Unified approach to bounded, periodic and almost periodic solutions of differential systems. (English) Zbl 0899.34029

The authors consider quasilinear systems \[ X'= AX+ F(t, X),\tag{1} \] where the \((n,n)\)-matrix \(A\) is hyperbolic and the nonlinearity \(F(t,X)\) satisfies some boundedness conditions. The authors aim at a unified approach to the existence problem of a bounded or periodic or almost-periodic solution, if \(F(\cdot,X)\) is assumed to have the respective property. As for bounded and periodic solutions, known results on the interrelation with the analogous problems for the linear equation \[ X'= AX+ P(t)\tag{2} \] are used. In the case of almost periodicity, however, the correspondence between (1) and (2) is incomplete. To ensure the existence of an almost-periodic solution to (1), it has to be assumed that \(F(\cdot,X)\) is “uniformly almost-periodic” and \(F(t,\cdot)\) is Lipschitzian with a sufficiently small Lipschitz constant \(L\). In some special cases, an explicit upper bound for \(L\) is given.
Finally, the system \[ X'= A(t)X+ F(t, X)\tag{3} \] with time-variable \(A(t)\) is briefly considered. Applying again known results concerning its connection with the linear equation \(X'= A(t)X+ P(t)\), the authors prove the existence of bounded or periodic solutions to (3), if all eigenvalues of \(A(t)+ A^T(t)\) have a negative time-independent upper bound, and \(f(t, X)\) is somehow bounded.


34C25 Periodic solutions to ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations