Abstract
We study the solvability of a nonlocal boundary-value problem for a differential equation with nonlinearity. The linear part of the equation has complex coefficients which together with the coefficient in nonlocal conditions are considered to be the parameters of the problem. The area of change of each parameter is limited by a complex circle with its center at the origin. The nonlinear part of the equation is given by a smooth function that satisfies, together with its derivatives, some conditions of growth in the Dirichlet–Fourier space scale. The proofs are based on the differentiable Nash–Moser iteration scheme, where the main difficulty is to get estimates of the interpolation type for the inverse linearized operators obtained at each step of the iteration. The estimation is connected with the problem of small denominators which is solved, by using a metric approach on the set of parameters of the problem.
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 1, pp. 37–59, January–March, 2021.
Translated from Ukrainian by V.V. Kukhtin
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Il’kiv, V.S., Strap, N.I. & Volyanska, I.I. Conditions of solvability of the nonlocal boundary-value problem for a differential-operator equation with weak nonlinearity. J Math Sci 256, 753–769 (2021). https://doi.org/10.1007/s10958-021-05458-4
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DOI: https://doi.org/10.1007/s10958-021-05458-4