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On Opial's type inequalities for an integral operator with homogeneous kernel
Journal of Inequalities and Applications volume 2012, Article number: 123 (2012)
In this article, we establish some new Opial's type integral inequalities for an integral operator with homogeneous kernel. The results in special cases yield some of the interrelated results and provide new estimates on inequalities of this type.
MS (2000) Subject Classification: 26D15.
In 1960, Opial  established the following inequality:
Theorem 1.1 Suppose f ∈ C1[0, h] satisfies f(0) = f(h) = 0 and f(x) > 0 for all x ∈(0, h). Then
The Opial's type inequality was first established by Willett :
Theorem 1.2 Let x(t) be absolutely continuous in [0, a], and x(0) = 0. Then
A non-trivial generalization of Theorem 1.2 was established by Hua :
Theorem 1.3 Let x(t) be absolutely continuous in [0, a], and x(0) = 0. Further, let l be a positive integer. Then
A sharper inequality was established by Godunova :
Theorem 1.4 Let f(t) be convex and increasing functions on [0, ∞) with f(0) = 0. Further, let x(t) be absolutely continuous on [α,τ], and x(α) = 0. Then, following inequality holds
Opial's inequality and its generalizations, extensions, and discretizations play an important role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations [5–13]. For Opial-type integral inequalities involving high-order partial derivatives, we refer the readers to see [14, 15]. For an extensive survey on these inequalities, see . Mitrinović and Pečarić  proved some new extensions of Opial's type inequalities. The aim of this article is to establish some Opial-type inequalities, which are some extensions of Godunova and Levin's, Mitrnović and Pečarić inequalities.
2 Statement of results
Here, we shall extend some of the previous results for the singular integral operator T which have an integral representation. For this, we say that the integral operator T (f)y belongs to the class U(f, K) if it can be represented in the following form
where f(x) is a continuous function on [0, + ∞), and K(y, x) is a non-negative homogeneous kernel function defined on [0, + ∞) × [0, + ∞), such that T(f)y > 0 if f(x) > 0, where x ∈ [0, + ∞).
Theorem 2.1 For i = 1, 2, 3, let T(f i )y ∈ U(f i , K), where f2(x) > 0, x ∈ [0, + ∞). Further, let p(x) > 0, x ∈ [0, + ∞), and let g(y, x) be convex and increasing on [0, ∞) × [0, ∞). Then the following inequality holds
Remark 2.2 Let the singular integral operator T(f)y change to a function x(t), so we say that the function x(t) belongs to the class if it can be represented in the following form
where y(t) is a continuous function on [α, τ], and K(t, s) is a non-negative kernel defined on [α, τ] × [α, τ]; such that x(t) > 0 if y(t) > 0, t ∈ [α, τ]. Taking these in Theorem 2.1 and with suitable modifications, then (2.2) becomes the following established by Mitrinović and Pečarić .
Remark 2.3 In particular, if λ > 0 we have
then y(t) is the derivative of order λ of x(t) in the sense of Riemann-Liouville. Thus, if x(t) is differentiable, then for λ = 1, it follows that y(t) = x'(t).
Further, taking for f(x, y) = f(x) and K(t, s) = K λ (t, s) in (2.3), (2.3) reduces to a result of Godunova and Levin .
Now, let T(f)y ∈ U(f, K), where K(y, x) = 0 for x > y. Such the singular integral operator we shall say belong to the class U1(f, K). It is clear that in the case, we have
Theorem 2.4 Let the function f(x) be differentiable on [0, ∞), and that for v > 1 the function f(x1/v) be convex and f(0) = 0, and. Further, let T(f)y ∈ U1(f, K), where
Remark 2.5 Let the singular integral operator T(f)y change to a function x(t), so we say that the function x(t) belongs to the class . Taking these in Theorem 2.4 and with suitable modifications, then (2.5) becomes the following result.
This is just a new result established by Mitrinović and Pečarić .
3 Proofs of results
Proof of Theorem 2.1 From the hypotheses of Theorem 2.1, it turn out that
By using the Jensen integral inequality, we have
This completes the proof.
Proof of Theorem 2.4 From the hypotheses of Theorem 2.4 and in view of Hölder's inequality,
Moreover, notes that
Further, from the convexity of f(t1/v) it follows that the function t1-vf'(t) is non-decreasing, thus we have
This completes the proof.
Opial Z: Sur une inégalité. Ann Polon Math 1960, 8: 29–32.
Willett D: The existence-uniqueness theorem for an n th order linear ordinary differential equation. Am Math Monthly 1968, 75: 174–178. 10.2307/2315901
Hua LK: On an inequality of Opial. Sci China 1965, 14: 789–790.
Godunova EK: Integral'nye neravenstva s proizvodnysi i proizvol'nymi vypuklymi funkcijami. Uc Zap Mosk Gos Ped In-ta im Lenina 1972, 460: 58–65.
Rozanova GI: On an inequality of Maroni (Russian). Math Zametki 1967, 2: 221–224.
Das KM: An inequality similar to Opial's inequality. Proc Am Math Soc 1969, 22: 258–261.
Agarwal RP, Thandapani E: On some new integrodifferential inequalities. Anal sti Univ ''Al I Cuza'' din Iasi 1982, 28: 123–126.
Yang GS: A note on inequality similar to Opial inequality. Tamkang J Math 1987, 18: 101–104.
Agarwal RP, Lakshmikantham V: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific, Singapore; 1993.
Bainov D, Simeonov P: Integral Inequalities and Applications. Kluwer Academic Publishers, Dordrecht; 1992.
Li JD: Opial-type integral inequalities involving several higher order derivatives. J Math Anal Appl 1992, 167: 98–100. 10.1016/0022-247X(92)90238-9
Cheung WS: On Opial-type inequalities in two variables. Aequationes Math 1989, 38: 236–244. 10.1007/BF01840008
Cheung WS: Some generalized Opial-type inequalities. J Math Anal Appl 1991, 162: 317–321. 10.1016/0022-247X(91)90152-P
Zhao CJ, Cheung WS: Sharp integral inequalities involving high-order partial derivatives. J Inequal Appl 2008, 2008: 1–10. Article ID 571417
Agarwal RP, Pang PYH: Sharp opial-type inequalities in two variables. Appl Anal 1996, 56(3):227–242.
Agarwal RP, Pang PYH: Opial Inequalities with Applications in Differential and Difference Equations. Kluwer Academic Publishers, Dordrecht; 1995.
Mitrinović DS, Pečarić JE: Generalizations of two inequalities of Godunova and Levin. Polish Acad Sci Math 1988, 36: 645–648.
Godunova EK, Levin VI: O neikotoryh integralnyh nerabenstvah, soderzascih proizvodnye (Russian). Izv vuzov Matem 1969, 91: 20–24.
The research of Chang-Jian Zhao was supported by the National Natural Science Foundation of China (10971205), and the research of Wing-Sum Cheung was partially supported by a HKU URC grant.
The authors declare that they have no competing interests.
C-JZ and W-SC jointly contributed to the main results Theorems 2.1 and 2.4. All authors read and approved the final manuscript.
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Zhao, C., Cheung, W. On Opial's type inequalities for an integral operator with homogeneous kernel. J Inequal Appl 2012, 123 (2012). https://doi.org/10.1186/1029-242X-2012-123
- Opial's inequality
- Hölder's inequality
- Jensen's inequality
- integral operator
- homogeneous kernel