On the solvability of a nonlinear difference boundary value problem in the case of parametric resonance

2021 ◽  
Vol 34 ◽  
pp. 153-163
Author(s):  
Sergei Chuiko ◽  
Yaroslav Kalinichenko
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Parametric Resonance ◽  
Parametric Excitation ◽  
Critical Case ◽  
Boundary Value ◽  
Point Boundary ◽  
Periodic Problems

We construct necessary and sufficient conditions for the existence of solution of seminonlinear boundary value problem for a parametric excitation system of difference equations. The convergent iteration algorithms for the construction of the solutions of the semi-nonlinear boundary value problem for a parametric excitation system difference equations in the critical case have been found. The investigation of periodic and Noetherian boundary-value problems in the critical cases is traditionally performed under the assumption that the differential equation and boundary conditions are known and fixed. As a rule, the study of periodic problems in the case of parametric resonance is reduced to the investigation of the problems of stability. At the same time, due to numerous applications in electronics, geodesy, plasma theory, nonlinear optics, mechanics, and machine-building, the analysis of periodic boundary-value problems in the case of parametric resonance requires not only to find the solutions but also to determine the eigenfunctions of the corresponding differenсе equation. The investigation of autonomous Noetherian boundary-value problems is also reduced to the study of Noetherian boundary-value problems in the case of parametric resonance because the change of the independent variable in the critical case gives a nonautonomous boundary-value problem with an additional unknown quantity. The aim of the present paper is to construct the solutions of Noetherian boundary-value problems in the case of parametric resonance whose solvability is guaranteed by the corresponding choice of the eigenfunction of the analyzed boundary-value problem. The applied classification of Noetherian boundary-value prob\-lems in the case of parametric resonance depending on the simplicity or multiplicity of roots of the equation for generating constants noticeably differs from a similar classification of periodic problems in the case of parametric resonance and corresponds to the general classification of periodic and Noetherian boundary-value problems. The equation for generating constants obtained for the Noetherian boundary-value problems in the case of parametric resonance strongly differs from the conventional equation for generating constants in the absence of parametric resonance by the dependence both of the equation and of its roots on a small parameter, which leads to noticeable corrections of the approximate solutions as compared with the approximations obtained by the Poincare method. Using the convergent iteration algorithms we expand solution of seminonlinear two-point boundary value problem for a parametric excitation Mathieu type difference equation in the neighborhood of the generating solution. Estimates for the value of residual of the solutions of the seminonlinear two-point boundary value problem for a parametric excitation Mathieu type difference equation are found.

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.

The Weighted Error Estimation of Finite Element Method for Two-Point Boundary Value Problem

2012 ◽  
Vol 182-183 ◽  
pp. 1571-1574
Author(s):  
Qi Sheng Wang ◽  
Jia Dao Lai
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Error Estimation ◽  
Boundary Value ◽  
Point Boundary ◽  
Weighted Error ◽  
Element Method

In this paper, the weighed error estimation of finite element method for the two-point boundary value problems are discussed. Respectively, the norm estimation of the H1 and L2 are obtained.

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.

scholarly journals Existence theorems for a second orderm-point boundary value problem at resonance

1995 ◽  
Vol 18 (4) ◽  
pp. 705-710 ◽  
Author(s):  
Chaitan P. Gupta
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence Theorems ◽  
Boundary Value ◽  
Continuation Theorem ◽  
Point Boundary ◽  
Existence Of A Solution ◽  
At Resonance ◽  
Problem At Resonance

Letf:[0,1]×R2→Rbe function satisfying Caratheodory's conditions ande(t)∈L1[0,1]. Letη∈(0,1),ξi∈(0,1),ai≥0,i=1,2,…,m−2, with∑i=1m−2ai=1,0<ξ1<ξ2<…<ξm−2<1be given. This paper is concerned with the problem of existence of a solution for the following boundary value problemsx″(t)=f(t,x(t),x′(t))+e(t),0<t<1,x′(0)=0,x(1)=x(η),x″(t)=f(t,x(t),x′(t))+e(t),0<t<1,x′(0)=0,x(1)=∑i=1m−2aix(ξi).Conditions for the existence of a solution for the above boundary value problems are given using Leray Schauder Continuation theorem.

