# Tagged Questions

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### Are there methods to determine the stability of functional differential equations?

For nonlinear differential equations there are methods to determine the stability of fixed points and limit cycles using a phase plane analysis. If I have a functional differential equation with two ...
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### Finding function given its Jacobian and the initial condition

Consider continuously differentiable function $f:\mathbb{R}^k\mapsto \mathbb{R}^k$. We know that $f(x_0)=y_0$ and the Jacobian matrix is given for all $x$. I'd like to know the explicit for of the ...
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### Studiy of a differential operator

Let $V=W^{1,p}_0(\Omega)$ and this dual space $V'=W^{-1,p'}(\Omega)$ with $p'$ the conjugate of $p$. Let $A(u)=-\mathrm{div}(|\nabla u|^{p-2} \cdot \nabla u)$. How we can prove that $A$ is monotone? ...
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### operator onto-theorem

I have this theorem: Let $V$ a Banach space, reflexive,separable, and let $A$ an operator monotonic, bounded, semi-continuos, coercive. Then, $A$ is onto. Where we can find the proof of this ...
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### Geometric series of an operator

In solving a first order linear differential equation $(1-D)y=x^2$ where $D\equiv \frac{d}{dx}$ the way I learnt was that we proceed as ...
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### Existence of solutions to linear evolution equation with a noncoercive operator

Consider the Gelfand triple $V\hookrightarrow H \hookrightarrow V'$ and, for given $T>0$, the Sobolev-Bochner space $$\mathcal W(0,T) := \{ v \in L^2(0,T;V): \dot v \in L^2(0,T;V')\}.$$ Consider ...
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### Understand Picard-Lindelöf Proof

I am trying to understand the Picard-LindelĂ¶f from my book which uses the fixed point theorem. The task is trying to find $x \in C(a,b)$ in open interval $(a, b)$ containing $t_0$ such that it ...
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### Proof of an application of the contraction mapping theorem to differential equations

Please consider the theorem below together with the first part of its proof. 1) Why is M closed? 2) Why is M complete? 3) Why is the final integral a continuous function? (The curvy C denotes the ...
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### Existence results for this ODE? (periodic)

Are there any existence/uniqueness results for solutions to the ODE $$y'(t) = f(y(t),t)$$ $$y(0) = y(T)$$ on the time interval $[0,T]$ where $f$ is Caretheodory and $T$-periodic in $t$. I am looking ...
Let $q, r, f \in C[0,1], (\alpha, \beta) \in \mathbb{R}^2$ and show that the the inhomogenous lineair boundary value problem: $$y''(t) + q(t)y'(t) + r(t)y(t) = f(t)$$ $$y(0) = \alpha, y'(0) = \beta$$ ...