Necessary and sufficient conditions for solvability of the inhomogeneous Riemann boundary-value problem with minus infinite index of logarithmic order min (?, ?)?1 for a half-plane

1979 ◽  
Vol 18 (3) ◽  
pp. 313-319
Author(s):  
P. Yu. Alekna
Keyword(s):  
Boundary Value Problem ◽  
Half Plane ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Index ◽  
Riemann Boundary Value Problem ◽  
Logarithmic Order ◽  
Riemann Boundary ◽  
Necessary And Sufficient

On a Riemann Boundary Value Problem with Infinite Index in the Half-plane

2021 ◽  
pp. 221-241
Author(s):  
Hrachik M. Hayrapetyan ◽  
Smbat A. Aghekyan ◽  
Artavazd D. Ohanyan
Keyword(s):  
Boundary Value Problem ◽  
Half Plane ◽  
Boundary Value ◽  
Infinite Index ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary

scholarly journals Riemann boundary value problem for hyperanalytic functions

2005 ◽  
Vol 2005 (17) ◽  
pp. 2821-2840 ◽  
Author(s):  
Ricardo Abreu Blaya ◽  
Juan Bory Reyes ◽  
Dixan Peña Peña
Keyword(s):  
General Solution ◽  
Boundary Value Problem ◽  
Singular Integral ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Cauchy Kernel ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary ◽  
Necessary And Sufficient

We deal with Riemann boundary value problem for hyperanalytic functions. Furthermore, necessary and sufficient conditions for solvability of the problem are derived. At the end the explicit form of general solution for singular integral equations with a hypercomplex Cauchy kernel in the Douglis sense is established.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

On the solvability of the antiperiodic boundary value problem for systems of linear generalized differential equations

2017 ◽  
Vol 24 (2) ◽  
pp. 169-184
Author(s):  
Malkhaz Ashordia
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Spectral Type ◽  
Unique Solvability ◽  
Boundary Value ◽  
Type Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Differential ◽  
Necessary And Sufficient

AbstractAn antiperiodic boundary value problem is considered for systems of linear generalized differential equations. A Green-type theorem on the unique solvability of the problem and representation of its solution are established. Effective necessary and sufficient conditions (of spectral type) are given for the unique solvability of the problem as well.

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