periodic solutions of some second-order differential equations with desultorily sublinear nonlinearities

in this paper, we study the existence of periodic solutions of the rayleigh equations

x′′ + f (x′) + g(x) = e(t).

the nonlinear term g satisfies the following conditions:

lim inf_x→+∞g(x)/x⩽ 0, and

lim inf_{x→+∞,x∈sa}xg′(x)/g(x)> −∞,

where a is a non-negative constant, s_a = {x ∈ r⁺ : g(x) > a} with sup s_a = +∞ and f satisfies the sublinear condition. using the leray–schauder continuation theorem, we give sufficient conditions for the existence of periodic solutions of the given equations.