## 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.

### MSC:

 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