Acta Appl Math (2010) 110: 477–497 DOI 10.1007/s10440-009-9456-y Ostrowski Type Inequalities on Time Scales for Double Integrals ´ Anh Ngô · Wenbing Chen Wenjun Liu · Quôc Received: December 2008 / Accepted: 16 January 2009 / Published online: February 2009 © Springer Science+Business Media B.V 2009 Abstract In this paper we first derive an Ostrowski type inequality on time scales for double integrals via -integral which unify corresponding continuous and discrete versions We then replace the -integral by the ∇∇-, ∇-, and ∇ -integrals and get completely analogous results Keywords Ostrowski inequality · Double integrals · Time scales Mathematics Subject Classification (2000) 26D15 · 39A10 · 39A12 · 39A13 Introduction In 1938, Ostrowski [27] proved the following interesting integral inequality which has received considerable attention from many researchers [11, 12, 21, 22, 25, 26, 32] Theorem Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b) whose derivative function f : (a, b) → R is bounded on (a, b), i.e., f ∞ = supt∈(a,b) |f (t)| < ∞ W Liu ( ) · W Chen College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China e-mail: wjliu@nuist.edu.cn W Chen e-mail: chenwb@nuist.edu.cn Q.A Ngô Department of Mathematics College of Science, Vietnam National University, Hanoi, Vietnam e-mail: bookworm_vn@yahoo.com Q.A Ngô Department of Mathematics, National University of Singapore, Science Drive 2, Singapore 117543, Singapore 478 W Liu et al Then f (x) − b b−a f (t)dt ≤ a x − a+b + (b − a)2 (b − a) f (1) ∞ for all x ∈ [a, b] In [13], Dragomir et al proved the following Ostrowski type inequality for double integrals Theorem Let f : [a, b] × [c, d] → R be such that the partial derivatives ∂ f (t,s) ∂t∂s ∂f (t,s) ∂f (t,s) , ∂s , ∂t exist and are continuous on [a, b] × [c, d] Then f (x, y) − ≤ x − a+b + (b − a)2 + b (b − a)(d − c) + d f (t, s)dtds a c (b − a) x − a+b (b − a)2 + ∂f ∂t ∞ + y − c+d + (d − c)2 y − c+d (d − c)2 (b − a)(d − c) ∂f ∂s (d − c) ∂ 2f ∂t∂s ∞ (2) ∞ for all (x, y) ∈ [a, b] × [c, d] The development of the theory of time scales was initiated by Hilger [15] in 1988 as a theory capable to contain both difference and differential calculus in a consistent way Since then, many authors have studied certain integral inequalities or dynamic equations on time scales [1, 7, 8, 16, 28–31, 36] In [8], Bohner and Matthews established the following so-called Ostrowski inequality on time scales which was later generalized by the present authors [18–20, 23] Theorem (See [8], Theorem 3.5) Let a, b, x, t ∈ T, a < b and f : [a, b] → R be differentiable Then f (x) − b−a b f σ (t) t ≤ a M h2 (x, a) + h2 (x, b) , b−a (3) where M = supa t , then we say that t is rightscattered, while if ρ(t) < t then we say that t is left-scattered Points that are right-scattered and left-scattered at the same time are called isolated If σ (t) = t , t is called right-dense, and if ρ(t) = t then t is called left-dense Points that are right-dense and left-dense at the same time are called dense Let t ∈ T, then two mappings μ, ν : T → [0, +∞) satisfying μ(t) := σ (t) − t , (t) := t − ρ(t) are called forward and backward graininess functions, respectively We now introduce the set Tκ , Tκ and