Existence and Stability of Almost Periodic Solutions in Almost Periodic Systems

where x$ are real, k is a positive integer and a^(f) are almost periodic functions of t. Under some conditions on ##(£), Theorem 1 shows that the trivial solution of the first approximation of system (1-1) is uniformly asymptotically stable in a subspace IT of R (see Definition 2) . Using this fact, we obtain a nontrivial almost periodic solution of system (1 • 1) which is uniformly asymptotically stable in a compact set @ and whose module is contained in the module of ai3(£) . This is shown in Theorem 2. Especially, in the case where <2y(£) in system (1-1) are constants, the system governs one of mathematical models of gas dynamics (cf. [2, p. 104] ) and has been studied by Jenks [4] . One of Jenks' results is a special case of Theorem 2. We denote by R the real Euclidean ;?-space. Let R= ( — oo, oo) and J? = [0, oo ). For x in R, let \x\ be the Euclidean norm of x and Xi be the z'-th component. We let