IJRRAS (4) ● March 2011 www.arpapress.com/Volumes/Vol6Issue4/IJRRAS_6_4_06.pdf DISCRETE INEQUALITY OF GRONWALL – BELLMAN TYPE E.M.Roshdya & M.S Mousab Department of Mathematics, Military Technical College, Cairo, Egypt b Department of Basic Sciences, Arab academy for Science, Technology and Maritime Transport a ABSTRACT Some new discrete inequalities of Gronwall – Bellman type that have a wide range of application in the theory of finite difference equations are given INTRODUCTION A powerful technique in the study of many problems concerning the behavior of solutions of discrete time systems is to use the recurrent inequalities involving sequences of real numbers, which may be considered as discrete analogues of Gronwall – Bellman inequality [3] or its variants In recent years there have several linear and nonlinear discrete generalization of this useful inequality for instance see [1, 2, 4, 5].The aim of this paper is to establish some useful discrete inequalities which claim the following as their origin Lemma a [sygiyama (5)] :let x (n) and f (n) be real valued functions defined for 𝑛 ∈ 𝑁 and suppose that f (n) ≥ 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑛 ∈ 𝑁 𝑖𝑓 𝑛−1 𝑥 𝑛 ≤ 𝑥0 + 𝑓 𝑠 𝑛 𝑠 , 𝑛∈𝑁 𝑠=𝑛 Where N is the set of positive 𝑛0 + 𝑘 𝑘 = 0,2, … , 𝑛0 ≥ 𝑜 is a given integer and 𝑥0 is a nonnegative constant then 𝑛−1 𝑥 𝑛 ≤ 𝑥0 1+𝑓 𝑠 , 𝑛∈𝑁 𝑠=𝑛 Many application of this lemma or its variants may be found in [4], [5] as in numerous books and papers MAIN RESULTS Before giving the main results we need the following notions and definitions: Let N be a set of points 𝑛0 + 𝑘 𝑘 = 0,1,2 … 𝑤𝑒𝑟𝑒 𝑛0 ≥ is given integer The expression 𝑛−1 𝑠=𝑛 𝑏 𝑠 represents a solution of linear difference equation ∆𝑥 𝑛 = 𝑏 𝑛 , ∀ 𝑛 ∈ 𝑁 such that 𝑥 𝑛0 = ∆𝑥 𝑛 = 𝑥 𝑛 + − 𝑥 𝑛 It is supposed that 𝑛−1 𝑠=𝑛 𝑏 𝑠 = 𝑛 −1 The expression 𝑠=𝑛 𝑐(𝑠) represents a solutions of the linear difference equation x n+1 =c n x n , n∈𝑁 𝑛 −1 Where x (𝑛0 ) = it is supposed that 𝑠=𝑛 𝑐(𝑠) = THEOREM 1: Let x (n), f (n) and g (n) be real -valued non negative functions defined on N, for which the