1. The well-known Gronwall-Bellman inequality has been employed extensively in a variety of problems in the study of ordinary differential equations. This inequality has been generalized and extended in various contexts, in particular to vector forms by Opial [5] and others. While considering these inequalities, the central problem is always to estimate a function satisfying a differential inequality by the maximal and minimal solutions of a related differential system. It was proved by Kamke [4] that in case of non-uniqueness of solutions, extremal solutions do exist under certain monotonicity conditions. Burton and Whyburn [2], and more recently Ziebur [7], have proved the existence of these extremal solutions even under the more general mixed monotonicity conditions. In this paper we show that in all these problems, the existence of extremal solutions is a consequence of a lattice fixed-point theorem-a technique employed by Hanson and Waltman [3] in the context of another problem. The lattice fixed-point theorem proves not only the existence of a solution but also the existence of extremal solutions at once, and is thus ideally suited for applications to problems of this kind. In Section 2, we give Theorem (2.2) which is a generalization of that used in [3] and in Sections 3 and 4, we will apply this theorem to the problems considered in [2,4, 5, 71. We believe that this gives a unified approach to several problems and furthermore has the advantages of brevity and elegance.