On Certain New Gronwall-ou-iang Type Integral Inequalities in Two Variables and Their Applications

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