Questions tagged [nonlinear-analysis]
For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.
554 questions
0
votes
0
answers
30
views
How to estimate the integral $\displaystyle\int |v|^\alpha|Sv| $?
Let $n\geq3$ and $1\leq\alpha<2^*-1\ \ \left(2^*=\displaystyle\frac{2n}{n-2}\right)$. Let $p=\alpha(2^*)'$. Define the operator
\begin{align*}
S:\ L^2\big((0,T),L^p(\Omega)\big)&\longrightarrow ...
- 651
0
votes
0
answers
23
views
How to expand the assumptions for the nonlinear heat equation to obtain uniqueness?
Let $n\geq3$, $\Omega\subset\mathbb R^n$ be bounded, $g\in L^2(\Omega)$ and $f:\ L^2\big((0,T),L^2(\Omega) \big) \longrightarrow L^2\big((0,T),H^{-1}(\Omega) \big)$ satisfies
\begin{align*}
\exists L&...
- 651
2
votes
0
answers
32
views
Show the continuity of solution operator in the nonlinear heat equation
Let $\Omega\subset\mathbb R^n$. We consider the nonlinear heat equation
\begin{align*}
(P)\left\{\begin{aligned}
u_t-\Delta u&=f(u), &&\Omega\times(0,T),
\\ u&=0, &&\partial\...
- 651
0
votes
1
answer
66
views
Method to solve $0= \frac{x-1}{x} + f(x)$
Given an equation of the form:
$$0= \frac{x-1}{x} + f(x)$$
Knowing what $x_0$ makes $f(x_0)=0$, is it possible to determine when the full equation vanish? Are there particular techniques to study such ...
- 83
0
votes
0
answers
50
views
Question about regularity of a nonlinear PDE
Suppose $\Omega$ is a bounded domain with Lipschitz boundary. I know that u is a solution to a nonlinear pde which takes the form $Lu+u^2=0$ and $u\in H^{1}_{0}$, where $L$ is a linear elliptic ...
0
votes
0
answers
69
views
Envelope solutions
This is my first post. I try to calculate the curve length of an envelope.
The equation of the curve is giving by :
$\begin{pmatrix}
x_M \\
y_M
\end{pmatrix}
=
\begin{pmatrix}
x_T(\phi) \\
y_T(\phi)
\...
0
votes
0
answers
68
views
One-dimensional monotone operator is cyclically monotone
I try to understand the proof in the book of Brezis: "Opérateurs Maximaux Monotones" in which it is claimed that a multi-valued monotone operator $T:D(T)\subset\mathbb{R}\to\mathbb{R}$ is ...
- 1,483
0
votes
0
answers
29
views
Reference for the Sobolev $L^2$ trace lemma for an arbitrary line $C\subset \mathbb R^d$?
I am trying to find a reference for the following Lemma found in a paper:
Let $g\in H^s(\mathbb R^d)$ with $s>\frac{d-1}{2}$ and $C\subset\mathbb R^d$ be an arbitrary straight line. Then, there ...
- 329
0
votes
1
answer
86
views
Show that the $T$ is a compact operator
Let $\Omega\subset\mathbb R^2$ be bounded. Consider the partial differential equation $(P)$
\begin{align}
-\Delta u+u^5&=h \ \ \text{on }\Omega,
\newline u&-0\ \ \text{on }\partial\Omega.
\end{...
- 651
1
vote
0
answers
53
views
How many solutions can an equation with negative (but integer) exponents have?
A polynomial of degree $n$ cannot have more than $n$ solutions. However, a polynomial can only have nonnegative powers. What about an equation that has negative integer powers? Is there any analogous ...
- 390
0
votes
0
answers
26
views
type of solutions of $-u^{\prime\prime}=\lambda e^{u}$ based on the value of the parameter $\lambda$. (Gelfand problem).
My question comes from the book Stable Solutions of Elliptic Partial Differential Equations Louis Dupaigne, pages 30-31. \
Question: In the book, it defines $\lambda^{*}=\lambda^{*}(N)>0$ but dont ...
- 125
0
votes
0
answers
75
views
Regarding the asymptotic solution of the nonlinear ODE
I want to find the asymptotic solution to the non-linear ODE:
\begin{equation}
\frac{d^2f}{dr^2}+\frac{1}{r}\frac{df}{dr}-\frac{n^2}{r^2}f-\frac{\lambda}{2}f(f^2-1)=0,
\end{equation}
with the boundary ...
