Skip to main content

Questions tagged [nonlinear-analysis]

For questions on nonlinear analysis, a branch of mathematical analysis that deals with nonlinear mappings.

Filter by
Sorted by
Tagged with
0 votes
0 answers
53 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} ...
Dennis Marx's user avatar
0 votes
0 answers
43 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)...
Dennis Marx's user avatar
1 vote
0 answers
35 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 ...
UltrashortGiraffe's user avatar
2 votes
0 answers
45 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?
AMath91's user avatar
  • 153
5 votes
0 answers
86 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 >...
jacktrnr's user avatar
  • 273
0 votes
0 answers
32 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 ...
RiaDoog's user avatar
2 votes
0 answers
36 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 ...
Toco's user avatar
  • 21
2 votes
1 answer
35 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 $X,Y$ ...
Bigalos's user avatar
  • 394
0 votes
0 answers
26 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 ...
G2MWF's user avatar
  • 1,381
2 votes
1 answer
30 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 ...
Noumeno's user avatar
  • 363
0 votes
0 answers
19 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 ...
G2MWF's user avatar
  • 1,381
1 vote
1 answer
43 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 ...
poisson's user avatar
  • 1,015
0 votes
0 answers
22 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 ...
Rika's user avatar
  • 35
0 votes
0 answers
44 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 ...
Elliot's user avatar
  • 31
0 votes
0 answers
31 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. ...
phy_math's user avatar
  • 6,490
2 votes
1 answer
101 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 ...
Rika's user avatar
  • 35
0 votes
0 answers
61 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 ...
Rika's user avatar
  • 35
0 votes
0 answers
42 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 ...
Fathi's user avatar
  • 25
0 votes
0 answers
24 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 ...
G2MWF's user avatar
  • 1,381
1 vote
1 answer
63 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 ...
Rika's user avatar
  • 35
0 votes
0 answers
17 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 ...
Tan1278's user avatar
  • 393
0 votes
0 answers
10 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 ...
SS1's user avatar
  • 79
2 votes
2 answers
219 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\...
cbi's user avatar
  • 59
0 votes
0 answers
16 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(...
G2MWF's user avatar
  • 1,381
0 votes
0 answers
34 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 ...
Charles6's user avatar
0 votes
0 answers
32 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 ...
ran's user avatar
  • 3
1 vote
1 answer
30 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, ...
cmp4's user avatar
  • 11
0 votes
0 answers
37 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: $$ ...
SS1's user avatar
  • 79
0 votes
1 answer
41 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 ...
Ayush Sharma's user avatar
0 votes
0 answers
75 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 ...
SR9054505's user avatar
3 votes
1 answer
41 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 ...
1Teaches2Learn's user avatar
1 vote
0 answers
58 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 ...
a06e's user avatar
  • 6,761
2 votes
0 answers
94 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} ...
Falcon's user avatar
  • 4,072
0 votes
0 answers
22 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 ...
Axel Wang's user avatar
1 vote
0 answers
246 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 ...
boundary's user avatar
1 vote
0 answers
36 views

Updated Gorelik principle

One version of the Gorelik principle is the following: Let $E,X$ be Banach spaces and suppose $U:E\to X$ is a Lipschitz isomorphism (that is, a Lipschitz bijection whose inverse is also Lipschitz). ...
user avatar
3 votes
1 answer
84 views

Inequalities for the solution of $x = (x-a) e^{x+a}$.

Let $a > 0$. The equation $$(x-a) \, e^{x+a} = x $$ must be solved for $x > 0$. Since the solution does not have a closed form, I would like to obtain bounds for the solution. Until now, I was ...
P.S. Dester's user avatar
  • 1,043
0 votes
1 answer
111 views

Define a compact set $\mathcal{X}$ such that $x(t) \in \mathcal{X}$ for all $t \geq t_0$ [closed]

I have state vector $x \in \mathbb{R}^{\text{n}} $, that behaves following this inequality $ \|x(t)\|\leq c_1\|x(t_0)\|\exp(-c_2(t-t_0))+c_3 $ where $c_1, c_2, c_3$ are positive constants. This ...
SpaceTAKA's user avatar
  • 165
0 votes
1 answer
85 views

