Solvability of the Nonlocal Boundary-Value Problem for a System of Differential-Operator Equations in the Sobolev Scale of Spaces and in a Refined Scale

Ільків, В. С.; Strap, N. I.

doi:10.1007/s11253-015-1108-y

Cited by 9 publications

(7 citation statements)

References 7 publications

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“…In view of the value of β, equality N q+1 = N 2 q , and formula (10) and |ε| < ε 0 , we can write the inequality…”

Section: The Norm Of the Solution H Q+1 Of The Equation H Q+1mentioning

confidence: 99%

“…Returning to the estimate of the norm of operator L −1 Nq , from Lemmas 2 and 3 and formulas (6) and (10) we can show that, for anyd ∈…”

Section: Introduce the Sets G Q Corresponding To The Sets G Q By The mentioning

confidence: 99%

“…. , p, were considered in the papers [7][8][9][10]. By using of the Nash-Mozer iteration scheme, conditions of the solvability of the problems in the Sobolev spaces were established for the functions of several complex variables, in the Hörmander-Hilbert spaces and in the scale of spaces of functions, which are Dirichlet-Taylor series with fixed spectrum.…”

mentioning

confidence: 99%

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Two-point nonlocal problem for a weak nonlinear differential-operator equation

Volyanska

Ільків

Strap³

2018

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We study the solvability of a two-point nonlocal boundary-value problem for an operatordifferential equation with weakly nonlinear right-hand side. The proof of theorems is carried out within the Nash-Moser iterative scheme. In this scheme, the important point is a construction of estimates to norms of inverse linearized operators arising at each step of this scheme as well as the related problem of small denominators. The inverse operator for linearized operator is obtained by the method developed in the work of M. Berti, P. Bolle (Duke Math. J., 134 (2), 359-419 (2006)). The problem of small denominators is solved by using the metric approach.

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“…In view of the value of β, equality N q+1 = N 2 q , and formula (10) and |ε| < ε 0 , we can write the inequality…”

Section: The Norm Of the Solution H Q+1 Of The Equation H Q+1mentioning

confidence: 99%

“…Returning to the estimate of the norm of operator L −1 Nq , from Lemmas 2 and 3 and formulas (6) and (10) we can show that, for anyd ∈…”

Section: Introduce the Sets G Q Corresponding To The Sets G Q By The mentioning

confidence: 99%

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Two-point nonlocal problem for a weak nonlinear differential-operator equation

Volyanska

Ільків

Strap³

2018

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“…Their theory [7] is supplemented in [22][23][24][25][26][27][28][29] for a more extensive class of Hörmander inner product spaces. The refined Sobolev scale and other classes of Hörmander spaces are applied to the spectral theory of elliptic differential operators on manifolds [7,Section 2.3], theory of interpolation of normed spaces [23,30], to some differential-operator equations [31], parabolic initial-boundary value problems [32][33][34][35], and in mathematical physics [36,37]. However, elliptic operators on vector bundles have not been covered by this theory.…”

Section: Introductionmentioning

confidence: 99%

Elliptic operators on refined Sobolev scales on vector bundles

Zinchenko

2017

Open Mathematics

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: We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.

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“…Ptashnyk et al [45,46,47]. The problems with the local multipoint conditions were considered in [48,50], with the nonlocal conditions in [22,23,27,47] (in the complex domain in [28]), with the integral conditions in [25,26]. The question of how to estimate the small denominators with complicated nonlinear structure is common for all types of multipoint problems ( [24,27,47]).…”

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confidence: 99%

Two-point boundary value problem for a partial differential equation in spaces of periodic functions

et al. 2020

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We investigate the two-point in time boundary value problem for the partial differential equations of the second-order with one spatial variable and constant coefficients. The problem is considered in in the spaces of functions which Fourier coefficients are characterized by exponential behavior on the Cartesian product of the time interval and spatial domain $\mathbb{R}/2\pi\mathbb{Z}$. The correct solvability of the problem is established, the formulas for solutions are presented, the kernel is described and the smoothness of the solution is established in the spaces of functions that are periodic in one spatial variable. We have established the conditions which are close to the necessary conditions of solvability of the problem in scale of spaces of functions with exponentially increasing (or decreasing) Fourier coefficients.We also found the asymptotic estimates demonstrating the absence of the problem of small denominators, which arises of many spatial variables and makes the boundary value problem incorrect. We have established sufficient conditions of the finite-dimensionality of the kernel of the problem and found upper bounds for its dimension. The results are obtained under the condition of minimum smoothness on the right-hand sides of two-point conditions, which is close to the necessary condition.

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Solvability of the Nonlocal Boundary-Value Problem for a System of Differential-Operator Equations in the Sobolev Scale of Spaces and in a Refined Scale

Cited by 9 publications

References 7 publications

Two-point nonlocal problem for a weak nonlinear differential-operator equation

Two-point nonlocal problem for a weak nonlinear differential-operator equation

Elliptic operators on refined Sobolev scales on vector bundles

Two-point boundary value problem for a partial differential equation in spaces of periodic functions

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