- 25
2
votes
0
answers
34
views
Solving PDE $u_{t} + u^3u_{x} = 0$ with different initial conditions
I'm trying to solve this excercise:
$u_{t} + u^3u_{x} = 0 \quad,x\in \mathbb{R} \space,t>0$
with the following initial conditions:
$(i)\quad u(x,0) = \sqrt[3]{x} \quad \quad \quad (ii)\quad u(x,0) =...
- 21
0
votes
0
answers
11
views
Rewriting the distance based function ( Proposition 24.27) nonlinear analysis
This is from the proof of Proposition 24.27 in Convex Analysis and Monotone Operator Theory.
C is a nonempty closed, convex subset of a Hilbert space $H$. $p=Prox_{f}(x)$, where $f=\phi ( d_C)$, where ...
- 327
0
votes
0
answers
43
views
Analogues of elliptic functions for generalisations of elliptic integrals
I'm looking at the following integral
$t = \int_{x_0}^{x(t)}
\frac{dx'}
{\sqrt{A+Bx'+Cx'^2+Dx'^3+Ex'^4+Fx'^6}}$
and want to know what the inverse function $x(t)$ is, knowing $x(0)=x_0$ and $A=-...
- 21
0
votes
0
answers
57
views
Variable coefficient differential equation
I am absolutely not familiar with differential equations. However, I am facing the following differential equation:
\begin{equation}
a(x)y^{\prime}(x)+b(x)y(x)=c(x)\sqrt{1-d(x)y^{2}(x)}
\end{equation}
...
- 313
0
votes
0
answers
63
views
Solving Abel equation of the second kind
I am absolutely not familiar with differential equations. However, I am facing the following non-linear variable coefficient ordinary differential equation:
\begin{equation}
a(x)y^{3}(x) + b(x)y^{2}(x)...
- 313
1
vote
0
answers
47
views
How to accurately average a function with a nonlinear response?
I am a physics PhD student working in optics and I have a bit of a weird problem that I am trying to sort out and I'm hoping you math folks can help me with.
Without boring you with the experimental ...
2
votes
0
answers
62
views
Holomorphic implicit function theorem
Is there a version of the implicit function theorem for holomorphic functions between complex Banach spaces? If yes, do you know any reference?
- 153
5
votes
0
answers
100
views
Characterize Bifurcation in Nonlinear ODE
Consider the mapping
$F: H^1_0(\mathbb R_+) \times \mathbb R \to H^1_0(\mathbb R_+)$
for an ODE with solutions $(u_E,E)$ that satisfy
$$ F(u_E,E) = 0.$$
Suppose there is a solution iff $E > E_0 >...
- 303
0
votes
0
answers
40
views
Exercise 7.6 of Robinson, Rodrigo, Sadowski: Smoothness of Navier-Stokes on Bounded Domains
My question is about Exercise 7.6 of the excellent book 'The Three-Dimensional Navier-Stokes Equations' by Robinson, Rodrigo and Sadowski. More generally, it is about higher regularity in space of ...
- 3
3
votes
0
answers
49
views
Reference request: uniformly continuous semigroups of nonlinear (Lipschitz) operators
Consider a Banach space $X$ with norm $\vert\cdot\vert$, and call an operator $A\colon X\to X$ Lipschitz whenever
$$\sup_{f\neq g} \frac{\vert Af-Ag\vert}{\vert f-g\vert}<+\infty;$$
the Lipschitz ...
- 31
2
votes
1
answer
59
views
Higher order Frechet derivatives viewed as bilinear maps, on Taylors theorem
So I have been studying some introductory non-linear analysis. I am currently looking at higher order Frechet derivatives and I want to proof-check/ make sure I got something right. So given $X,Y$ ...
- 394
0
votes
0
answers
40
views
Degree of a smooth map is nonzero implies that it is surjective
I would like to prove that if $f: M\to N$ is a smooth map between oriented manifolds with $M$ compact, then $\text{deg}f\neq0$ implies that $f$ is surjective.
Here is my attempt :
Let $y\in N\setminus ...
- 1,641
2
votes
1
answer
48
views
Does the presence of neural bias change the hypothesis space of a NN?
This question has to do with neural networks, but it is a purely mathematical question, so I think it belongs here.
