Generalized Ostrowski type inequalities for multiple points on time scales involving functions of two independent variables
© Feng and Meng; licensee Springer. 2012
Received: 14 December 2011
Accepted: 29 March 2012
Published: 29 March 2012
In this article, we establish some new Ostrowski type integral inequalities on time scales involving functions of two independent variables for k2 points, which on one hand unify continuous and discrete analysis, on the other hand extend some known results in the literature. The established results can be used in the estimate of error bounds for some numerical integration formulae, and some of the results are sharp.
Mathematical Subject Classification 2010: 26E70; 26D15; 26D10.
Recently many authors have studied various inequalities, among which the Ostrowski type inequalities have attracted much attention in the literature. The Ostrowski inequality was originally presented in  (see also in [, pp. 468]) as stated in the following theorem.
for x ∈ [a, b].
In recent years, various generalizations of the Ostrowski inequality including continuous and discrete versions have been established (for example, see [3–14] and the references therein). On the other hand, Hilger  initiated the theory of time scales as a theory capable of treating continuous and discrete analysis in a consistent way, based on which some authors have studied the Ostrowski type inequalities on time scales (see [16–24]). The established Ostrowski type inequalities on time scales unify continuous and discrete analysis, and can be used to provide explicit error bounds for some known and some new numerical quadrature formulae.
In this article, we will establish some new Ostrowski type inequalities on time scales involving functions of two independent variables for k2 points, which on one hand extend some known results in the literature, on the other hand unify continuous and discrete analysis.
First we will give some preliminaries on time scales. A time scale is an arbitrary nonempty closed subset of the real numbers. If denotes an arbitrary time scale, then on we define the forward and backward jump operators and such that .
Definition 1.2. A point is said to be left-dense if ρ(t) = t and , right-dense if σ(t) = t and , left-scattered if ρ(t) < t and right-scattered if σ(t) > t.
Definition 1.4. A function is called rd-continuous if it is continuous at right-dense points and if the left-sided limits exist at left-dense points, while f is called regressive if 1 + μ(t)f(t) ≠ 0, where μ(t) = σ(t) - t. C rd denotes the set of rd-continuous functions.
Remark 1.6. If , then fΔ(t) becomes the usual derivative f'(t), while fΔ(t) = f(t + 1) - f(t) if , which represents the forward difference.
The following two theorem include some important properties for delta derivative on time scales.
if f(t) ≥ 0 for all a ≤ t ≤ b, then .
where h0(t, s) = 1.
For more details about the calculus of time scales, we refer to .
Throughout this article, ℝ denotes the set of real numbers and ℝ+ = [0,∞), while ℤ denotes the set of integers, and ℕ0 denotes the set of nonnegative integers. For a function f and two integers m0, m1, we have provided m0 > m1. denote two arbitrary time scales, and for an interval . Finally, for the sake of convenience, we denote the forward jump operators on by σ uniformly.
2. Main results
The inequality (1) is sharp in the sense that the right side of (1) can not be replaced by a smaller one.
Combining (3) and (4) we get the desired inequality (1).
On the other hand, Using K = 1, the right side of (1) reduces to h2 (a, b)h2 (c, d), which implies (2) holds for equality form, and then the sharpness of (1) is proved.
Remark 2.2. Theorem 2.1 is the 2D extension of [, Theorem 3].
If we furthermore take in the inequality above, then we obtain the time scale version of Simpson's inequality , which is omitted here.
In Theorem 2.1, if we take for some special time scales, then we immediately obtain the following three corollaries.
where K denotes the maximum value of the absolute value of the difference Δ1Δ2f over [m1, m2-1]ℤ × [n1, n2-1]ℤ.
As long as we notice for ∀t, s ∈ ℤ, we can easily get the desired result.
where K denotes the maximum value of the absolute value of the q1 q2-difference over .
Substituting (8) into (1) we get the desired result.
Then combining (4), (10) and (11) we obtain the desired inequality (9).
Remark 2.7. If we take , where λ ∈ [0,1], then Theorem 2.6 reduces to [, Theorem 4].
In the following, we will establish a generalized Ostrowski-Grüss type integral inequality on time scales based on the result of Theorem 2.1.
The proof for Lemma 2.6 is similar to [, pp. 295-296], and we omit it here.
Then combining (3), (10), (14), (15) we get the desired result.
Remark 2.10. If we take k = 2, then Theorem 2.9 becomes the 2D extension on time scales of [, Theorem 4]. If we take , then Theorem 2.9 becomes the 2D extension in the continuous case of [, Theorem 2.1].
Remark 2.11. For Theorems 2.6 and 2.9, we can also obtain similar results as shown in Corollaries 2.3-2.5, which are omitted here.
In this article, we establish some new generalized Ostrowski type inequalities on time scales involving functions of two independent variables for k2 points, which unify continuous and discrete analysis. We note that the presented inequalities in Theorems 2.6 and 2.9 are not sharp, and the sharp version of them are supposed to further research.
This study was partially supported by the National Natural Science Foundation of China (11171178), Natural Science Foundation of Shandong Province (ZR 2009 AM011) (China) and Specialized Research Fund for the Doctoral Program of Higher Education (20103705110003)(China). The authors thank the referees very much for their careful comments and valuable suggestions on this article.
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