Periodic solutions of nonlinear differential equations involving the method of scalar nonlinearities (Q1823369)

A new type of proof for the existence of periodic solutions for equations of type \(\dot x=Ax+F(x)x+r(t)\) where F(x) is nonlinear. The method lies in constructing a homotopy (parameter \(0\leq p<\infty)\) between integral operators and applying fixed point (Leray-Schauder) theory; the operators are based on the equation above, a form of Laplace transforms, and the following equation \[ \dot x=(A+\bar F)x+r(t),\bar F\equiv \frac{1}{T}\int^{T}_{0}F(x(s))ds \] (x, r real n-vectors; \(A,F,\bar F\) \(n\times n\) matrices). Since one of the operators represents the equation \(d\bar F/dt=0\) there is some resemblance to Dirac's delta- function; the homotopy seems to be a new idea; the proof is simple and is applied to Duffing's equation.