Consider $f: \mathbb{R}^N \to \mathbb{R}^M$ such as to be implementable with a ...
- 383
0
votes
0
answers
24
views
Extension of degree theory on sub-manifolds to manifolds
I am learning the Brouwer degree in the settings of sub-manifolds sztisfying hood assumptions (compactness, orientation, connectedness). I would like to know if there is a reasonable way to extend it ...
- 1,641
1
vote
1
answer
47
views
normalizable solution of a nonlinear equation
How to find a normalizable solution of the nonlinear differential equation below?
$$ R'' + \frac{R'}{r} - R + R^3 =0 . $$
The domain is $[0,\infty ]$ and we want the norm of the solution to be ...
- 1,035
0
votes
0
answers
33
views
Volterra integral operator is completly continuous, given that its kernel is continuous
Prove that Volterra integral operator on $C[a,b]$ is completly continuous, knowing its kernel $K:C[a,b]\times[a,b]\times\mathbb{R}\rightarrow\mathbb{R}$ is continuous.
I used this definition, the ...
- 35
0
votes
0
answers
47
views
How to calculate the non-linear function
Consider the following function:
$$\sum\limits_{i = 1}^{14} {{A_i}} {\alpha _i}^{n - 1} = {1 \over {4n - 3}}$$
$$\left| {{\alpha _j}} \right| <= 1$$
To determine the value of ${A_i}$ and ${\alpha ...
- 31
0
votes
0
answers
33
views
System of non-linear differential equations, $\dot{\vec{\theta}} = K^{-1} \hat{J}^{T} \vec{h}$
Suppose I have following system $\hat{J} \dot{\vec{\theta}} = \vec{h}$ with $\hat{J}$ is a function of $\theta$. I want to solve $\vec{\theta}$.
Naively, one starts with the following construction. ...
- 6,540
2
votes
1
answer
105
views
Consider $x^5-5x+1=0$. Show by Contraction Principle that there exists a unique solution in the interval $[-1,1]$ & find it with an error $<10^{-3}$
I thought of using $f′(x)=5x^4−5$. Then $5x^4−5=0$ and determine the critical points $1$ and $−1$. I know that between consecutive real roots of $f$ there is a real root of $f'$, but I'm not sure what ...
- 35
0
votes
0
answers
94
views
Consider the equation $x^5-5x+1=0$, find the number of real roots and indicate the intervals where these roots belong.
I thought of using $f'(x)=5x^4-5$. Then $5x^4-5=0$ and determine the critical points $1$ and $-1$. I know that between consecutive real roots of $f$ there is a real root of $f′$. So there should be 3 ...
- 35
0
votes
0
answers
43
views
Convergence analysis for $x_{k+1}=A\lvert x_k\rvert+c$
I have the following iteration
$$x_{k+1}=A\lvert x_k\rvert+c $$ where $x_k \in \mathbb R^n$ and $A \in \mathbb R^{n \times n}$ is a square matrix. The absolute value if taking over the elements. I ...
- 27
0
votes
0
answers
47
views
Topological degree of a continuous mapping
I am discovering the topological degree and while I can see why it’s construction is legit for smooth map $f : M\to N$ where $N$ and $M$ are differentiable manifolds, the later being compact. I cannot ...
- 1,641
1
vote
1
answer
72
views
Using the Contraction Principle, show that the sequence given by $x_{n+1}=\ln(\sqrt{1 + x_n^2})$ is convergent and find its limit.
Show (using the Contraction Principle) that the sequence $(x_n)_{n \in \mathbb{N}}$ given by $x_{n+1}=\ln(\sqrt{1 + x_n^2})$, $n \in \mathbb{N}$ and $x_0 = 1$ is convergent and find its limit.
Ps. I ...
- 35
0
votes
0
answers
20
views
Solution to a system of nonlinear equations with certain conditions
I am working in a model and I found a problem relating a nonlinear system of equations. Let $\mathbf{D}(\mathbf{Q})\in \mathbb{R}^N$ for $\mathbf{Q}\in \mathbb{R}^N$ be a continously differentiable ...
- 403
0
votes
0
answers
13
views
Example 1.2 Nonlinear Control Khalil
$f( x) =\begin{bmatrix}
x_{2}\\
-sat( x_{1} +x_{2})
\end{bmatrix}$
is not continuously differentiable on $R^2$. Using the fact that the saturdation function sat(.) satisfies $|sat(\eta)-sat{\xi}|$, we ...
