On Opial's type inequalities for an integral operator with homogeneous kernel
© Zhao and Cheung; licensee Springer. 2012
Received: 8 February 2012
Accepted: 30 May 2012
Published: 30 May 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:
The Opial's type inequality was first established by Willett :
A non-trivial generalization of Theorem 1.2 was established by Hua :
A sharper inequality was established by Godunova :
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
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, + ∞).
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 .
This is just a new result established by Mitrinović and Pečarić .
3 Proofs of results
This completes the proof.
Proof of Theorem 2.4 From the hypotheses of Theorem 2.4 and in view of Hölder's inequality,
This completes the proof.
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.
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