On some new integral inequalities and their applications☆
Introduction
The integral inequalities involving functions of one and more than one independent variables which provide explicit bounds on unknown functions play a fundamental role in the development of the theory of differential equations. During the past few years, many such new inequalities have been discovered, which are motivated by certain applications. For example, see [1], [2], [3], [4], [5], [6], [7], [8] and the references therein. In the qualitative analysis of some classes of partial differential equations, the bounds provided by the earlier inequalities are inadequate and it is necessary to seek some new inequalities in order to achieve a diversity of desired goals. In this paper, we establish some new integral inequalities involving functions of two independent variables. Our results generalize some results in [6].
Section snippets
Main results
In what follows, R denotes the set of real numbers and R+=[0,∞) is the given subset of R. The first order partial derivatives of a z(x,y) defined for x,y∈R with respect to x and y are denoted by zx(x,y) and zy(x,y) respectively. Throughout this paper, all the functions which appear in the inequalities are assumed to be real-valued and all the integrals involved exist on the respective domains of their definitions.
The following lemmas are useful in our main results. Lemma 1 Let u(t), a(t), b(t) be[6]
Some applications
Example 1 Consider the partial differential equationwhere h:R+2×R→R, r:R+2→R, σ, τ:R+→R are continuous functions and d is a real constant, p⩾1 is a constant.
Suppose thatwhere a(x,y), c(x,y) are nonnegative continuous functions for x,y∈R+. Let u(x,y) be a solution of , for x,y∈R+, thenfor x,y∈R
References (8)
On a certain inequality arising in the theory of differential equations
J. Math. Anal. Appl.
(1994)On some new inequalities related to certain inequalities in the theory of differential equations
J. Math. Anal. Appl.
(1995)- et al.
On some new nonlinear delay integral inequalities
J. Math. Anal. Appl.
(2000) On some integral inequalities and their applications
J. Math. Anal. Appl.
(1997)
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Supported by the natural science foundation of Shandong province, China (Y2001A03).