Coercive estimate and separation theorem for one nonlinear differential operator in a Hilbert space

Problem of separation for differential operators was first investigated by W. N. Everitt and M. Giertz in the beginning of seventieth of the last century. They mainly have investigated, in their works, separation of Sturm–Liouville operator operator and its powers. Later, this problem was investigated by K. Kh. Boimatov, M. Otelbaev, F. V. Atcinson, W. D. Evans, A. Zettl and others. The main part of papers published in this direction concerns with the case of linear operators(both ordinary differential operators and partial differential operators). Separation of nonlinear differential operators was mainly investigated in case when operator under consideration was a weak perturbation of linear one. The case when operator under consideration is not a weak perturbation of linear one was investigated only in some works. Results of this paper also concerns with this poorly studied case. The paper is devoted to studying coercive properties of nonlinear differential operator of the form
$$L[u(x)]=-u^{VI}(x)+V(x,u(x))u(x)$$
in Hilbert space $L_2(R)$ and separation theorem for this operator is proved.
The investigated operator $L[u(x)]$ is strictly nonlinear, in the sense that in the general case it cannot be represented as a weak perturbation of a linear operator.