scholarly journals Solvability of singular second order m-point boundary value problems of Dirichlet type

2005 ◽  
Vol 71 (1) ◽  
pp. 41-52 ◽  
Author(s):  
Ruyun Ma ◽  
Bevan Thompson
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Second Order ◽  
Continuation Theorem ◽  
Point Boundary ◽  
Dirichlet Type ◽  
Carathéodory Conditions

Let f: [0, 1] × ℝ2 → ℝ be a function satisfying the Carathéodory conditions and t (1 − t) e (t) ∈ L1(0, 1). Let ai ∈ ℝ and ξi ∈ (0, 1) for i = 1, …, m − 2 where 0 < ξ1 < ξ2 < … < ξm−2 < 1. In this paper we study the existence of C[0, 1] solutions for the m-point boundary value problem The proof of our main result is based on the Leray-Schauder continuation theorem.

scholarly journals Sign-variations of solutions of nonlinear discrete boundary value problems

2007 ◽  
Vol 76 (1) ◽  
pp. 33-42 ◽  
Author(s):  
Ruyun Ma
Keyword(s):  
Difference Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The One ◽  
The Relationship

In this paper, we study two-point boundary value problems for the nonlinear second order difference equation We establish the relationship between the number of sign-variation of f on {0,…, T + 2} and the one of the solution u of the above problem.

scholarly journals Boundary Value Problems for Fourth Order Nonlinearp-Laplacian Difference Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Qinqin Zhang
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Mountain Pass ◽  
Mountain Pass Lemma

We consider the boundary value problem for a fourth order nonlinearp-Laplacian difference equation containing both advance and retardation. By using Mountain pass lemma and some established inequalities, sufficient conditions of the existence of solutions of the boundary value problem are obtained. And an illustrative example is given in the last part of the paper.

Existence and uniqueness for two-point boundary value problems

1981 ◽  
Vol 88 (3-4) ◽  
pp. 219-234 ◽  
Author(s):  
John V. Baxley ◽  
Sarah E. Brown
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Shooting Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Boundary Conditions ◽  
Point Boundary ◽  
Initial Value

SynopsisBoundary value problems associated with y″ = f(x, y, y′) for 0 ≦ x ≦ 1 are considered. Using techniques based on the shooting method, conditions are given on f(x, y,y′) which guarantee the existence on [0, 1] of solutions of some initial value problems. Working within the class of such solutions, conditions are then given on nonlinear boundary conditions of the form g(y(0), y′(0)) = 0, h(y(0), y′(0), y(1), y′(1)) = 0 which guarantee the existence of a unique solution of the resulting boundary value problem.

ON A METHOD FOR SOLVING A FAMILY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 73 (1) ◽  
pp. 70-75
Author(s):  
S.M. Temesheva ◽  
◽  
P.B. Abdimanapova ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Point Boundary ◽  
Local Boundary ◽  
The Family ◽  
Non Local

In this paper, we consider a boundary value problem for a family of linear differential equations that obey a family of nonlinear two-point boundary conditions. For each fixed value of the family parameter, the boundary value problem under study is a nonlinear two-point boundary value problem for a system of ordinary differential equations. Non-local boundary value problems for systems of partial differential equations, in particular, non-local boundary value problems for systems of hyperbolic equations with mixed derivatives, can be reduced to the family of boundary value problems for ordinary differential equations. Therefore, the establishment of solvability conditions and the development of methods for solving a family of boundary value problems for differential equations are actual problems. In this paper, using the ideas of the parametrization method of D. S. Dzhumabaev, which was originally developed to establish the signs of unambiguous solvability of a linear two-point boundary value problem for a system of ordinary equations, a method for finding a numerical solution to the problem under consideration is proposed.

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