# Tagged Questions

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### Qualitative properties of solutions to a ordinary differential equation.

I have this problem : $$\begin{cases} -(p(t)u'(t))'=f(t,u(t))\\u(0)=u(+\infty)=0\end{cases}$$ we have that $u$ is continues, $f:\mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and ...
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### Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} It is well known that if $f$ is ...
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### The existence of integral equations solution for a 2-dimensional unknown function

Suppose $f(\cdot,\cdot)\in C[0,1]^2$ is a kernel. $f$ is integratble $\int_0^1\int_0^1 f(x,y)dxdy<\infty$. $a,b\in C[0,1]$ are known functions, and $z(\cdot,\cdot)\in C[0,1]^2$ is a 2-dimensional ...
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### Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form ...
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### Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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### Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
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### How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
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### Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
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### Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
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### Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix ...
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### Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
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### Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
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### ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
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### Impulsive Boundary value problems

I have this paper They consider this impulsive problem i dont understand this : Proof. First, suppose that $x\in E\cap C^2[J',R]$ is a solution of problem $(1.5)$. It is easy to see by ...
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### Gronwall inequality for $\frac{d}{dt}u(t) \leq C_1u(t) + C_2\sqrt{u(t)}$

I have the inequality $$\frac{d}{dt}u(t) \leq C_1u(t) + C_2\sqrt{u(t)}$$ for a positive function $u$. Is there a Gronwall inequality that I can use to write $$u(t) \leq C_3u(0)?$$ or something ...
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how to find the Green function of this problem : $$\begin{cases} -(p(t)u'(t))'=\lambda f(t,u(t)) ~ \text{a.e.} ~t>0\\ u(0)=u(+\infty)=0 \end{cases}$$ Thank you
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### Is the particle in a ring a regular Sturm-Liouville problem?

The problem of a particle in a ring is a well-known eigenvalue problem $$\frac{d^2}{d\theta^2} \psi(\theta) + V_0 \psi(\theta) = \lambda \psi(\theta)$$in physics and the SchrÃ¶dinger equation has a ...
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### Properties of this Sturm-Liouville problem.

Given the ODE $$f''(x) + \left(\alpha_1 \cos(x) + \cos^2(x) - \lambda \right) f(x)= 0,$$ where $\theta \in [-\pi,\pi]$, $||f||_{L^2} < \infty$. I was wondering whether there is anything we ...
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### Second Level Operators:

What would be an example of an Operator $$H$$ such that for any and all explicit functions U $$H[u] = I$$ where I is some other function However, for some other Operator W ex: [d/dx] ...
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### Matrix norm in Banach space

How can I calculate the following matrix norm in a Banach Space: $$A=\begin{pmatrix} 5 & -2 \\ 1 & -1 \\ \end{pmatrix} ?$$ I have tried ...
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### Theorem with an example

I have this theorem In the paper they give an example: But here $H_1$ is not satisfied ! How to correct it please? http://mathoverflow.net/questions/163788/theorem-with-an-example
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### First eigenvalue of Laplacian and Poincaré inequality

Any idea on how to solve: $\int_{\Omega} |\nabla u|^2 d^n x=\lambda_1\int_{\Omega}u^2 d^n x$, with $u\in H^1_0(\Omega)$ and $\Omega\subset\mathbb{R}^n$, and $\lambda_1$ the first eigenvalue of the ...
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### The index of some perturbation about elliptic operator with Robin boundary condition

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all ...