On certain new Gronwall-Ou-Iang type integral inequalities in two variables and their applications
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Some new Gronwall-Ou-Iang type integral inequalities in two independent variables are established. These integral inequalities can be applied as tools to the study of certain class of integral and differential equations. Some applications to a terminal value problem are also indicated. 1. Introduction In his study of boundedness of solutions to linear second order differential equations, Ou-Iang [12] established and applied the following useful nonlinear integral inequality. Theorem 1.1. Let u and h be real-valued, nonnegative and continuous functions defined on R+ = [0, ∞) and let c ≥ 0 be a real constant. Then the nonlinear integral inequality 2
2
u (x) ≤ c + 2
x 0
h(s)u(s)ds,
x ∈ R+ ,
(1.1)
implies u(x) ≤ c +
x 0
h(s)ds,
x ∈ R+ .
(1.2)
As indicated by Pachpatte [15], this result has been frequently used by authors to obtain global existence, uniqueness and stability of solutions of various nonlinear integral and differential equations. On the other hand, Theorem 1.1 has also been extended and generalized by many authors; see, for example, the reference [2, 3, 6, 7, 8, 9, 13, 14, 15, 17, 18]. Like Gronwall type inequalities, (1.1) is also used to obtain a priori bounds to the unknown function. Therefore, integral inequalities of this type are usually known as Gronwall-Ou-Iang type inequalities. In recent years, Pachpatte [16] discovered some new integral inequalities involving functions in two independent variables. These inequalities are applied to study the boundedness and uniqueness of the solutions of the following terminal value problem Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:4 (2005) 347–361 DOI: 10.1155/JIA.2005.347
348
Gronwall-Ou-Iang type integral inequalities
for the hyperbolic partial differential equation (1.3) with conditions (1.4)
D1 D2 u(x, y) = h x, y,u(x, y) + r(x, y), u(x, ∞) = σ∞ (x),
u(∞, y) = τ∞ (y),
(1.3)
u(∞, ∞) = k.
(1.4)
Recently, Cheung [2] and Dragomir-Kim [4, 5] established additional new GronwallOu-Iang type integral inequalities involving functions of two independent variables, and Meng and Li [10] generalized the results of Pachpatte to certain new inequalities. Our main aim here, motivated by the works of Cheung, Dragomir-Kim and Meng-Li, is to establish some new and more general Gronwall-Ou-Iang type integral inequalities with two independent variables which are useful in the analysis of certain classes of partial differential equations. 2. Main results In what follows, we define R = (−∞, ∞), R1 = [1, ∞), R+ = [0, ∞), and for any k ∈ N, Rk+ = (R+ )k . Denote by C i (M,S) the class of all i-times continuously differentiable functions defined on set M with range in set S (i = 1,2,...) and C 0 (M,S) = C(M,S). The first-order partial derivatives of a function z(x, y) for x, y ∈ R with respect to x and y are denoted as usual by D1 z(x, y) and D2 z(x, y), respectively. We also assume that all improper integrals appeared in the sequel are always convergent. We need the following lemmas in the discussion of our
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