- 79
2
votes
2
answers
223
views
Prove these equations have only zero solution.
Original problem: consider the function $f = f_{a,b,c}(u,v,w)$:
$$
f_{a,b,c}(u,v,w) = (v + T)^3 + v T (v+ T) - u^2 T - v w^2, \quad u,v,w \in\mathbb{R},
$$
where
$$
T = -a u -b v- c w,
$$
and $a,b,c\...
- 133
0
votes
0
answers
22
views
Multi valued function and lower semi continuity
I consider $X$ a metric space and $F_1,F_2$ two disjoint subsets of $X$. Let $T : X\rightrightarrows\mathbb{R}$ be a multi valued function defined by :
$T(x) =\{0\}$ on $F_1$
$T(x)=\{1\}$ on $F_2$
$T(...
- 1,641
0
votes
0
answers
36
views
How to assign optimal coefficients to the time-derivative terms so that the PDE will quickly evolve into a time independent one?
I am trying to solve a set of nonlinear time-independent PDEs, e.g.,
$$L{[\bf{u}]=0}……(1)$$
where $L$ is a nonlinear differential operator and $\bf{u}$ is the unknowns. The specific form of $L$ is too ...
- 65
0
votes
0
answers
42
views
The definition of a continuous semigroup
Here is the definition of a continuous semigroup
Let $C$ be a subset of a Banach space $X$. A semigroup on $C$ is a group $\{S(t):t\geq 0\}$ of a self-maps defined on the subset $C$ which satisfies ...
- 3
1
vote
1
answer
71
views
One-phase association fit / rate constant value comparison
Currently, I am writing my thesis (in molecular biology - not mathematics), and I am puzzled over the results.
I measured an increase in a signal and did a one-phase association fit in GraphPad. Now, ...
- 11
0
votes
0
answers
39
views
Nonlinear Dynamics and Chaos Strogatz Question 4.4.3
Over dampened Pendulum System:
$$
mL^{2}\ddot{\theta } +b\dot{\theta } +mgL\sin \theta =\Gamma
$$
First order approximation:
$$
b\dot{𝜃}+mgL\sin{}𝜃=Γ
$$
Nondimensionalize, diving through by mgL:
$$
...
- 79
0
votes
1
answer
52
views
Multiplication of a time-domain sinusoid to a s-domain (Laplace) signal?
I am confused between the transformations between the time-domain and the frequency domain. I have a signal y(t) which is a sum of multiple sinusoids. I band-pass filter this signal to extract one ...
- 93
0
votes
0
answers
92
views
Singularity of a non- linear second order ODE
I have the encountered a singularity in the equation below .
$$
y^{\prime \prime}(x)+\frac{2}{x} y^{\prime}+\left[y-\left(1+\frac{2}{x^2}\right)\right] y(x)=0, \quad 0<x<+\infty,
$$
with ...
3
votes
1
answer
47
views
Seeking name of "trick" involving operators like $A + \tau B$, where $B$ is Lipschitz.
Theorem.
On Hilbert space $V$, suppose $T: V \to V$ is nonlinear and that $T = A + \tau B$ where $A$ is linear and strongly monotone, $B$ is nonlinear and Lipschitz, and $\tau > 0$ can be made ...
- 2,499
1
vote
0
answers
59
views
Given a posdef matrix $M$, find $x$ such that $x_i = \operatorname{sign}\left(\sum_j M_{ij}x_j\right)$
Let $M$ be a real symmetric positive definite matrix. Can we characterize the sign vectors $x$, that satisfy the condition:
$$x_i = \operatorname{sign}\left(\sum_j M_{ij}x_j\right)$$
That is, this ...
- 6,891
2
votes
0
answers
99
views
Estimate for a second order non-linear ODE
I am considering the following non-linear ODE
\begin{cases}
\ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\\\
y(0) = 0\\\\
\dot y(T) = c
\end{cases}
...
- 4,368
0
votes
0
answers
29
views
How to visualize low-dimensional torus in a high-dimensional system?
I have a system of very high-dimensions (1000s of independent variables), but I could show that the dynamics is attracted to a 1D limit cycle or a 2D torus (with commensurate frequencies, so still ...
- 101
1
vote
0
answers
248
views
How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, I heard that $\operatorname{Id}-K$ is a proper map. Proper map ...
- 11