λ n, n Z, 2.12 where K 1 max i r {u hi,ρ 1 v hi }, and K 2 ρk 1 with ρ e f r λ h i. Moreover, λ can be chosen as the smallest root in the interval, 1 of equation F λ, where F λ λ 1 a ce [ b cf d ( e fλ rn h r 1] λ rn h 2.13 with n Z,h r h i. Proof. Let {x n,y n } n Z hr inequalities be a sequence of nonnegative real numbers satisfying the system of Δx n ax n b x n hi cy n d y n hi, n Z, y n ex n f x n hi, n Z Since 1 a, it is easy to prove by induction that if u n x n and v n y n for n h r,...,, then u n x n and v n y n for all n Z. Next we prove that, under the assumptions of the theorem, there exists a solution {x n,y n } n Z hr to system 2.14 in the form x n λ n,y n ρλ n with ρ>, λ, 1. Indeed, such {x n,y n } n Z hr is a solution of 2.14 if and only if λ n 1 1 a λ n b λ n h i cρλ n d ρλ n h i, n Z, ρλ n eλ n f λ n h i, n Z. 2.15

6 6 Journal of Inequalities and Applications This is equivalent to the existence of a solution λ, 1 of equation F λ, where F is the polynomial defined in Now, in view of ce >, we have F 1 a ac < in case rn > h, F 1 a ac b cf d e f r 1 < in case rn h, and F lim λ F λ < in case rn < h. On the other hand, F 1 a b c e f d e f r 1 > inviewof 2.1. Asa consequence, there exists λ, 1 such that F λ. Hence, λ,ρ is a solution of 2.15 with ρ e f r λ h i >. For this value of λ, the pair {Kλ n,kρλn } is a solution of 2.14 for every K. Thus, choosing K max i r {u hi,ρ 1 v hi }, we have u n Kλ n,andv n Kρλ n for all n h r,...,. These imply u n x n,andv n y n for all n h r,...,. Hence, using the first part of the proof, we can conclude that u n x n,andv n y n for all n Z. Remark 2.3. In 1, a discrete Halanay-type inequality was given as in Theorem A, where the inequalities were replaced by Δu n au n bũ n cv n d v n, n, v n eu n fũ n, n, 2.16 where Δu n u n 1 u n, ũ n max{u n,...,u n r }, v n max{v n 1,...,v n r }, and r 1isa natural number. Note that if a sequence {u n } n Z r of positive real numbers satisfies 2.16, then it also satisfies 2.4. On the other hand, let p r 1, h i i; a 1 a i 1, b b b 1 1/7, c c c 1, d d d 1 1/7, e e e 1 1/7, and f f f 1 1/7. Then we might easily show that the sequence {1/2 n } n Z 1 satisfies 2.4 but not Indeed, Δu n n 1 2 n 1 2, n 1 < 1 2 n 1 ( n 1 1 ( n n n n, 2.17 with 1 b i 1 c i 1 d i 1 e i 1 f i < 1 a i 1. On the other hand, Δu n 1 2 n 1 > 1 2 n 1 7 max { 1 2 n, 1 2 n n. } 1 ( n 1 7 max { } 1 2 n Therefore, in the case of positive sequences, the discrete inequality 2.4 is less conservative than the discrete Halanay-type inequality given by 2.16.

7 Journal of Inequalities and Applications 7 3. Global Stability of Difference Equations In order to show the applicability of the previous result, in this section we consider the generalized difference equation Δx n ax n bf ( n, x n,x n h1,...,x n hr, 3.1 where n, h i Z, i 1,...,r, and b>. Although, for every initial string {x hr,x hr 1,...,x }, the solution {x n } of 3.1 can be explicitly calculated by a recurrence formula similar to 2.2, it is in general difficult to investigate the asymptotic behavior of the solutions using that formula. The next result gives an asymptotic estimate by a simple use of the discrete Halanay inequality. Theorem 3.1. For all n, x n,x n h1,...,x n hr Z R r 1, assume that f satisfies the following inequalities: ( f n, xn,x n h1,...,x n hr β xn hj j p, f ( n, x n,x n h1,...,x n hr xn γ j Δx n hj, where β j,γ j,p R, r γ i a >, h j Z j,...,r 1, and h r Z with h <h 1 < <h r. If either a a 1 b, <bγ<1, and <β 1, or b a<and <bγ< a b a bβ 1 hold, then there exists a constant λ, 1 for every solution {x n } of 3.1 such that ( { } xn max xi,α 1 1 Δxi λ n, n h r i Z, 3.4 where α 1 a b r β iλ n n h i p,β r β i,γ r γ i, and λ can be chosen as the smallest root in the interval, 1 of equation g 1 λ, where g 1 λ λ 1 a b b a γ i λ h i ( bγ i bβ j λ n h j h i p n 3.5 with n Z. As a consequence, the trivial solution of 3.1 is globally asymptotically stable. Proof. Let {x n } be a solution of 3.1. Equation 3.1 can be written in the form Δx n a b x n b [ f ( n, x n,x n h1,...,x n hr xn ]. 3.6

