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Equation data
Category:5. Integral Equations
Subcategory:5.1. Linear Equations of the First Kind with Variable Limit of Integration
Equation(s):1. $\displaystyle\int_0^{x^\alpha}y(t)dt=\lambda y(x),\ \alpha\in (0,1), y\in L_2[0,1]$.

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2. $\displaystyle\int_{x^{1/\alpha}}^1 y(t)dt=\lambda y(x),\ \alpha\in (0,1), y\in L_2[0,1]$.
Solution(s),
Transformation(s),
Integral(s)
:
1. If $\lambda\not\in\{(1-\alpha)\alpha^n, \ \hbox{where} \ n=0,1,\dots\}$ then $y(x)= 0$.

If $\lambda=\lambda_{n+1}=(1-\alpha)\alpha^n$ then $\displaystyle y(x)=y_{n+1}(x)
=x^{\frac{\alpha}{1-\alpha}}\left(\ln^n x+ \sum\limits_{k=1}^{n}\frac{n!}{(n-k)!}
 \frac{\alpha^{k(k-1)/2}(1-\alpha)^k}{(1-\alpha)\dots
 (1-\alpha^k)}\ln^{n-k}x\right)$.

Solutions $\{y_{n+1}\}_{n=0}^\infty$ form a complete system in $L_2[0,1]$.
\medskip

2. If $\lambda\not\in\{(1-\alpha)\alpha^n, \ \hbox{where} \ n=0,1,\dots\}$ then $y(x)= 0$.

If $\lambda=\lambda_{n+1}=(1-\alpha)\alpha^n$ then $\displaystyle y(x)=y_{n+1}(x)=
1+\sum\limits_{k=2}^\infty(-1)^{k-1}\frac{\alpha^{(k-1)(k-2-2n)/2}}{(1-\alpha)\dots
 (1-\alpha^{k-1})}x^{\frac{1-\alpha^{k-1}}{(1-\alpha)\alpha^{k-1}}}$.

 Solutions $\{y_{n+1}\}_{n=0}^\infty$ do not form a complete system in $L_2[0,1]$.

It is interesting to note that $\sum_{n=0}^\infty\lambda_n$ does not depend on $\alpha$, namely
$\sum_{n=0}^\infty\lambda_n=\sum_{n=0}^\infty(1-\alpha)\alpha^n=1$.
Remarks:More general result.

Let $\phi :  [0,1]\longrightarrow [0,1]$ be a  nondecreasing
continuous function such that $\phi(x)>x$ for $x\in (0,1)$ and
$V_{\phi}$ be defined on $L_2[0,1]$ by 
$V_{\phi}:\
f(x)\rightarrow\int\limits_0^{\phi(x)}f(t)dt$.
Let also
$\sigma_p(V_\phi)\setminus\{0\}=\{\lambda_n\}_{n=1}^\omega$ $(
1\le\omega\le\infty)$ denotes a point spectrum of $V_\phi$. Then

 $(1)$ $\omega<\infty$ if and only if $\phi(0)>0$ and $\phi(1-\varepsilon)=1$ for some
$0<\varepsilon<1$;

 $(2)$  $\lim\limits_{\varepsilon\rightarrow
  0}\sum\limits_{|\lambda_n|>\varepsilon}\lambda_n=1$;

  $(3)$ $\sum\limits_{n=1}^\omega|\lambda_n|^{1+\varepsilon}<\infty$ for all
  $\varepsilon>0$.
Novelty:Material has been fully published elsewhere
References:1. I. Yu. Domanov, On the spectrum of the operator which is a composition of integration and substitution, Studia Math., 2008, Vol. 185, No.1, pp. 49--65.
2. I. Yu. Domanov, On the spectrum and eigenfunctions of the operator $(Vf)(x)=\int\limits_0^{x^\alpha}f(t)dt$, Banach Center Publ., 2007, Vol. 75, pp. 137--142.
Author/Contributor's Details
Last name:Domanov
First name:Ignat
Middle(s) name:Yurii
Country:Ukraine
City:Donetsk
Affiliation:Institute of Applied Mathematics and Mechanics
Statistic information
Submission date:Sat 21 Jun 2008 10:31
Edits by author:0

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