1
vote
0answers
24 views

The uniqueness of a solution of a system of differential equations

Suppose I have a system of delay differential equations $\frac{d G_1(\phi_1(s))}{ds}= F_1(\phi_1(s),\phi_2(1-s),1-s)$ $\frac{d G_2(\phi_2(1-s))}{ds}= F_2(\phi_1(s),\phi_2(1-s),s)$ where $G_1, G_2, ...
0
votes
0answers
24 views

General solution of the recurrence equation with real shifts

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
1
vote
1answer
27 views

Linear after change of variables

Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, ...
0
votes
1answer
28 views

Similar outputs from different transfer functions

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, ...
2
votes
1answer
31 views

About the space $u \in C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})$

I am reading Taylor's Partial differential equations III (nonlinear equations) (Section 1 of Chapter 16, Theorem 1.2), and Sogge's Lectures on Non-linear wave equatuions. I notice that in the energy ...
1
vote
1answer
65 views

A fairly difficult differential equation

I was thinking what if I had a differential equation of the form: $$\frac{d^2u}{dx^2} + vu(x) = 0 $$ where $v(y(x))$, that is $y$ is a function of $x$. What are the possible restrictions that I can ...
1
vote
1answer
48 views

When can an arbitrary function be put into this form?

A problem I've been trying to address but not been able to get very far with is to devise a method to check whether or not some function $q(x)$ can be written like $$ q(x) = ...
0
votes
2answers
84 views

What method is used to find the expression of a function?

Hi everybody I've found some difficulties in this exercise please could you give me help: let $f$ continuous function in $\mathbb R$ $$\forall(x;y)\in\mathbb R, f (x+y) + f(x-y) = 2(f(x)+f(y))$$ 1 - ...
2
votes
2answers
92 views

Many other solutions of the Cauchy's Functional Equation

By reading the Cauchy's Functional Euqations on the Wiki, it is said that On the other hand, if no further conditions are imposed on f, then (assuming the axiom of choice) there are infinitely ...
1
vote
0answers
42 views

Does one have to check some additional hypothesis to apply the implicit function theorem for infinite dimensional spaces?

Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces (i.e., L is a bijective bounded linear operator) and $A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation ...
0
votes
0answers
100 views

integral equations - i need help to expand the function [duplicate]

I have the following integral equation to solve: $$\int_{0}^{2\pi} (\cos^2(x+y)+1/2) \phi (y) dy$$ So, I need to find $\lambda$ where $\lambda$ is the eigenvalues's function. Well, my main goal is ...
-2
votes
2answers
117 views

Integral equation that's cant solve… Need a hand [closed]

Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations. $$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi ...
0
votes
1answer
130 views

Finding the Extremals of a Functional J.

The functional $J$ is defined on smooth functions $y \colon [a,b] \to \mathbb{R}$ satisfying $y(a) = u$, $y(b) = v$ and is given by $$J[y]=\int_a^b \sqrt{y} \sqrt{1+(y')^2}\, dx.$$ I have found ...
4
votes
0answers
77 views

$f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$

Consider the equation: $f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$, for $a \geq 0$ and $b \geq 0$. Is my understanding that this simple functional equation is important in analysis. Can ...
2
votes
1answer
77 views

Functional inequality

Let $S$ be a semigroup such that $S\ne S+S$ and let $f:S\to \Bbb C$ be an unbounded function satisfying $$ |f(s_1)f(s_2)-f(t_1)f(t_2)|\le 1 $$ for all $s_1, s_2, t_1, t_2 \in S$ such that ...
0
votes
1answer
250 views

Find all positive functions of a positive real such that $ f(xf(y))=yf(x) $ and $\lim_{x\to\infty}f(x)=0$

Find all functions defined on the set of positive reals which take positive real values and satisfy: $$ f(xf(y))=yf(x) $$ for all ; $ f(x)\to0 $ and as $ x\to\infty $
37
votes
2answers
630 views

Looking for a function such that…

There was this question on one of the whiteboards at my company, and I found it intriguing. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Anyway, here it is: ...
1
vote
0answers
47 views

solving this recurrence equation

is it possibel to solve the equation g$(x)= \sum_{n=1}^{\infty}f(x/n) $ for $ f(x)$ with other methods different from taking the Mellin transform on both sides ?? thanks.
1
vote
3answers
181 views

Solution of functional equation $f(x)=-f(x-a)$

I have a problem with finding solution. I suppose it will be something like $f(x) =G(x)\Re(e^{\frac{x\pi}{a}})$, where $\Re$ is real part of a complex number, $G(x)$ periodic function whith period ...
2
votes
2answers
154 views

Positive twice differential decreasing function, is it convex?

If g is a positive, twice differentiable function that is decreasing and has limit zero at infinity, does g have to be convex? I am sure, from drawing a graph of a function which starts off as being ...
4
votes
0answers
170 views

Vector valued contraction

I am familiar with the Banach Fixed Point Theorem and I have used it to prove existence and uniqueness of functional equations in Banach spaces like $C(X)$, the space of bounded continuous function ...