# Questions tagged [integral-operators]

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### Bifurcation points in parametric Hammerstein Integral equation

I am a theoretical physicist working on integrable systems, hence I preemptively ask forgiveness for the eventual lack of mathematical rigor. My question concerns the properties of a particular ...
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### Eigenfunctions and eigenvalues of an operator defined by a certain integral

Let $k :[0,1]^2 \rightarrow \bf{R}$ be a kernel function definded by $k(x,y)= (1- max(x,y))^2 .$ Now, let $L$ be a linear operator defined on $L^2 [0,1]$ by $$Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...
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### A Fredholm equation with a particular kernel

How to solve this kind of Fredholm’s equation? $$x(t)+\lambda \int\limits_{0}^{1}\! \big[ts - \min\{t,s\}\big]x(s)ds=t$$ Thanks for any help.
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### Delta-distribution composed with a function from the Fourier representation

A well known representation of Dirac's delta-distribution is via the Fourier transform of distributions: \begin{equation} \delta[f]:=f(0)=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}xk}f(k)\mathrm{...
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### Integral equation with kernel defined in a rectangle

Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$ Observe that the kernel is not defined on a square. My question: Can I apply the classical ...
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### Uniqueness of solution of Volterra Integral Equation with deviating argument

In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$: \begin{equation}...
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### A special integral equation of Volterra type

Let $a,f \in L^2(0,t)$ (where $t \leqslant 1$), and consider the following integral equation: $$f(t)\int_0^t a(s)\,ds + \int_0^t a(t - s) f(s) \, ds = 0$$ My question is : under what condition ...
Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-...