8 8 Journal of Inequalities and Applications Hence, we know that x n [ 1 a b ] n 1 n [ ] n i 1 b [ ( ] x 1 a b f i, xi,x i h1,...,x i hr xi, 3.7 where n Z. Thus, using inequality 3.3, weobtain [ ] xn n n 1 1 a b x [ ] n i 1bγj 1 a b Δxi hj. 3.8 Denote u n x n for n h r,...,, and u n [ 1 a b ] n n 1 x [ ] n i 1bγj 1 a b Δxi hj 3.9 for n Z. Then we have x n u n and, from inequality 3.9,weobtain Δu n a b u n bγ Δxn hj j 3.1 for n Z. On the other hand, using hypothesis 3.2 in 3.1, we have Δxn a xn b β xn hj j p a u n b β j u p n h j Denote v n Δx n. We can apply Theorem 2.1 to the system of inequalities 3.1 and 3.11 with r a i a b, b i, c i, d i bγ j, r e i a, and f i bβ j. Consequently, Theorem 2.1 ensures the validity of the following inequality: ( { x n max x i,α 1 } 1 Δx i λ n, n h r i Z, 3.12 where λ and α 1 arechosenasintheorem 3.1. This completes the proof of the theorem. Next, we obtain new conditions for the asymptotic stability of 3.1 using inequality 3.13 instead of 3.3.

9 Journal of Inequalities and Applications 9 Corollary 3.2. For all n, x n,x n h1,...,x n hr Z R r 1, assume that f satisfies inequality 3.2 and the following condition: f ( n, x n,x n h1,...,x n hr xn ( γj x n hj p δ j Δx n η j Δx n hj, 3.13 where γ j,δ j,η j,p R, r η j a >, h j Z j,...,r 1, and h r Z with h <h 1 < <h r. If bγ b δ η a bβ <a b holds, where β r β i, γ r γ i,δ r δ i, and η r η i, then there exists a constant λ, 1 for every solution {x n } of 3.1 such that ( { } xn max xi,α 1 1 Δxi λ n, n h r i Z, 3.15 where α 1 a b r β iλ n n h i p, and λ can be chosen as the smallest root in the interval, 1 of equation g 2 λ, where g 2 λ λ ( 1 a b b a δ b a η i λ h i b ( γ i bδβ i λ n h i p n ( 3.16 bη i bβ j λ n h j h i p n with n Z. As a consequence, the trivial solution of 3.1 is globally asymptotically stable. Similarly, using Theorem 2.2 instead of Theorem 2.1, we obtain the following result. Theorem 3.3. For all n, x n,x n h1,...,x n hr Z R r 1, assume that f satisfies the following inequalities: ( f n, xn,x n h1,...,x n hr β xn hj, ( f n, xn,x n h1,...,x n hr xn γ xn hj δδxn η Δxn hj, 3.17 where β, γ, δ, η R,h j Z,j,...,r 1, and h r Z with h <h 1 < <h r. If a δ > and bγ bδ ( a bβ bη ( a bβ r 1 <a b 1, 3.18

10 1 Journal of Inequalities and Applications then there exists a constant λ, 1 for every solution {x n } of 3.1 such that ( xn max { xi,ρ 1 1 Δxi} λ n, n h r i Z, 3.19 where ρ 1 a bβ r λh i, and λ can be chosen as the smallest root in the interval, 1 of equation F 1 λ, where F 1 λ λ ( 1 a b a bδ [ b γ bβδ η ( a bβλ rn h ] r 1 λ rn h 3.2 with n Z,h r h i. As a consequence, the trivial solution of 3.1 is globally asymptotically stable. Remark 3.4. Equation 3.1 covers a variety of difference equations. For instance, we can consider the following difference equation: Δx n ax n bf ( x n k, b > Next, we study the asymptotic behavior of the solutions of We can apply Theorem 3.1, Corollary 3.2,or Theorem 3.3 to obtain some relations between coefficients a and b that ensure the global asymptotic stability of the zero solution. Moreover, from Theorem 3.1 we know that if there exists β, γ R such that f x β x p, f x x γ Δx for all x, and if either a <a 1 b, <bγ<1, and <β 1, or b a<and<bγ< a b a bβ 1 hold, then all solutions of 3.21 converge to zero. Acknowledgment The authors thank the referees of this paper for their careful and insightful critique. References 1 R. P. Agarwal, S. Deng, and W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Applied Mathematics and Computation, vol. 165, no. 3, pp , R. P. Agarwal, Y.-H. Kim, and S. K. Sen, New discrete Halanay inequalities: stability of difference equations, Communications in Applied Analysis, vol. 12, no. 1, pp. 83 9, R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, vol. 44 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, D. Baĭnov and P. Simeonov, Integral Inequalities and Applications, vol. 57 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, N. S. Bay and V. N. Phat, Stability analysis of nonlinear retarded difference equations in Banach spaces, Computers & Mathematics with Applications, vol. 45, no. 6 9, pp , K. L. Cooke and A. F. Ivanov, On the discretization of a delay differential equation, Journal of Difference Equations and Applications, vol. 6, no. 1, pp , 2. 7 K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.

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