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Math Help - Homogeneous Fredholm equation of the second kind

  1. #1
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    Homogeneous Fredholm equation of the second kind

    Hi,
    during the analysis of a problem in my phd thesis
    I have resulted in the following equation.

    \varphi(x)= \int_a^b K(x,t)\varphi(t)dt

    which is clearly a homogeneous Fredholm equation of the second kind

    The problem is that I can't find in any text any way of solving it.
    Solutions are provided only for special cases like when the kernel K
    is symmetric

    K(x,t)=K(x,t)
    or when it is separable which are both not my case.

    The particular form of the equation I am dealing with is
    \varphi(x)= \int_a^b \Lambda(x,t)g(x)\varphi(t)dt

    where \Lambda(x,t) is symmetric and depends only in the difference x-t, \Lambda(x,t)=\Lambda(x-t) , and g(x) a known function involving logarithm.

    Any ideas of how to deal with this kind of form?
    Thank you in advance
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  2. #2
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    I don't really know about integral equations, but it is an easy consequence of Banach's fixed point theorem that if 1> \| K \| _{\infty} (b-a) then your equation only has \varphi (x)=0 as solution. In general the eigenvalue problem \lambda \varphi (x) = \int_a^b K(x,t)\varphi (t)dt has only the trivial solution if |\lambda | > \| K\| _{\infty} (b-a) where K: [a,b]\times [a,b] \rightarrow \mathbb{R} is continous and we consider the mapping F(f)(x)= \int_a^b K(x,t)f(t)dt from C[a,b] to itself.
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  3. #3
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    Hey, I've been thinking about this problem and here's some things that I came up with: We can consider the same operator F, only this time defined in other spaces.

    1.- F: L^1[a,b] \rightarrow L^1[a,b] and g\in L^1[a,b] \cap L^{\infty} [a,b] and we then have (noting that K(x,t)=K(x-t)) \| F(\varphi ) \| _1 \leq \| g\| _{\infty } \|K\| _{1 } \| \varphi \| _1 and so if \| g\| _{\infty } \|K\| _{1 } \leq \alpha \in (0,1) we get only the trivial solution. What I did here was note that \int_a^b K(x,t)\varphi (t)dt is just a convolution, and use said operation's properties.

    2.- F:L^2[a,b] \rightarrow L^2[a,b] and K \in L^1 \left( [a,b]\times [a,b] \right) \cap L^{\infty } \left( [a,b] \times [a,b] \right) we have then \| F(\varphi ) \| _2 \leq c\| \varphi \| _2 \| \overline{K} g \| _2 where \overline{K} (x)= \mbox{ess} \sup_t  \ K(x,t). This one's a little harder to get (I'm still not sure I have all the calculations right) and uses Jensen's inequality (if you want the details just ask).

    If the equation does (or should) have a non-trivial solution well...
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  4. #4
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    Thank you for your reply.
    I think that the trivial solution always exists as a consequence of the Fredholm theory.

    Anyway, the trivial solution is definitely not what I am looking for.
    The φ(x) function corresponds to a physical quantity in my problem
    and it should not be zero.

    In fact I expect a family of solutions depending on some parameter.
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  5. #5
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    Quote Originally Posted by yiorgos View Post
    Thank you for your reply.
    I think that the trivial solution always exists as a consequence of the Fredholm theory.

    Anyway, the trivial solution is definitely not what I am looking for.
    The φ(x) function corresponds to a physical quantity in my problem
    and it should not be zero.

    In fact I expect a family of solutions depending on some parameter.
    Well, the trivial solution exists simply because F is a linear operator.

    If you want a non-trivial solution first check that none of the above inequalities hold, and maybe if K is smooth (or C^1) you could differentiate the equation to get some sort of differential equation for \varphi. If all else fails maybe a qualitative approach suffices for you.
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