All Questions
13 questions
7
votes
2
answers
697
views
Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)
According to the entry "Differential inequality" of the Encyclopedia of Mathematics
http://www.encyclopediaofmath.org/index.php/Differential_inequality
the following result is due to Chaplygin (1919)...
- 1,371
7
votes
1
answer
235
views
A criterion on a vector field for its flow to extend continuously at $t=\infty$
In my work in algebraic topology I need to build a special homotopy and I came up with a construction based on some ordinary differential equation in which I am not an expert. I miss some argument to ...
6
votes
1
answer
2k
views
Gronwall's inequality for higher order derivatives
Gronwall's inequality says that solutions to the initial value problem $u'(t) \leq \beta(t)u(t)$ with $u(0)=u_0$ are bounded by solutions to the problem with inequality replaced with equality for $t\...
- 205
4
votes
1
answer
524
views
Controlling subsolutions of a second order linear ODE
Let $f:[0,\infty) \to \mathbb{R}$ obey the differential inequality
$$f'' - 2\alpha f' + 2\alpha f \leq 0$$
where $0 < \alpha < 2$ is some constant. If $f(0) = 0$ and $f'(0) = 1$, can I say that $...
- 205
3
votes
1
answer
303
views
On Wazewski's theorem on system of differential inequalities
According to Springer's Encyclopedia of Math entry on differential inequalities, T. Wazewski proved in 1950 the following theorem:
Consider the system of differential inequalities given by
$$ \...
- 1,590
3
votes
1
answer
99
views
A bound on an oscillatory solution of an ODE
This question was restated as follows:
Let $V\colon[a,b]\to\mathbb{R}$ be smooth, strictly decreasing and
$V(b) = 0$. Suppose that $f\colon[a,b]\to\mathbb{R}$ is smooth and
satisfies $f''(x)+V(x) f(x)...
- 128k
2
votes
1
answer
138
views
The blow-up rate of a nonlinear oscillator
(Related to this Math.SE question.)
For $p>1$, let $u$ be a solution to $$\tag{1}\frac{d^2 u}{dt^2} + u = |u|^{p-1}u$$ that blows up at $T>0$, that is $$\lim_{t\nearrow T}u(t)=+\infty.$$
...
- 692
2
votes
1
answer
253
views
Differential inequalities for a strictly diagonal dominant system of linear ODEs
Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column ...
- 135
1
vote
1
answer
109
views
Bound on $L^1$ norm of solution of two-point boundary value problem
This has to be known, but I have not been able to find it in the literature (probably due to not being too familiar with two-point boundary value problems). I have a function $u:[0,1]\to\mathbb{R}$ ...
- 3,065
1
vote
2
answers
345
views
how to solve a singular integral equation involving the kernel $1/x$
Dear all,
Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that
$$
f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0,
$$
...
- 1,649
1
vote
2
answers
204
views
One inequality connected with the linear second order ODE
Is the following statement true?
Let $ a>0, b>0, h>0 $, $x(t)$ be the solution of the differential equation
$ \ddot{x}+a \dot{x}+bx=h$
with initial conditions $x(0)=u<0 , \dot{x}(0)...
- 97
1
vote
2
answers
404
views
Controlling solutions of a second order linear differential inequality
A slightly less general version of this question was asked, in a subsequent comment, by the OP of the question at
Controlling subsolutions of a second order linear ODE
Let $f:[0,\infty) \to \mathbb{...
- 128k
0
votes
0
answers
103
views
Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity
Problem:
Consider the autonomous ODE system
\begin{align*}
\dot{x} &= (1-x) (z-xy)\\
\dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\
\dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z
\end{...
- 291