Abstract
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 V I (x) + V (x,u(x))u(x) in Hilbert space L2(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.
Highlights
Results of this paper concerns with this poorly studied case
The paper is devoted to studying coercive properties of nonlinear differential operator of the form in Hilbert space L2(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
Summary
Впервые разделимость дифференциальных выражений с частными производными исследовалась К.Х. В работах [13] и [16] исследовалась разделимость линейного бигармонического оператора. Что разделимость нелинейных дифференциальных операторов, в основном, рассматривалась тогда, когда исследуемый оператор является слабым возмущением линейного оператора (см., например, [8] и [10]). Лишь в отдельных работах (см., например, [7], [17] — [19]) изучалась разделимость строго нелинейных дифференциальных операторов, то есть операторов не представляющихся в виде слабого возмущения линейного оператора. В работах [7] и [8] изучались коэрцитивные свойства нелинейного оператора Шредингера и Дирака, в статье [17] рассматривается вопрос о разделимости нелинейного дифференциального операторов второго порядка с матричными коэффициентами. Как в [7], [17], [18], [19] рассматриваемый нами оператор является строго нелинейным
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