inequality 𝑠−1 𝑛−1 𝛼 𝑥 𝑛 ≤ 𝑥0 + 𝑛+1 𝑠=𝑛 𝑓 𝑠 𝑥 𝑠 + 𝑠=𝑛 𝑓 𝑠 ( 𝑡=𝑛 𝑔 𝑡 𝑥 𝑡 ) , 𝑛 ∈ 𝑁 Holds, where 𝑥0 is a nonnegative constant and ≤ 𝛼 ≤ 𝑠−1 Then 𝑥 𝑛 ≤ 𝑥0 + 𝑛−1 𝑠=𝑛 𝑓 𝑡 𝑡=𝑛 + 𝑓 𝜏 ∗ 𝑥0 1−𝛼 + (1 − 𝛼) 𝑠−1 𝜏=𝑛 𝜏 𝑡=𝑛 [1 𝑔(𝜏) + 𝑓 𝑡 ]𝛼 −1 𝑡−𝛼 (1) (2) Where all 𝑛 ∈ 𝑁 Proof: Define a function u (n) by right member of (1) thus 𝑛−1 ∆𝑢 𝑛 = 𝑓 𝑛 𝑔 𝑡 𝑥𝛼 𝑡 𝑥 𝑛 + , 𝑢 𝑛0 = 𝑥0 𝑡=𝑛 Which in view of (1) implies ∆𝑢 𝑛 ≤ 𝑓 𝑢 [𝑢 𝑛 + 𝛼 𝛼 Since 𝑥 𝑛 ≤ 𝑢 𝑛 if we put 416 𝑛 −1 𝑡=𝑛 𝑔(𝑡)𝑢 𝑡 𝛼 ] (3) IJRRAS (4) ● March 2011 Roshdy & Mousa ● Discrete Inequality of Gronwall – Bellman Type 𝛼 V (n) =u (n) + 𝑛−1 (4) 𝑡=𝑛 𝑔 𝑡 𝑢 𝑡 , 𝑣 𝑛0 = 𝑢 𝑛0 = 𝑥0 It follows from (3) and (4) and the fact that u n ≤ 𝑣 𝑛 the inequality 𝑣 𝑛 + − 𝑣 𝑛 ≤ 𝑓 𝑛 𝑣 𝑛 + 𝑔(𝑛)𝑣 𝛼 (𝑛) (5) is satisfied Define −1 𝑒 𝑛 = 𝑛𝑠=𝑛 + 𝑓 𝑠 −1 , 𝑒 𝑛0 = Then 𝑒 𝑛 + − 𝑒 𝑛 = −𝑓 𝑛 𝑒 𝑛 + (6) Multiplying both sides of (5) of by e (n+1) and using (6) we obtain 𝑣 𝑛 + 𝑒 𝑛 + − 𝑣 𝑛 𝑒 𝑛 ≤ 𝑔 𝑛 𝑒 𝑡−𝛼 (𝑛 − 1)[𝑣 𝑛 𝑒 𝑛 + ]𝛼 (7) Because v (n) is monotonic increasing, e (n) is monotonic decreasing 𝛼 ≤ we know that [𝑣 𝑛 𝑒 𝑛 + ]𝛼 ≥ 𝑧 −𝛼 , For all values of z between v (n) e (n) and v (n+1) e (n+1) So if we apply the mean value the theorem to the function 𝑓 𝑧 = 𝑧 1−𝛼 1−𝛼 We see that 𝑣 𝑛+1 𝑒 𝑛+1 1−𝛼 − 𝑣 𝑛 𝑒 𝑛 1−𝛼 1−𝛼 −𝛼 ≤ 𝑣 𝑛 𝑒 𝑛+ [𝑣 𝑛 + 𝑒 𝑛 + − 𝑣 𝑛 𝑒 𝑛 ] (8) From (7) & (8) we obtain 𝑣 𝑛 + 𝑒 𝑛 + 1−𝛼 − [𝑉 𝑛 𝑒 𝑛 ]1−𝛼 ≤ − 𝛼 𝑔(𝑛)𝑒(𝑛 + 1)1−𝛼 Summing up both sides of (9) from 𝑛0 𝑡𝑜 𝑛 − , 𝑤𝑒 𝑎𝑣𝑒 1−𝛼 [𝑉 𝑛 𝑒 𝑛 ]1−𝛼 − 𝑥𝑜 1−𝛼 ≤ (1 − 𝛼) 𝑛−1 (𝑠 + 1) 𝑠=𝑛 𝑔 𝑠 𝑒 From (10) we have (9) (10) 𝑉 𝑛 ≤ 𝑛−1 𝑠=𝑛 + 𝑓 𝑠 Substituting the value of v (n) in (3) we have 𝑥0 1−𝛼 + − 𝛼 𝑛−1 𝑛−1 𝑠=𝑛 𝑛−1 + 𝑓 𝑠 [𝑥𝑜 1−𝛼 + − 𝛼 ∆𝑢 𝑛 ≤ 𝑓(𝑛) 1+𝑓 𝑡 𝑠 𝛼 −1 1−𝛼 (11) [(1 + 𝑓(𝑡)]𝛼−1 ]1−𝛼 𝑔 𝑠 𝑠=𝑛 𝑠 𝑡=𝑛 𝑔 𝑠 𝑠=𝑛 𝑡=𝑛 Which implies the statements for u (n) such that 𝑛−1 𝑢 𝑛 ≤ 𝑥𝑜 + 𝑠−1 𝑠=𝑛 𝑠−1 𝜏 [1 + 𝑓(𝜏)] ∗ 𝑥0 1−𝛼 + (1 − 𝛼) 𝑓(𝑢) 𝜏=𝑛 [1 + 𝑓 𝑡 ]𝛼 −1 𝑔(𝜏) 𝑡=𝑛 1−𝛼 𝑡=𝑛 Now substituting into this value of u (n) in (1) we obtain the desired boned (2) Another interesting and useful discrete inequality which may be considered as discrete generalization of theorem A is embodied in the following theorem THEOREM 