# A slightly generalized existence and uniqueness theorem for integral equations (reference request)

I want to use the following statement without including the proof, which is completely straightforward but rather tedious:

Let $G_0:\mathbb R\times\mathbb R^m\to\mathbb R^m$, $\Theta_0:\mathbb R\times\mathbb R\times\mathbb R^m\to\mathbb R^m$, $H_0:\mathbb R\to\mathbb R^m$ be $C^\infty$ functions.

Suppose that a $C^\infty$ function $Z_0:[0,T]\to\mathbb R^m$ satisfies the integral equation $$G_0(t,Z_0(t))=H_0(t)+\int_{0}^t \Theta_0(t,s,Z_0(s))ds$$ for $0\le t\le T$ and $\operatorname{det}D_ZG_0(t,Z_0(t))\ne 0$ for all $t\in[0,T]$. Then for all $C^\infty$ functions $G,\Theta,H$ sufficiently close to $G_0,\Theta_0,H_0$ with many derivatives, there exists a unique solution $Z(t)$, $t\in[0,T]$ of the equation $$G(t,Z(t))=H(t)+\int_{0}^t \Theta(t,s,Z(s))ds$$ that is close to $Z_0$ uniformly and that solution is automatically close to $Z_0$ with many derivatives.

I suspect that it is all written in some classical and easy to access textbook on integral equations but my and my co-authors' attempts to search our library for a suitable reference failed. Of course, we always have an option to say "The proof is similar to ..." but, if the exact reference can be given, it would be much better.

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I've never seen this before, but it appears to be a rather straightforward variant of standard results. What I've done under similar circumstances is to state and prove the precise result in an appendix to my paper. –  Deane Yang Oct 28 '11 at 1:36
Yeah, that IS a completely straightforward variant of standard results and we have used the approach you suggested in a couple of our previous papers. We can use it again and again but it makes me feel like I'm in the business of rewriting textbooks for undergraduate students instead of that of producing research articles. Well, I'm, at least, happy to hear that somebody else has been in the same shoes :). –  fedja Oct 28 '11 at 2:24