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Consider the integral equation

$$\phi(x) = x + \lambda\int_0^1 \phi(s)\,ds$$

Integrating with respect to $x$ from $x=0$ to $x=1$:

$$\int_0^1 \phi(x)\,dx = \int_0^1x\,dx + \lambda \int_0^1\Big[\int_0^1\phi(s)\,ds\Big]\,dx$$

which is equivalent to

$$\int_0^1 \phi(x)\,dx = \frac{1}{2} + \lambda \int_0^1\phi(s)\,ds$$

How can I go from here in order to solve the problem for the homogeneous case and find the corresponding characteristic values and associated rank?

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    $\begingroup$ What is $\lambda$? What do you mean by "solve the problem"? I don't see what "the problem" is supposed to mean. Of which object do you want to find the characteristic values and ranks? Have you checked your definition of $\phi$? $\endgroup$ – James Mar 25 at 17:50
  • $\begingroup$ My apologies, $\lambda$ is an arbitrary constant. In essence I want to obtain an expression of $\phi(x)$ which does not contain a function of $s$, which the initial integral equation has. $\endgroup$ – LightningStrike Mar 25 at 17:54
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Relabelling the dummy variable $x\mapsto s$ on the LHS of your final equation, $$\int_0^1\phi(s)\,ds-\lambda\int_0^1\phi(s)\,ds=\frac12\\\implies \int_0^1\phi(s)\,ds=\frac1{2(1-\lambda)}$$

Thus $$\phi(x)=x+\frac\lambda{2(1-\lambda)}$$

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Note $\int_{0}^{1}{\phi(s)ds}$ is a constant say $a$. Your functional equation (FE) can be rewritten as: $$\phi(x)=x+a\lambda$$ Putting into FE yields:

$$x+a\lambda=x+\lambda\int_{0}^{1}{(s+a\lambda )ds }\iff a\lambda=\lambda\big(\frac{1}{2}+\lambda a\big)$$ If $\lambda=0$ then $\phi(x)=x$

if $\lambda\ne 1$ $a=\frac{1}{2}+\lambda a\iff ( 1-\lambda)a=\frac{1}{2}\iff a=\frac{1}{2-2\lambda}$ and then $\phi(x)=x+\frac{\lambda}{2-2\lambda}$

If $\lambda=1$ there won’t besuch $\phi$.

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If you are after finding $\phi(x)$, one approach that comes to mind is to assume it is smooth enough to have a normally convergent (so we can interchange series summation and integration) Taylor expansion on $[0, 1]$: $$ \phi(x) = \sum_{n \geq 0} a_{n} x^{n}. $$ Substituting it into your equation, we get: $$ \sum_{n \geq 0} a_{n} x^{n} = x + \lambda \sum_{n \geq 0}{a_{n} \over n+1}. $$ Matching up the coefficients of the difference powers of $x$, we get: $$ a_{n} = 0 \quad \mbox{ for } n \geq 2, $$ $$ a_{1} = 1, $$ and $$ a_{0} = \lambda \left(a_{0} + {a_{1} \over 2}\right). $$ This gives a relationship between $a_{0}$ and $\lambda$.

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Note that since $\lambda\int_0^1 \phi(s)\,ds$ is a constant (with respect to $x$), then we can write$$\phi(x)=x+a$$and by substitution we conclude that $$x+a=x+\lambda\int _{0}^{1}x+adx\implies\\a=\lambda({1\over 2}+a)\implies\\a={\lambda\over 2-2\lambda}$$ and we obtain$$\phi(x)=x+{\lambda\over 2-2\lambda}\quad,\quad \lambda\ne 1$$The case $\lambda=1$ leads to no solution.

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