On the Boundedness of Solutions of Nonlinear Differential and Difference Equations

as 23i-i lz*l +|«'t|—»0, for fixed /. Systems of the form (1.1) are of considerable interest in dynamics, and play an important role in many branches of applied mathematics. Usually the right-hand side does not involve any derivatives. In dynamics, where t represents the time, a natural problem is the determination of the behavior of the solutions for large values of the time, and this is the central problem considered in the paper. The behavior of the solution turns out to depend critically upon the initial values, and thus the question becomes one of stability in the sense of Liapounoff. A solution, 5, is said to be stable in the sense of Liapounoff if every solution, s', whose initial values are "close" to those of 5 remains "close" to s for all subsequent values of /. The word "close" is defined by a suitable metric. If the two solutions are given by z¿, z[, i = 1, 2, • • • , N, the distance between them will be taken to be ]C£-i |z< — z«' I • Ln our case, since fi(0, 0, • • • , 0, t) = 0, Zi = 0, i = l, 2, ■ • • , N, is a solution of (1.1). Letting a< = Zi(0) be the initial values of any other solution of (1.1), we shall show that provided that Sí¡-i Ia«! 's sufficiently small this solution remains small for all t>0. This investigation, for the case where the /¿ are power series in the z* beginning with second degree terms, and the a¿y are constants, was initiated by Poincaré, and pursued extensively by Liapounoff. Subsequent researches