BULL. AUSTRAL. MATH. SOC. MOS 45J05, (45D05) VOL. 3 (1970), 9-22. Integro-differential equations of Volterra type M. Rama Mohana Rao and Chris P. Tsokos The aim of this paper is concerned with studying the stability properties of an integro-differential system "by reducing it into a scalar integro-differential equation. A theorem is stated about the existence of a maximal solution of such systems and a "basic result on integro-differential inequalities. Utilizing these results we obtain sufficient conditions for uniform asymptotic stability of the trivial solution of the integro-differential system of the form x'{t) = F(t, x(t), Ax) , V=jfc) where F i C[R + X 5 ff x C(J)] , A i C[C ff , C(J)] with C E = \se € CU) : \\X\\ < fij , J=05t5a<«>, S H = {x € iP : \\x(t)\\ < H , H > 0 for t i j] , C{J) denotes the space of continuous functions, A a continuous operator such that A maps C{J) into C(J) . The fruitfulness of the results of the paper are illustrated with two applications. 1. Corduneanu [/], Levin [3] and Nohel [6] among others have studied the stability properties of solutions of integro-differential equations of Volterra type and many interesting results have been accumulated. Quite recently Lakshmikantham and Rama Mohana Rao [Z] investigated such a Received h April 1970. The authors thank Professor V. Lakshmikantham for his valuable suggestions during the progress of this research. 9 available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700045603 Downloaded from https://www.cambridge.org/core. IP address: 3.92.57.205, on 24 May 2020 at 15:18:22, subject to the Cambridge Core terms of use,