Tκκ , which are derived from the time scales T as follows If T has a left-scattered maximum t1 , then Tκ := T −{t1 }, otherwise Tκ := T If T has a right-scattered minimum t2 , then Tκ := T −{t2 }, otherwise Tκ := T Finally, we define Tκκ = Tκ ∩ Tκ Given a function f : T → R, we define the function f σ : T → R by f σ (t) = f (σ (t)) for all t ∈ T and define the function f ρ : T → R by f ρ (t) = f (ρ(t)) for all t ∈ T Let f : T → R be a function on time scales Then for t ∈ Tκ , we define f (t) to be the number, if one exists, such that for all ε > there is a neighborhood U of t (i.e., U = (t − δ, t + δ) ∩ T, for some δ > 0) such that for all s ∈ U f σ (t) − f (s) − f (t) (σ (t) − s) ≤ ε |σ (t) − s| We say that f is -differentiable on Tκ provided f (t) exists for all t ∈ Tκ Similarly, for t ∈ Tκ , we define f ∇ (t) to be the number, if one exists, such that for all ε > there is a neighborhood V of t (i.e., V = (t − δ, t + δ) ∩ T, for some δ > ) such that for all s ∈ V f ρ (t) − f (s) − f ∇ (t) (ρ(t) − s) ≤ ε |ρ(t) − s| We say that f is ∇-differentiable on Tκ provided f ∇ (t) exists for all t ∈ Tκ A function f : T → R is called rd-continuous, provided it is continuous at all right-dense points in T and its left-sided limits exist at all left-dense points in T A function f : T → R is called ld-continuous, provided it is continuous at all left-dense points in T and its right-sided limits exist at all right-dense points in T Definition A function F : T → R is called a -antiderivative of f : T → R provided F (t) = f (t) holds for all t ∈ Tκ Then the -integral of f is defined by b f (t) t = F (b) − F (a) a Definition A function G : T → R is called a ∇-antiderivative of g : T → R provided G∇ (t) = g(t) holds for all t ∈ Tκ Then the ∇-integral of g is defined by b g (t) ∇t = G (b) − G (a) a 480 W Liu et al Definition Let hk : T2 → R, k ∈ N0 be defined by h0 (t, s) = for all s, t ∈ T and then recursively by t hk+1 (t, s) = hk (τ, s) τ s for all s, t ∈ T Definition Let jk : T2 → R, k ∈ N0 be defined by j0 (t, s) = for all s, t ∈ T and then recursively by t jk+1 (t, s) = jk (τ, s) ∇τ s for all s, t ∈ T The Two-Variable Time Scales Theory The two-variable time scales calculus and multiple integration on time scales were introduced in [4, 5] (see also [6]) Let T1 and T2 be two given time scales and put T1 × T2 = {(x, y) : x ∈ T1 , y ∈ T2 }, which is a complete metric space with the metric d defined by d (x, y) , x , y = (x − x )2 + (y − y )2 , ∀ (x, y) , x , y ∈ T1 × T2 For a given δ > 0, the δ-neighborhood Uδ (x0 , y0 ) of a given point (x0 , y0 ) ∈ T1 × T2 is the set of all points (x, y) ∈ T1 × T2 such that d((x, y), (x0 , y0 )) < δ Let σ1 , ρ1 and σ2 , ρ2 be the forward jump and backward jump operators in T1 and T2 , respectively In the first part of this section, followed from [5], we recall in brief the so-called Riemann -integrals Then followed by a note in [5] (see also in [6]), we develop Riemann ∇∇integrals, Riemann ∇-integrals and Riemann ∇ -integrals 3.1 Riemann -Integrals In this subsection, we will recall the so-called double delta integrals from [5, Sects and 3] Definition The first order partial delta derivatives of f : T1 × T2 → R at a point (x0 , y0 ) ∈ Tκ1 × Tκ2 are defined to be ∂f (x0 , y0 ) f (σ1 (x0 ) , y0 ) − f (x, y0 ) = lim , x→x0 ,x=σ1 (x0 ) σ1 (x0 ) − x 1x f (x0 , σ2 (y0 )) − f (x0 , y) ∂f (x0 , y0 ) = lim y→y0 ,y=σ2 (y0 ) σ2 (y0 ) − y 2y Then, the authors defined the second order partial derivatives ∂ ∂ f (x, y) = 1x 1x ∂f (x, y) 1x and ∂ ∂ f (x, y) = 2y 1x 2y ∂f (x, y) 1x Ostrowski Type Inequalities on Time Scales for Double Integrals 481 Next, we recall the so-called double Riemann -integrals (which will be denoted by integrals) over regions T1 × T2 and present some properties of it over rectangles Suppose a < b are points in T1 , c < d are points in T2 , [a, b) is the half-closed bounded interval in T1 , and [c, d) is the half-closed bounded interval in T2 Let us introduce a -rectangle in T1 × T2 by R = [a, b) × [c, d) = {(t, s) : t ∈ [a, b), s ∈ [c, d)} Let {x0 , x1 , , xn } ⊂ [a, b] , where a = x0 < x1 < · · · < xn = b {y0 , y1 , , yk } ⊂ [c, d] , where c = y0 < y1 < · · · < yk = d and We call the collection of intervals P1 = {[xi−1 , xi ) : ≤ i ≤ n} a -partition of [a, b) and denote the set of all -partitions of [a, b) by P ([a, b)) Similarly, the collection of intervals P2 = {[yi−1 , yi ) : ≤ i ≤ k} is called a -partition of [c, d) and the set of all -partitions of [c, d) is denoted by P ([c, d)) Set Rij = [xi−1 , xi ) × [yj −1 , yj ), where ≤ i ≤ n, j ≤ ≤ k -partition of R , generated We call the collection P = {Rij : ≤ i ≤ n, ≤ j ≤ k} a by the -partition P1 = {[xi−1 , xi ) : ≤ i ≤ n} and -partition P2 = {[yi−1 , yi ) : ≤ j ≤ k} of [a, b) and [c, d), respectively, and write P = P1 × P2 The rectangles Rij , ≤ i ≤ n, -partitions of ≤ j ≤ k, are called the subrectangles of the partition P The set of all R is denoted by P (R) We need the following auxiliary result See [10, Lemma 5.7] for the proof Lemma For any δ > there exists at least one P1 ∈ P ([a, b)) generated by a set {x0 , x1 , , xn } ⊂ [a, b], where a = x0 < x1 < · · · < xn = b so that for each i ∈ {1, 2, , n} either xi − xi−1 δ or xi − xi−1 > δ and σ1 (xi−1 ) = xi We denote by (P )δ ([a, b)) the set of all P1 ∈ P ([a, b)) that possess the property indicated in Lemma Similarly, we define (P )δ ([c, d)) Further, by (P )δ (R) we denote the set of all P ∈ P (R) such that P = P1 × P2 where P1 ∈ (P )δ ([a, b)) and P2 ∈ (P )δ ([c, d)) Definition Let f be a bounded function on R and P ∈ P (R) be given as above In each rectangle Rij with ≤ i ≤ n, ≤ j ≤ k, choose an arbitrary point (ξij , ηij ) and form the sum n k f ξij , ηij (xi − xi−1 ) yj − yj −1 S= i=1 j =1 We call S a Riemann -sum of f corresponding to P ∈ P (R) Definition We say that f is Riemann -integrable over R if there exists a number I with the following property: For each ε > there exists δ > such that |S − I | < ε for every Riemann -sum S of f corresponding to any P ∈ (P )δ (R) independent of the way in which we choose (ξij , ηij ) ∈ Rij for ≤ i ≤ n, ≤ j ≤ k The number I is the Riemann -integral of f over R, denoted by 482 W Liu et al f (x, y) 1x y R We write I = limδ→0 S It is worth recalling from [5, Theorems 3.4 and 3.10] the following propositions -integrable functions on R = [a, b) × [c, d) and Proposition (Linearity) Let f, g be let α, β ∈ R Then [αf (x, y) + βg(x, y)] 1x 2y =α f (x, y) R 1x 2y +β R g(x, y) 1x y R An effective way for evaluating multiple integrals is to reduce them to iterated (successive) integrations with respect to each of the variables Proposition Let f be -integrable on R = [a, b) × [c, d) and suppose that the sind gle integral I (x) = c f (x, y) y exists for each x ∈ [a, b) Then the iterated integral b a I (x) x exists, and b f (x, y) 1x R 2y = d 1x a f (x, y) y c Remark The notation means that we take the first as the differentiation of the first variable of function under the integral sign and then we take the second as the differentiation of the second variable of function under the integral sign In the following parts of this section, for example, by ∇-integral, we mean that we take the first as the differentiation of the first variable of function under the integral sign and then we take the second ∇ as the differentiation of the second variable of function under the integral sign 3.2 Riemann ∇∇-Integrals Riemann ∇∇-integrals can be defined similarly to Riemann -integrals as following Definition The first order partial nabla derivatives of f : T1 × T2 → R at a point (x0 , y0 ) ∈ (T1 )κ × (T2 )κ are defined to be f (ρ1 (x0 ) , y0 ) − f (x, y0 ) ∂f (x0 , y0 ) , = lim x→x0 ,x=ρ1 (x0 ) ∇1 x ρ1 (x0 ) − x f (x0 , ρ2 (y0 )) − f (x0 , y) ∂f (x0 , y0 ) = lim y→y0 ,y=ρ2 (y0 ) ∇2 y ρ2 (y0 ) − y Then, we define the second order partial nabla derivatives as follows ∂ ∂ f (x, y) = ∇1 x ∇2 x ∂f (x, y) ∇1 x and ∂ ∂ f (x, y) = ∇2 y∇1 x ∇2 y ∂f (x, y) ∇1 x Next, we define the double Riemann ∇∇-integrals (which will be called by ∇∇integrals) over regions T1 × T2 and present some properties of it over rectangles Suppose Ostrowski Type Inequalities on Time Scales for Double Integrals 483 a < b are points in T1 , c < d are points in T2 , (a, b] is the half-closed bounded interval in T1 , and (c, d] is the half-closed bounded interval in T2 Let us introduce a ∇∇-rectangle in T1 × T2 by R∇∇ = (a, b] × (c, d] = {(t, s) : t ∈ (a, b], s ∈ (c, d]} Let {x0 , x1 , , xn } ⊂ [a, b] , where a = x0 < x1 < · · · < xn = b {y0 , y1 , , yk } ⊂ [c, d] , where c = y0 < y1 < · · · < yk = d and We call the collection of intervals P1 = {(xi−1 , xi ] : ≤ i ≤ n} a ∇-partition of (a, b] and denote the set of all ∇-partitions of (a, b] by P∇ ((a, b]) Similarly, the collection of intervals P2 = {(yi−1 , yi ] : ≤ i ≤ k} is called a ∇-partition of (c, d] and the set of all ∇-partitions of (c, d] is denoted by P∇ ((c, d]) Set Rij = xi−1 , xi × yj −1 , yj , where ≤ i ≤ n, j ≤ ≤ k We call the collection P∇∇ = {Rij : ≤ i ≤ n, ≤ j ≤ k} a ∇∇-partition of R∇∇ , generated by the ∇-partition P1 = {(xi−1 , xi ] : ≤ i ≤ n} and ∇-partition P2 = {(yi−1 , yi ] : ≤ j ≤ k} of (a, b] and (c, d], respectively, and write P∇∇ = P1 × P2 The rectangles Rij , ≤ i ≤ n, ≤ j ≤ k, are called the subrectangles of the partition P The set of all ∇∇-partitions of R∇∇ is denoted by P∇∇ (R) Similar to Lemma 1, we obtain the following lemma Lemma For any δ > there exists at least one P1 ∈ P∇ ((a, b]) generated by a set {x0 , x1 , , xn } ⊂ [a, b], where a = x0 < x1 < · · · < xn = b so that for each i ∈ {1, 2, , n} either xi − xi−1 δ or xi − xi−1 > δ and ρ1 (xi ) = xi−1 We denote by (P∇ )δ ((a, b]) the set of all P1 ∈ P∇ ((a, b]) that possess the property indicated in Lemma Similarly, we define (P∇ )δ ((c, d]) Further, by (P∇∇ )δ (R) we denote the set of all P∇∇ ∈ P∇∇ (R) such that P∇∇ = P1 × P2 where P1 ∈ (P∇ )δ ((a, b]) and P2 ∈ (P∇ )δ ((c, d]) Definition Let f be a bounded function on R and P ∈ P∇∇ (R) be given as above In each rectangle Rij with ≤ i ≤ n, ≤ j ≤ k, choose an arbitrary point (ξij , ηij ) and form the sum n k f ξij , ηij (xi − xi−1 ) yj − yj −1 S= i=1 j =1 We call S a Riemann ∇∇-sum of f corresponding to P ∈ P∇∇ (R) Definition 10 We say that f is Riemann ∇∇-integrable over R if there exists a number I with the following property: For each ε > there exists δ > such that |S − I | < ε for every Riemann ∇∇-sum S of f corresponding to any P ∈ (P∇∇ )δ (R) independent of the way in which we choose (ξij , ηij ) ∈ Rij for ≤ i ≤ n, ≤ j ≤ k The number I is the double Riemann ∇-integral of f over R, denoted by f (x, y)∇1 x∇2 y R We write I = limδ→0 S 484 W Liu et al Similarly to Proposition 1, we obtain Proposition (Linearity) Let f, g be ∇∇-integrable functions on R = (a, b] × (c, d] and let α, β ∈ R Then [αf (x, y) + βg(x, y)]∇1 x∇2 y = α f (x, y)∇1 x∇2 y + β R R g(x, y)∇1 x∇2 y R An effective way for evaluating ∇∇-integrals is to reduce them to iterated (successive) integrations with respect to each of the variables which can be proved similarly to Proposition Proposition Let f be ∇∇-integrable on R = (a, b]×(c, d] and suppose that the single ind b tegral I (x) = c f (x, y)∇2 y exists for each x ∈ [a, b) Then the iterated integral a I (x)∇x exists, and b f (x, y)∇1 x∇2 y = a R d ∇1 x f (x, y)∇2 y c In the next subsection, we can define ∇-integral over [a, b) × (c, d] by using partitions consisting of subrectangles of the form [α, β) × (γ , δ] 3.3 Riemann Riemann ∇-Integrals ∇-integrals can be defined similarly to Riemann -integrals as following Definition 11 The first order partial nabla derivatives of f : T1 × T2 → R at a point (x0 , y0 ) ∈ (T1 )κ × (T2 )κ are defined to be ∂f (x0 , y0 ) f (σ1 (x0 ) , y0 ) − f (x, y0 ) = lim , x→x ,x=σ (x ) σ1 (x0 ) − x 1x f (x0 , ρ2 (y0 )) − f (x0 , y) ∂f (x0 , y0 ) = lim y→y0 ,y=ρ2 (y0 ) ∇2 y ρ2 (y0 ) − y Then, we define the following mixed derivatives obtained by combining both delta and nabla differentiations ∂ f (x, y) ∂ = x∇2 y 1x ∂f (x, y) ∇2 y Next, we define the double Riemann ∇-integrals (which will be called by ∇integrals) over regions T1 × T2 and present some properties of it over rectangles Suppose a < b are points in T1 , c < d are points in T2 , [a, b) is the half-closed bounded interval in T1 , and (c, d] is the half-closed bounded interval in T2 Let us introduce a ∇-rectangle in T1 × T2 by R ∇ = [a, b) × (c, d] = {(t, s) : t ∈ [a, b), s ∈ (c, d]} Let {x0 , x1 , , xn } ⊂ [a, b] , where a = x0 < x1 < · · · < xn = b {y0 , y1 , , yk } ⊂ [c, d] , where c = y0 < y1 < · · · < yk = d and Ostrowski Type Inequalities on Time Scales for Double Integrals 485 We call the collection of intervals P1 = {[xi−1 , xi ) : ≤ i ≤ n} a -partition of [a, b) and denote the set of all -partitions of [a, b) by P ([a, b)) Similarly, the collection of intervals P2 = {(yi−1 , yi ] : ≤ i ≤ k} is called a ∇-partition of (c, d] and the set of all ∇-partitions of (c, d] is denoted by P∇ ((c, d]) Set Rij = xi−1 , xi × yj −1 , yj , where ≤ i ≤ n, j ≤ ≤ k We call the collection P ∇ = {Rij : ≤ i ≤ n, ≤ j ≤ k} a ∇-partition of R ∇ , generated by the -partition P1 = {[xi−1 , xi ) : ≤ i ≤ n} and ∇-partition P2 = {(yi−1 , yi ] : ≤ j ≤ k} of [a, b) and (c, d], respectively, and write P ∇ = P1 × P2 The rectangles Rij , ≤ i ≤ n, ≤ j ≤ k, are called the subrectangles of the partition P The set of all ∇-partitions of R ∇ is denoted by P ∇ (R) We denote by (P )δ ([a, b)) the set of all P1 ∈ P ([a, b)) that possess the property indicated in Lemma Similarly, we define (P∇ )δ ((c, d]) to be the set of all P2 ∈ P∇ ((c, d]) that possess the property indicated in Lemma Further, by (P ∇ )δ (R) we denote the set of all P ∇ ∈ P ∇ (R) such that P ∇ = P1 × P2 where P1 ∈ (P )δ ([a, b)) and P2 ∈ (P∇ )δ ([c, d)) Definition 12 Let f be a bounded function on R and P ∈ P ∇ (R) be given as above In each rectangle Rij with ≤ i ≤ n, ≤ j ≤ k, choose an arbitrary point (ξij , ηij ) and form the sum n k S= f ξij , ηij (xi − xi−1 ) yj − yj −1 i=1 j =1 We call S a Riemann ∇-sum of f corresponding to P ∈ P ∇ (R) Definition 13 We say that f is Riemann ∇-integrable over R if there exists a number I with the following property: For each ε > there exists δ > such that |S − I | < ε for every Riemann ∇-sum S of f corresponding to any P ∈ (P ∇ )δ (R) independent of the way in which we choose (ξij , ηij ) ∈ Rij for ≤ i ≤ n, ≤ j ≤ k The number I is the double Riemann ∇-integral of f over R, denoted by f (x, y) x∇2 y R We write I = limδ→0 S Proposition (Linearity) Let f, g be let α, β ∈ R Then [αf (x, y) + βg(x, y)] R x∇2 y ∇-integrable functions on R = [a, b) × (c, d] and =α f (x, y) R x∇2 y +β g(x, y) x∇2 y R An effective way for evaluating ∇-integrals is to reduce them to iterated (successive) integrations with respect to each of the variables which can be proved similarly to Proposition Proposition Let f be ∇-integrable on R = [a, b)×(c, d] and suppose that the single ind b tegral I (x) = c f (x, y)∇2 y exists for each x ∈ [a, b) Then the iterated integral a I (x)∇x 486 W Liu et al exists, and b f (x, y) x∇2 y = d 1x f (x, y)∇2 y a R c 3.4 Riemann ∇ -Integrals Riemann ∇ -integrals which is denoted by f (x, y)∇1 x 2y R ∇-integrals where R is a ∇ -rectangle of the form can be defined similarly to Riemann (a, b] × [c, d) We omit it in details Ostrowski’s Inequality on time Scales for Double Integrals In this section, we suppose that (a) T1 is a time scale, a < b are points in T1 ; (b) T2 is a time scale, c < d are points in T2 4.1 Ostrowski’s Inequality for Double Integrals via -Integral We first derive the following Ostrowski type inequality on time scales for double integrals via -integral Theorem Let x, t ∈ T1 , y, s ∈ T2 and f : [a, b] × [c, d] → R be such that the partial , ∂f (t,s) , ∂ 2fs(t,s) exist and are continuous on [a, b] × [c, d] Then derivatives ∂f (t,s) 1t 2s 1t f (x, y) − ≤ (b − a)(d − c) b d f (σ1 (t), σ2 (s)) a 1t 2s c M2 M1 h2 (x, a) + h2 (x, b) + h2 (y, c) + h2 (y, d) b−a d −c M3 + h2 (x, a) + h2 (x, b) h2 (y, c) + h2 (y, d) (b − a)(d − c) (4) for all (x, y) ∈ [a, b] × [c, d], where M1 = sup a