We discuss the existence of periodic solution for the doubly nonlinear evolution equation A(u'(t))+∂Φ(u(t))∋f(t) governed by a maximal monotone operator A and a subdifferential operator ∂ Φ in a Hilbert space H. As the corresponding Cauchy problem cannot be expected to be uniquely solvable, the standard approach based on the Poincaré map may genuinely fail. In order to overcome this difficulty, we firstly address some approximate problems relying on a specific approximate periodicity condition. Then, periodic solutions for the original problem are obtained by establishing energy estimates and by performing a limiting procedure. As a by-product, a structural stability analysis is presented for the periodic problem and an application to nonlinear PDEs is provided.
- Doubly nonlinear evolution equations
- Periodic problem
- Periodic solution