Sufficient conditions for convergence of the Newton-Kantorovich method in solving boundary value problems for quasilinear equations of elliptic type

1971 ◽  
Vol 12 (3) ◽  
pp. 408-412
Author(s):  
�. M. Mukhamadiev ◽  
V. Ya. Stetsenko
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Elliptic Type ◽  
Quasilinear Equations ◽  
Kantorovich Method ◽  
Sufficient Conditions For Convergence

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.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

scholarly journals Blowing-up solutions of the time-fractional dispersive equations

2021 ◽  
Vol 10 (1) ◽  
pp. 952-971
Author(s):  
Ahmed Alsaedi ◽  
Bashir Ahmad ◽  
Mokhtar Kirane ◽  
Berikbol T. Torebek
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Nonlinear Capacity ◽  
Blowing Up ◽  
De Vries ◽  
Initial Boundary Value

Abstract This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

On Solvability and Well-Posedness of Boundary Value Problems for Nonlinear Hyperbolic Equations of the Fourth Order

2008 ◽  
Vol 15 (3) ◽  
pp. 555-569
Author(s):  
Tariel Kiguradze
Keyword(s):  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Hyperbolic Equation ◽  
Well Posedness ◽  
Nonlinear Hyperbolic Equations ◽  
Right Hand

Abstract In the rectangle Ω = [0, a] × [0, b] the nonlinear hyperbolic equation 𝑢(2,2) = 𝑓(𝑥, 𝑦, 𝑢) with the continuous right-hand side 𝑓 : Ω × ℝ → ℝ is considered. Unimprovable in a sense sufficient conditions of solvability of Dirichlet, Dirichlet–Nicoletti and Nicoletti boundary value problems are established.

scholarly journals Integrable boundary value problems for elliptic type Toda lattice in a disk

2007 ◽  
Vol 48 (10) ◽  
pp. 102702 ◽  
Author(s):  
Metin Gürses ◽  
Ismagil Habibullin ◽  
Kostyantyn Zheltukhin
Keyword(s):  
Boundary Value Problems ◽  
Toda Lattice ◽  
Boundary Value ◽  
Elliptic Type

scholarly journals Existence and Uniqueness of Solutions to Third-Order Boundary Value Problems

2021 ◽  
Vol 22 (2) ◽  
pp. 221-240
Author(s):  
S. S. Almuthaybiri ◽  
J. M. Jonnalagadda ◽  
C. C. Tisdell
Keyword(s):  
Fixed Point ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Third Order ◽  
Fixed Point Methods ◽  
Bounded Sets

The purpose of this research is to connect fixed point methods with certain third-order boundary value problems in new and interesting ways. Our strategy involves an analysis of the problem under consideration within closed and bounded sets. We develop sufficient conditions under which the associated mappings will be contractive and invariant in these sets, which generates new advances concerning the existence, uniqueness and approximation of solutions.

scholarly journals NONSINGULAR INTEGRO-DIFFERENTIAL BOUNDARY VALUE PROBLEM NOT SOLVED WITH RESPECT TO THE DERIVATIVE

2020 ◽  
Vol 8 (2) ◽  
pp. 127-138
Author(s):  
S. Chuiko ◽  
O. Chuiko ◽  
V. Kuzmina
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Systematic Study ◽  
Critical Case ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Linear Boundary ◽  
Actual Problem ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boi- chuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of linear boundary-value problems for ordinary di- fferential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the linear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear osci- llations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the linear boundary value problems for the integro-differential boundary value problem not solved with respect to the derivative, in parti- cular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear integro-differential boundary value problem not solved with respect to the derivative. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian integro-differential boundary value problem not solved with respect to the derivative. The proposed scheme of the research of the nonlinear Noetherian integro-differential boundary value problem not solved with respect to the derivative in the critical case in this article can be transferred to the seminonlinear integro-differential boundary value problem not solved with respect to the derivative.

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