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Volterra Integral Equation as a Generalisation of Picard Theorem

  1. Oct 10, 2012 #1
    Hi Physics Forums, this is my first post here thanks in advance for any help hopefully I'll be able to return the favor.

    1. The problem statement, all variables and given/known data

    As a generalisation of the Picard Theorem: An integral equation of the form:

    [itex]y(x) = f(x) + \int_0^xK(x,x')y(x')dx'[/itex] [itex](0 \leq x \leq b)[/itex]

    where [itex]f(x)[/itex] and [itex]K(x,x')[/itex] are continuous is called a Volterra integral equation. Let [itex]||f||[/itex] and [itex]||K||[/itex] be upper bounds for
    [itex]0 \leq x \leq b [/itex] and for [itex]|K|[/itex] on [itex]0 \leq x' \leq x \leq b[/itex], respectively.
    Prove that the sequence [itex]\{y_n\}_{n\geq 0}[/itex] of functions defined by [itex]y_0(x) = f(x)[/itex] and

    [itex]y_{n+1} = f(x) + \int_0^xK(x,x')y_{n}(x')dx'[/itex] [itex]n\geq0[/itex]
    converges to a solution y(x) of the equation.

    2. Relevant equations

    3. The attempt at a solution
    I'm trying to show that the difference between successive approximations [itex]y_{n+1}-y_{n}[/itex] tends to [itex]0[/itex] as [itex]n\rightarrow \infty[/itex] But the problem is that when I try to formulate an induciton hypothesis to prove this, it appears to diverge...
    That is:

    [itex]\left|y_1 - y_0\right| = \left|\int_0^xK(x,x')y_0(x')dx'\right|[/itex]
    [itex]\leq \left|\int_0^xBdx'\right|[/itex]
    [itex]= \left|Bx\right|[/itex] where B = ||f||.||K||
    [itex]\left|y_2 - y_1\right| =\left|\int_0^xK(x,x')y_1(x') - K(x,x')y_0(x')dx'\right|[/itex]
    [itex]\leq \left|\int_0^xBx - Bdx'\right|[/itex]
    [itex]= \left|B(x^2 - x)\right|[/itex]

    But going this way, the difference appears to be getting bigger between successive approximations.
    I think my problems are stemming from integrating with respect to x' instead of x,
    can someone please show me where I'm going wrong?

    Thanks a lot for any help at all.
  2. jcsd
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