scholarly journals On the Modified Boundary Value Problem of De La Vallée-Poussin for Nonlinear Ordinary Differential Equations

1994 ◽  
Vol 1 (4) ◽  
pp. 429-458
Author(s):  
G. Tskhovrebadze
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Ordinary Differential Equations ◽  
De La Vallée Poussin

Abstract The sufficient conditions of the existence, uniqueness, and correctness of the solution of the modified boundary value problem of de la Vallée-Poussin have been found for a nonlinear ordinary differential equation u (n) = f(t, u, u′, … , u (n–1)), where the function f has nonitegrable singularities with respect to the first argument.

On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of even order

2021 ◽  
pp. 341-347
Author(s):  
Gusen E. Abduragimov ◽  
Patimat E. Abduragimova ◽  
Madina M. Kuramagomedova
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equation ◽  
Even Order

In the article, we consider a boundary value problem for a nonlinear ordinary differential equation of even order which, obviously, has a trivial solution. Sufficient conditions for the existence and uniqueness of a positive solution to this problem are obtained. With the help of linear transformations of T. Y. Na [T. Y. Na, Computational Methods in Engineering Boundary Value Problems, Acad. Press, NY, 1979, ch. 7], the boundary value problem is reduced to the Cauchy problem, the initial conditions of which make it possible to uniquely determine the transformation parameter. It is shown that the transformations of T. Y. Na uniquely determine the solution of the original problem. In addition, based on the proof of the uniqueness of a positive solution to the boundary value problem, a sufficiently effective non–iterative numerical algorithm for constructing such a solution is obtained. A corresponding example is given.

Lipschitz stability of nonlinear ordinary differential equations with non-instantaneous impulses in ordered Banach spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Pengyu Chen ◽  
Zhen Xin ◽  
Xuping Zhang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Banach Spaces ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Ordinary Differential Equations ◽  
Lipschitz Stability ◽  
Initial Value ◽  
Ordered Banach Spaces

Abstract We consider Lipschitz stability of zero solutions to the initial value problem of nonlinear ordinary differential equations with non-instantaneous impulses on ordered Banach spaces. Using Lyapunov function, Lipschitz stability of zero solutions to nonlinear ordinary differential equation with non-instantaneous impulses is obtained.

scholarly journals On a nonlinear boundary value problem involving a small parameter

1962 ◽  
Vol 2 (4) ◽  
pp. 425-439 ◽  
Author(s):  
A. Erdéyi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Small Parameter ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Boundary Value Problem ◽  
Image Position

In this paper we shall discuss the boundary value problem consisting of the nonlinear ordinary differential equation of the second order, and the boundary conditions.

Existence theorems for equations in normed spaces and boundary value problems for nonlinear vector ordinary differential equations

1984 ◽  
Vol 98 (1-2) ◽  
pp. 1-11 ◽  
Author(s):  
A. Cañada ◽  
R. Ortega
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Linear Part ◽  
Sufficient Conditions ◽  
First Eigenvalue ◽  
Normed Spaces ◽  
Nonlinear Vector

SynopsisThe existence of solutions to equations in normed spaces is proved when the nonlinear part of the equation satisfies growth and asymptotic conditions, whether the linear part is invertible or not. For this, we use the coincidence degree theory developed by Mawhin. We apply our abstract results to boundary value problems for nonlinear vector ordinary differential equations. In particular, we consider the Picard boundary value problem at the first eigenvalue and the periodic boundary value problem at resonance. In both cases, the nonlinear term can be of superlinear type. Also, necessary and sufficient conditions of Landesman-Lazer type are obtained.

A boundary value problem on the half-line for higher-order nonlinear differential equations

2009 ◽  
Vol 139 (5) ◽  
pp. 1017-1035 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Existence And Uniqueness ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Higher Order ◽  
Half Line ◽  
Nonlinear Ordinary Differential Equations ◽  
Specific Class

This article is devoted to the study of the existence of solutions as well as the existence and uniqueness of solutions to a boundary-value problem on the half-line for higher-order nonlinear ordinary differential equations. An existence result is obtained by the use of the Schauder–Tikhonov theorem. Furthermore, an existence and uniqueness criterion is established using the Banach contraction principle. These two results are applied, in particular, to the specific class of higher-order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of higher-order linear ordinary differential equations, respectively. Moreover, some (general or specific) examples demonstrating the applicability of our results are given.

Qualitative properties of nonlinear ordinary differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 79-85 ◽  
Author(s):  
Nelson Onuchic ◽  
Plácido Z. Táboas
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Ordinary Differential Equation ◽  
Point Of View ◽  
Nonlinear Ordinary Differential Equations ◽  
Qualitative Properties ◽  
Image Position

SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

On one two-point BVP for the fourth order linear ordinary differential equation

2017 ◽  
Vol 24 (2) ◽  
pp. 265-275
Author(s):  
Sulkhan Mukhigulashvili ◽  
Mariam Manjikashvili
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Fourth Order ◽  
Boundary Value ◽  
Homogeneous Problem ◽  
Sufficient Conditions ◽  
Linear Ordinary Differential Equation ◽  
Point Boundary ◽  
Order Linear

AbstractIn this article we consider the two-point boundary value problem\left\{\begin{aligned} &\displaystyle u^{(4)}(t)=p(t)u(t)+h(t)\quad\text{for }% a\leq t\leq b,\\ &\displaystyle u^{(i)}(a)=c_{1i},\quad u^{(i)}(b)=c_{2i}\quad(i=0,1),\end{% aligned}\right.where {c_{1i},c_{2i}\in R}, {h,p\in L([a,b];R)}. Here we study the question of dimension of the space of nonzero solutions and oscillatory behaviors of nonzero solutions on the interval {[a,b]} for the corresponding homogeneous problem, and establish efficient sufficient conditions of solvability for the nonhomogeneous problem.

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