On a kind of symmetric weakly non-linear ordinary differential systems

For a class of weakly non-linear ordinary differential equations, the existence of a unique symmetric solution is established and its stability is studied. The symmetry of a solution is understood in the sense of a certain linear functional equality which includes, in particular, the cases of periodic, anti-periodic, even, and odd solutions. Efficient stability conditions in terms of logarithmic norms and spectral stability conditions are obtained. The theory is illustrated by examples.