2: Let x (n), f (n) and g (n) be real valued nonnegative functions defined on N for which the inequality 𝑛−1 𝑥 𝑛 ≤ 𝑥0 + 𝑛−1 𝑓 𝑠 𝑥 𝑠 + 𝑠=𝑛 𝑠−1 𝑓 𝑠 [ 𝑠=𝑛 𝜏=𝑛 𝜏−1 𝑔(𝑡) 𝑥 𝛼 𝑡 }] 𝑓 𝜏 { 𝑡=𝑛 Holds for all n ∈ 𝑁 , 𝑥0 is a nonnegative constant 𝑜 ≤ 𝛼 ≤ Then 𝑛−1 𝑥 𝑛 ≤ 𝑥0 + 𝑠−1 𝑓 𝑠 {𝑥0 + 𝑠=𝑛 𝜏−1 + 𝑓(𝑡) ∗ 𝑥0 1−𝛼 + (1 − 𝛼) 𝑓(𝑧) 𝜏=𝑛 𝜏−1 𝑡=𝑛 𝑔(𝑡) 𝑡=𝑛 1−𝛼 𝑡 + 𝑓(𝜏) 𝛼 −1 𝜏=𝑛 For all 𝑛 ∈ 𝑁 Proof: The proof follows by a similar argument as in the proof of theorem with suitable modifications and hence omit the details Our next theorem is a slight variant of theorem this has an added advantage in establishing a more general discrete inequality THEOREM 3: Let x (n), f (n) and g (n) be real valued nonnegative function defined on N and h(n) be a positive monotonic non decreasing function defined on N for which the inequality 𝑛−1 𝑠−1 1−𝛼 𝛼 𝑥 𝑛 ≤ 𝑛 + 𝑛−1 𝑥 (𝑡)] (12) 𝑠=𝑛 𝑓 𝑠 𝑥 𝑠 + 𝑠=𝑛 𝑓 𝑠 [ 𝑠=𝑛 𝑔 𝑡 𝑡 Holds for all ∈ 𝑁 , where 𝑜 ≤ 𝛼 ≤ then 𝑥 𝑛 ≤𝑘 𝑛 𝑛 (13) 417 IJRRAS (4) ● March 2011 Roshdy & Mousa ● Discrete Inequality of Gronwall – Bellman Type 𝑠−1 𝑠−1 𝑡 Where 𝑘 𝑛 = + 𝑛−1 𝑠=𝑛 𝑓 𝑠 𝑡=𝑛 [ + 𝑓(𝑠)] ∗ + (1 − 𝛼) 𝑡=𝑛 𝑔(𝜏) 𝑡=𝑛 + 𝑓(𝜏) Proof: since h (n) is positive monotonic and non decreasing we observe from (12) that 𝑥(𝑛) (𝑛) ≤ 1+ ≤1+ 𝑛−1 𝑠=𝑛 𝑛−1 𝑠=𝑛 𝑓(𝑠) 𝑓 𝑠 𝑥(𝑠) (𝑛) 𝑥 𝑠 𝑠 + + 𝑛−1 𝑠=𝑛 𝑛−1 𝑠=𝑛 𝑓 𝑠 [ 𝑓 𝑠 ( 𝑥 𝑛 𝑛 𝑠−1 𝑡=𝑛 𝑠−1 𝑡=𝑛 ≤ 1+ 𝑔 𝑡 1−𝛼 (𝑡) 𝑔 𝑡 1−𝛼 (𝑡) 𝑛−1 𝑠=𝑛 𝑓 𝑠 𝑥 𝑠 𝑠 𝛼−1 1−𝛼 (14) 𝑥 𝛼 (𝑡) (𝑛) 𝑥𝛼 𝑡 𝑡 + ) 𝑛−1 𝑠=𝑛 𝑓 𝑠 𝑠−1 𝑡=𝑛 𝑔 𝑡 𝑥 𝑡 𝛼 (15) 𝑡 Now by application theorem (1) we obtain the desired bound in (13) We now apply the results in theorem (3) to establish the following More general inequality in which the monotone character of h (n) plays an important role THEOREM 4: Let z(n), f(n), g(n),p(n)and q(n) be real valued positive function defined on N ; W(n,r) be a positive, monotonic ,non decreasing in r where 𝑟 > For each fixed 𝑛 ∈ 𝑁 the function 𝑚(𝑢) > , 𝐻 𝑛 ≥ be non decreasing in n 𝐻 𝑜 = and suppose further that inequality 𝑛−1 𝑥 𝑛 ≤ 𝑚 𝑛 + 𝑝 𝑛 𝐻 𝑠−1 𝑠=𝑛 𝑞 𝑠 𝑊(𝑠, 𝑥 𝑠 ) + 𝑠=𝑛 𝑓 𝑠 𝑥(𝑠) 1−𝛼 𝑠−1 𝑡−1 + 𝑛−1 (16) 𝑠=𝑛 𝑓(𝑠) 𝑠=𝑛 𝑞 𝜏 𝑊(𝜏, 𝑥(𝜏) 𝑡=𝑛 𝑔(𝑡) 𝑚 𝑡 + 𝑝 𝑡 𝐻 is satisfied for