Periodic solutions of weakly nonlinear integrodifferential equations with impulses
The author investigates the equation \[ dx/dt=Ax+f(t,x,\int^{t}_{t- \eta}\phi (t,s,x(s))ds), \] where x,f,\(\phi\) are vectors (f is T-periodic in t, \(\phi\) is T-periodic in t and s) and A is a constant matrix, \(\eta >0\), under the condition that x experiences jumps at \(t=t_ i(x)\), i.e., \(\Delta x|_{t=t_ i(x)}=I_ i(x)\), \(i=0,\pm 1,\pm 2,...\), with \(t_{i+p}(x)=t_ i(x)+T\), \(I_{i+p}(x)=I_ i(x)\) for all i and p a fixed integer. Lipschitz and continuity properties on \(f,\phi,I_ i,t_ i\) are imposed. Under certain additional conditions the existence of T- periodic solutions is proved, first with \(t_ i(x)\equiv t_ i\), second, with \(t_ i(x)\) nonconstant. The proof is by iteration.
scientific article; zbMATH DE number 3985894