On the existence of variational principles for a differential-difference evolution operator (Q1760481)

The author describes the structure of operators \(P_{\lambda}\) and \(Q\) for which the equation \(N(u)\equiv \sum_{\lambda=-1}^{1} P_{\lambda} (t) u_t(t+\lambda \tau)-Q(t,u(t+\lambda \tau))=0\), \(u\in D(N), t\in [t_0,t_1]\) admits a direct variational statement. Here, \(P_{\lambda}(t)\) is a linear operator, \(Q:[t_0-\tau,t_1+\tau]\times U_1 \to V_1\) is a nonlinear operator, \(D(N)=\{ u\in C^1([t_0-\tau,t_1+\tau]; U_1): u(t)=\varphi_1(t), t \in [t_0-\tau,t_0], u(t)=\varphi_2(t), t \in [t_1,t_1+\tau] \}\) is the domain of the operator \(N: D(N) \subset C^1([t_0-\tau,t_1+\tau]; U_1)\to C([t_0-\tau,t_1+\tau]; V_1)\), and \(U_1 \subset V_1\) are real linear normed spaces.