all n ∈ N , ≤ α < then x n ≤ k n m n + p n H(r n ) , n ∈ N (17) Where k (n) is defined in (14) and r (n) is solution of ∆r n = q n W n, k n m n + p n H(r n ) , r n0 = (18) existing on N Proof: Define h(n) = m (n) + p (n) H 𝐧−𝟏 𝐬=𝐧𝟎 (𝐪 𝐬 𝐖(𝐬, 𝐱(𝐬) (19) 𝐧−𝟏 𝐧−𝟏 𝟏−𝛂 𝛂 Then (16) can be restated as 𝐱 𝐧 ≤ 𝐤 𝐧 + 𝐧−𝟏 𝐟 𝐬 𝐱 𝐬 + 𝐟(𝐬) 𝐠 𝐭 𝐡 𝐭 ∗ 𝐳 (𝐭) 𝐬=𝐧𝟎 𝐬=𝐧𝟎 𝐭=𝐧𝟎 Since h (n) is a positive, monotonic, non decreasing on N we have from theorem (3) 𝐱 𝐧 ≤ 𝐡 𝐧 𝐤(𝐧) (20) Where k (n) is given in (14) now from (19) and (20), we have 𝐱 𝐧 ≤ 𝐤(𝐧) 𝐦 𝐧 + 𝐩 𝐧 𝐇(𝐳(𝐧) (21) Where 𝐳 𝐧 = 𝐧−𝟏 𝐠 𝐬 𝐖 𝐬, 𝐱 𝐬 , 𝐳 𝐧 = 𝟎 𝐬=𝐧𝟎 𝟎 Consequently it follows that ∆𝐳 𝐧 ≤ 𝐠 𝐧 𝐖(𝐧, 𝐤(𝐧) 𝐦 𝐧 + 𝐩 𝐧 𝐇(𝐳(𝐧) (22) A suitable application of theorem (1) yields 𝐳 𝐧 ≤𝐫 𝐧 , 𝐧∈𝐍 (23) Where r (n) is the solution of (18) such that 𝐭 𝐧𝟎 = 𝐳 𝐧𝟎 = 𝟎 Now from (21) and (23) the desired bound is follows REMARKS: (1) In theorem 1, 2, and we presented inequalities which generalize theorem A (2)Theorem and can be used to establish similar results as boundedness and asymptotic behavior of solution of perturbed difference equations of the form 𝐱 𝐧 + 𝟏 = 𝐀 𝐧 𝐱 𝐧 + 𝐟 𝐧, 𝐱 𝐧 𝐓 𝐱 𝐱 𝐧𝟎 = 𝐱 𝟎 𝐲 𝐧+𝟏 =𝐀 𝐧 𝐲 𝐧 , 𝐲 𝐧𝟎 = 𝐱 𝟎 𝛂 Where T is the operator defined by 𝐓𝐱 𝐧 = 𝐧−𝟏 𝐬=𝐧𝟎 𝐤(𝐧, 𝐬, 𝐱 (𝐬)), 𝟎 ≤ 𝛂 < 𝟏 𝐬−𝟏 𝛂 Or 𝐓𝐱 𝐧 = 𝐧−𝟏 𝐬=𝐧𝟎 𝐤 𝐧, 𝐬, 𝐭=𝐧𝟎 𝐠(𝐬, 𝐭, 𝐱 𝐭 ) , 𝟎 ≤ 𝛂 < 𝟏 REFERENCES [1] [2] [3] [4] [5] H.el owaidy, A.ragab and A Abdediam on some new integral inequalities of Grronwall-Bellman type J.Math Comp 106(1999)289-303 G.Jones fundamental inequalities for discrete and discontinuous functional equations J.Soc.Ind Math 12(1964) 43-5-7 B.G.Pachpatte: finite difference inequalities and their applications.proc.Nat.Acad.sci India 43, (1973) 348-366 B.G.Pachpatte: inequalities for differential and integral equations, Academic press 1998 S.sugiyama: On stability problem of 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On stability problem of difference equations Bull.sci Engr researches lab Waseda univ 45 , (1959) 140 - 144 41 8 .. .IJRRAS (4) ● March 2011 Roshdy & Mousa ● Discrete Inequality of Gronwall – Bellman Type