scholarly journals EXISTENCE OF POSITIVE SOLUTION OF TWO-POINT BOUNDARY PROBLEM FOR ONE NONLINEAR ODE OF THE FOURTH ORDER

2017 ◽  
Vol 20 (10) ◽  
pp. 9-16
Author(s):  
E.I. Abduragimov
Keyword(s):  
Banach Space ◽  
Positive Solution ◽  
Boundary Problem ◽  
Fourth Order ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Second Derivative ◽  
Point Boundary ◽  
Strongly Nonlinear ◽  
Apriori Estimates

In the work sufficient conditions for existence at least one positive solution of two-point boundary problem for one class of strongly nonlinear differential equations of the fourth order are received. The problem is considered on a segment [0,1] (more general case of segment[0, a] is reduced to considered). On the ends of a segment the solution of y and its second derivative of y′′ areequal to zero. Right part of an equation f (x, y) isn’t negative at x\geq 0 andat all y. Performance of sufficient conditions is easily checked. Performance ofthese conditions is easily checked. In the proof of existence the theory of conesin banach space is used. Also apriori estimates of positive solution, which ispossible to use further at numerical construction of the solution are obtained.

A Two-Point Boundary Problem for Ordinary Self-Adjoint Differential Equations of Fourth Order

1954 ◽  
Vol 6 ◽  
pp. 416-419 ◽  
Author(s):  
H. M. Sternberg ◽  
R. L. Sternberg
Keyword(s):  
Boundary Problem ◽  
Fourth Order ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Matrix Functions ◽  
Point Boundary ◽  
Vector Identity ◽  
Class C ◽  
Image Position ◽  
Homogeneous Vector

The purpose of this note is to establish Theorem A below for the two-point homogeneous vector boundary problemwhere the Pi(x) are given real m × m symmetric matrix functions of x with P0(x) positive definite and Pi(x) of class C2−i on an infinite interval [a, ∞), and where by a solution of (1.1) — (1.2) for a ≤ x1 < x2 < ∞ we understand a real m-dimensional column vector u = u(x) of class C2 on [a, ∞) which is such that Pi(x)u(2−i) is of class C2−i on [a, ∞) and which satisfies (1.1) — (1.2) with the former a vector identity on [a, ∞).

scholarly journals Existence Results for Fourth Order Non-Homogeneous Three-Point Boundary Value Problems

2021 ◽  
pp. 162-172
Author(s):  
R. Ravi Sankar ◽  
N. Sreedhar ◽  
K. R. Prasad
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Nonlinear Differential Equations ◽  
Existence Results ◽  
Point Boundary ◽  
Real Numbers ◽  
Uniqueness Of Solutions ◽  
Order Four

The present paper focuses on establishing the existence and uniqueness of solutions to the nonlinear differential equations of order four y(4)(t) + g(t, y(t)) = 0, t ∈ [a, b], together with the non-homogeneous three-point boundary conditions y(a) = 0, y′(a) = 0, y′′(a) = 0, y(b) − αy(ξ ) = λ, where 0 ≤ a < b, ξ ∈ (a, b), α, λ are real numbers and the function g: [a, b] × R→R is a continuous with g(t, 0) ≠ 0. With the aid of an estimate on the integral of kernel, the existence results to the problem are proved by employing fixed point theorem due to Banach.

scholarly journals Positive solutions of fourth-order superlinear singular boundary value problems

2002 ◽  
Vol 66 (1) ◽  
pp. 95-104 ◽  
Author(s):  
Guoliang Shi ◽  
Shaozhu Chen
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Closed Interval ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Singular Boundary ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
Necessary And Sufficient

This paper investigates fourth-order superlinear singular two-point boundary value problems and obtains necessary and sufficient conditions for existence of C2 or C3 positive solutions on the closed interval.

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.

scholarly journals Stability of some differential equations of the fourth-order and fifth-order

2019 ◽  
pp. 18-27
Author(s):  
Boris S. Kalitine
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fourth Order ◽  
Classical Method ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equations ◽  
Necessary And Sufficient ◽  
Positive Functions ◽  
Fifth Order

The article is devoted to the study of the problem of stability of nonlinear ordinary differential equations by the method of semi-definite Lyapunov’s functions. The types of fourth-order and fifth-order scalar nonlinear differential equations of general form are singled out, for which the sign-constant auxiliary functions are defined. Sufficient conditions for stability in the large are obtained for such equations. The results coincide with the necessary and sufficient conditions in the corresponding linear case. Studies emphasize the advantages in using the semi-positive functions in comparison with the classical method of applying Lyapunov’s definite positive functions.

scholarly journals Oscillation Theorems for Nonlinear Differential Equations of Fourth-Order

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 520 ◽  
Author(s):  
Osama Moaaz ◽  
Ioannis Dassios ◽  
Omar Bazighifan ◽  
Ali Muhib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Middle Term ◽  
Oscillation Theorems ◽  
Fourth Order Differential Equation

We study the oscillatory behavior of a class of fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation with middle term. Our theorems extend and complement a number of related results reported in the literature. One example is provided to illustrate the main results.

scholarly journals Existence of solution for nonlinear fourth-order three-point boundary value problem

2018 ◽  
Vol 38 (1) ◽  
pp. 67-82
Author(s):  
Zouaoui Bekri ◽  
Slimane Benaicha
Keyword(s):  
Boundary Value Problem ◽  
Nontrivial Solution ◽  
Fourth Order ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Existence Of Solution ◽  
Point Boundary ◽  
Nonlinear Alternative

In this paper, we study the existence of nontrivial solution for the fourth-order three- point boundary value problem having the following formu(4) (t) + f (t, u(t)) = 0, 0 < t < 1,u(0) = α(η), u'(0) = u''(0) = 0, u(1) = βu(η),where η ∈ (0, 1), α, β ∈ R, f ∈ C ([0, 1] × R, R). We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.

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