On the construction of a variational principle for a certain class of differential-difference operator equations

In this paper, we obtain necessary and sufficient conditions for the existence of variational principles for a given first-order differential-difference operator equation with a special form of the linear operator $P_\lambda(t)$ depending on $t$ and the nonlinear operator $Q.$ Under the corresponding conditions the functional is constructed. These conditions are obtained thanks to the well-known criterion of potentiality. Examples show how the inverse problem of the calculus of variations is constructed for given differential-difference operators.