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I got this homework question that I am stuck on. Let $J = [0, a]$ (with $a > 0$ fixed). Let $g(x)$ be a function which is continuous at all $x \in J$ and let $k(x, t, z)$ be a function which is continuous and bounded in $\{(x, t, z) \in J\times J \times \mathbb{R} : t < x\}$. Prove that there exists at least one solution of the equation: $$y(x)=g(x)+\int\limits_{0}^{x}k(x,t,y(t))dt, \ \ \ x\in J$$ It can apparently be proven in a similar way as the Peano existence theorem http://en.wikipedia.org/wiki/Peano_existence_theorem I would show my workings so far but I just have no idea where to start on this one. I think that we need to differentiate the given equation to transform it into an equivalent differential equation and then show that a solution to the DE exists. Any help would be much appreciated, Thanks.

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I think you can make this work with successive approximations in a more direct way that what you're suggesting. Try the following: $$ y^N(0) := g(0)$$, and then $$y^N(\frac{i a}{N}) = g(\frac{i a}{N}) + \int_{0}^{\frac{i a}{N}} k(\frac{ia}{N}, t, y^N(t)) dt$$, where you linearly interpolate between successive values of $y^N$ at $t = i a/N$. The family $\{y^N\}$ is uniformly bounded and equicontinuous (show this using the slopes), and so possesses a uniformly convergent subsequence. –  A Blumenthal Jan 23 '13 at 22:25
    
So I would apply the Arzelà–Ascoli theorem on the function $y^N$ to show that it contains a convergent subsequence which would satisfy the required equation? –  Slugger Jan 23 '13 at 22:53

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