Existence and uniqueness of $-\Delta u+u^2=f $

My question is very similar to the one given here: Existence and uniqueness of $-\Delta u + u^3 =f$ If we consider the equation $-\Delta u+u^2=f$ in 3 dimensions with $f\in L^2(\Omega)$ is there ...
micha's user avatar
  • 41
0 votes
0 answers
21 views

Boundary of p(x)-triharmonic operator

Regarding the boundary conditions often associated with this operator, why is the assumption $\Delta^{2} u=0$ commonly made, and what are the implications of this hypothesis for the solution ...
Adnanovic's user avatar
1 vote
0 answers
53 views

Solving a super-quadric equation

I have to solve the following scalar non-linear equation. \begin{equation*} \xi^{\frac{2}{\varepsilon}}+(\xi-k)^{\frac{2}{\varepsilon}}=1 \end{equation*} with respect to $\xi\geq 0$. Here $\varepsilon&...
matteogost's user avatar
0 votes
0 answers
94 views

What should I prove to show the states lie within a compact set?

I'm trying to prove the local stability of a nonlinear system and got the following inequality. $ \|x(t)\|\leq c_1\|x(t_0)\|\exp(-c_2(t-t_0))+c_3\epsilon_m\cdots $(i) where $c_1, c_2, c_3$ are ...
SpaceTAKA's user avatar
  • 165
1 vote
0 answers
28 views

Nonlinear elliptic equation with Dirichlet conditions without weak solution

I am not familiar with the theory of nonlinear PDEs and wonder if there is a theorem that states sufficient and necessary conditions for the existence of weak solutions to the problem $\begin{cases} -\...
scottish's user avatar
2 votes
0 answers
54 views

Second-order Taylor expansion for Operators

Let $u(t)$ and $v(t)$ be functions in $C^{\infty}$. Then let $A(u)$ be an operator. A valid reference mentioned that the second-order Taylor expansion of the operator $A$ is: $$A(u+v) = A(u) + dA(u)[v]...
Redsbefall's user avatar
  • 4,855
0 votes
0 answers
42 views

Existence of solution to a advection-reaction equation with forcing term

Consider the advection-reaction equation in One-Dimension $\dfrac{\partial u}{\partial t} + \dfrac{\partial u}{\partial x} = u(1-u) + f(x,t); x\in\mathbb{R}, t>0$ with initial condition $u(x,0) = g(...
J.MD24's user avatar
  • 1
1 vote
0 answers
99 views

Solve for y: $\frac{d^2y}{dx^2} = A\sin(y)+B\cos(y)$

Current Progress I am currently unable to proceed further, any help is welcomed. Looks forward to seeing different approaches to this differential equation. $ {y}'' = A\sin(y) + B\cos(y) \\g(y)=A\sin{(...
zich's user avatar
  • 31
0 votes
0 answers
74 views

Reference proof from Zarantonello's 1960 Article

I cannot find anywhere an online version or reference proof of Zarantonello's 1960 article: E. H. Zarantonello. Solving functional equations by contractive averaging. Math. Research Center Report, 160,...
user avatar
0 votes
1 answer
79 views

Does this inequality guarantee the global stability in this paper?

I'm reading a very informative paper. But I met some formulations hard to understand. In the stability proof section (Sec. V, Theorem 5.2), they define a Lyapunov function as $V(s) = \frac{1}{2}m\|s\|^...
SpaceTAKA's user avatar
  • 165
0 votes
0 answers
46 views

How is it OK to safely neglect one-step difference of the system state in this paper?

I’m reading a very informative paper. But I met some formulations hard to understand. In assumption 2, they have the inequality of one step difference, $$\Delta u_k \leq \sigma(B_o^{-1})\big(L_a\Delta ...
SpaceTAKA's user avatar
  • 165
0 votes
0 answers
54 views

How is it possible to have discrete-mixed continuous Lyapunov function in this paper?

I'm reading a very informative paper. But I met some formulations hard to accept. In the stability proof section (Sec. V, Theorem 5.2), they define a Lyapunov function as $V(s) = \frac{1}{2}m\|s\|^2$ ...
SpaceTAKA's user avatar
  • 165

1
2 3 4 5
11