DISCRETE GENERALIZATION OF GRONWALL-BELLMAN INEQUALITY WITH MAXIMA AND APPLICATION * Snezhana Hristova, Kremena Stefanova, Liliana Vankova

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1 МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2012 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2012 Proceedings of the Forty First Spring Conference of the Union of Bulgarian Mathematicians Borovetz, April 9 12, 2012 DISCRETE GENERALIZATION OF GRONWALL-BELLMAN INEQUALITY WITH MAXIMA AND APPLICATION * Snezhana Hristova, Kremena Stefanova, Liliana Vankova Several new types of linear discrete inequalities containing the maximum of the unknown function over a past time interval are solved. Some of these inequalities are applied to difference equations with maximum and the continuous dependence of a perturbation is studied. 1. Introduction. The theory of finite difference equations has been rapidly developed in recent years and also it has been proved the fundamental importance of its applications in modeling of real world problems. At the same time there are many real world processes in which the present state depends significantly on its maximal value on a past time interval. Adequate mathematical models of these processes are the so-called difference equations with maximum. Meanwhile, this type of difference equations is not widely studied yet and there are only some isolated results [3]. Finite difference inequalities which exhibit explicit bounds of unknown functions provide, in generally, a very useful and important tool in the development of the theory of finite difference equations. During the past few years, motivated and inspired by their applications in various branches of difference equations, many such inequalities have been established [1, 2, 4]. The main purpose of the paper is solving of a new type of linear discrete inequalities containing the maximum over a past time interval. Also, they are applied to difference equations with maximum and the continuous dependence on a perturbation is studied. 2. Preliminary notes. Let R + = [0,, N be the set of nonnegative integers i.e. the numbers 0, 1, 2, 3,... and a, b N be such that a < b. Denote by Na, b = {k N : a k b} and Na = {k N : k a}. We assume that m = 0 and m = 1 for n > m. l=n l=n In the proofs of our main results we need the following lemma: Lemma 1 [1, Theorem 4.1.1].Let f, q : Na R +, p, u : Na R and let for all k Na uk pk + qk flul be satisfied. * 2000 Mathematics Subject Classification: 39A22, 26D15, 39B62, 39A10, 47J20. Key words: discrete Gronwall Bellman type inequality, difference equations with maxima, bounds. The research was partially supported by Grand NI11FMI004/ , Fund Scientific Research, Plovdiv University. 180

2 Then, for all k Na the inequality 1 uk pk + qk plfl holds. 1 + qτfτ Remark 1. Note that Lemma 1 is true if pk and uk change sign on Na. 3. Main results. Let a, h N be fixed so that a h. Theorem 1. Let the following conditions be fulfilled: 1. The functions p, g, G : Na R + are nondecreasing. 2. The functions q, Q : Na R +, the function ϕ : Na h, a R +, and max k Na h, a ϕk pa. 3. The function u : Na h R + satisfies the inequalities 2 uk pk + gk 3 uk ϕk, qlul + Gk Ql max us, k Na, k Na h, a. Then, for k Na the inequality 4 uk pk + Sk pl [ ql + Ql ] holds, where Sk = max gk, Gk for k Na. 1 + Sτ [ qτ + Qτ ] Proof. Define the function zk : Na h R + by the equalities pk + gk qlul for k Na, zk = +Gk Ql max us, pa for k Na h, a. For any k Na, k a 1 we have k k zk + 1 = pk gk + 1 qlul + Gk + 1 Ql max us pk gk + 1 pk + gk qlul + Gk + 1 qlul + Gk Ql max us Ql max us = zk. Therefore, the function zk is nondecreasing in Na h. From the definition of the function zk, its monotonicity, inequalities 2, 3 and 181

