Exponential dichotomies, the Fredholm alternative, and transverse homoclinic orbits in partial functional differential equations (Q818890)

The transversality of homoclinic orbits in partial functional-differential equations is investigated. The authors first discuss the exponential dichotomies for linear operator equations. Then, they show that the Fredholm alternative holds if the homogeneous equation has exponential dichotomies on \(\mathbb{R}\). Transversality of homoclinic orbits for periodically perturbed partial functional-differential equations is studied using the Lyapunov-Schmidt method and the Melnikov integral.