3 condition 2 it follows that uk zk, k Na h. Therefore, max zs = zk for k Na and for any k Na we obtain s [k h, k] 5 zk pk + gk qlzl + Gk Qlzl [ ] pk + Sk ql + Ql zl. According to Lemma 1, from inequality 5 we get for k Na 6 zk pk + Sk pl [ ql + Ql ] max us s [k h, k] 1 + Sτ [ qτ + Qτ ]. Inequality 6 implies the validity of the required inequality 4. Corollary 1. Let the conditions 2, 3 of Theorem 1 be satisfied where pk p, gk g, Gk G for k Na and p, g, G 0 are constants. Then, uk p 1 + S [ ql + Ql ] for k Na, where S = max g, G. Remark 2. The proof of Corollary 1 is based on the inequality b b b Al 1 + Aτ 1 + Al. Corollary 2. Let the conditions of Theorem 1 be satisfied and gk 1, Gk 1 for k Na. Then uk pksk 1 + Sl [ ql + Ql ] for k Na. Corollary 3. Let the conditions 2, 3 of Theorem 1 be satisfied where pk p = const 0 and gk = Gk = 1 for k Na. Then uk p [ ] 1 + ql + Ql for k Na. 4. Applications. Let a, b N : a < b and the function τ : Na, b N be given such that there exists h N, h a: k h τk k for k Na, b. Let u : N R. Denote uk = uk + 1 uk, k N. Consider the following difference equation with maxima : 7 uk = f k, uk, max us, k Na, b, s Nτk, k and its perturbed difference equation with maxima : 8 vk = f k, vk, max vs + g k, vk, k Na, b s Nτk, k with initial conditions 9 uk = ϕk, k Na h, a. 182

4 Remark 3. Note that in the case τk k the equation with maxima 7 reduces to a difference equation which is well-known in the literature [1]. In our further investigations we assume that the initial value problems 7, 9 and 8, 9 have solutions uk and vk for k Na h, b. Theorem 2 Continuous dependence on the perturbation. Let the following conditions be fulfilled: 1. The function f : Na, b R R R satisfies Liptschitz s condition fk, x 1, y 1 fk, x 2, y 2 λk x 1 x 2 + λk y 1 y 2, where x i, y i R, k Na, b and the functions λ, λ : Na, b R The function g : Na, b R R satisfies the condition 10 gk, ζ µk, k Na, b, ζ R, where the function µ : Na, b R + and b µk = M <. 11 k=a 3. The function ϕ : Na h, a R. Then, for k Na, b the following inequality holds [ ] uk vk M 1 + λl + λl. Proof. The functions uk and vk satisfy the equalities uk = vk for k Na h, a and for k Na, b: uk = ϕa + f l, ul, max us s Nτl, l 12 vk = ϕa + f l, vl, max vs + g l, vl. s Nτl, l Then, since Nτk, k [k h, k], we get for k Na uk vk M + λl ul vl + λl max us vk. According to Corollary 3, from inequality 12 we obtain inequality 11. Example. Consider the linear difference equation with maxima 13 uk + 1 = Auk + B max us, s Nτk, k and the perturbed difference equation with maxima k Na, b, 14 vk + 1 = Avk + B max s Nτk, k vs + e vk 2 k, k Na, b, with initial conditions uk = vk = C, k Na h, a, where A = const 1, B = const 0, C = const 0. Then, λk A 1 0 and λk B 0 for k Na, b. 183

5 Let g k, v = e v 2 k. Then µk = 1 2 k and b Theorem 2, we get uk vk 2 A + B k a 1, k Na + 1, b. k=a 1 2 = M <. According to 2k REFERENCES [1] R. P. Agarwal. Difference Equations and Inequalities: Theory, Methods and Applications. CRC Press, [2] S. Elaydi. Introduction to Difference Equations and Inequalities: Theory, Methods and Applications. CRC Press, [3] J. W. Luo, D. D. Bainov. Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima. J. Comput. Appl. Math., [4] C. Wing-Sum, M. Qing-Hua, J. Pecaric. Some discrete nonlinear inequalities and applications to difference equations, Acta Math. Sci., 28B2, 2008, Snezhana Hristova Kremena Stefanova Liliana Vankova Faculty of Mathematics and Informatics Plovdiv University 236, Bulgaria Blvd 4000 Plovdiv, Bulgaria snehri@uni-plovdiv.bg kstefanova@uni-plovdiv.bg lilqna.v@gmail.com ДИСКРЕТНО ОБОБЩЕНИЕ С МАКСИМУМИ НА НЕРАВЕНСТВОТО НА ГРОНУОЛ-БЕЛМАН И ПРИЛОЖЕНИЯ Снежана Христова, Кремена Стефанова, Лиляна Ванкова В работата са решени няколко нови видове линейни дискретни неравенства, които съдържат максимума на неизвестната функция в отминал интервал от време. Някои от тези неравенства са приложени за изучаване непрекъснатата зависимост от смущения при дискретни уравнения с максимуми. 184