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We consider perodic homogenization of boundary value problems for quasilinearsecond-order ODE systems in divergence form of the type$a(x,x/\varepsilon,u(x),u'(x))'= f(x,x/\varepsilon,u(x),u'(x))$ for $x \in[0,1]$. For small $\varepsilon>0$ we show existence of weak solutions$u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty\approx 0$, where $u_0$ is a given non-degenerate solution to the homogenizedboundary value problem, and we describe the rate of convergence to zero for$\varepsilon \to 0$ of the homogenization error $\|u_\varepsilon-u_0\|_\infty$.In particular, we show that this rate depends on the smoothness of the maps$a(\cdot,y,u,u')$ and $f(\cdot,y,u,u')$.  Our assumptions are, roughly speaking, as follows: The maps$a,f:[0,1]\times\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$are continuous, the maps $a(x,y,\cdot,\cdot)$ and $f(x,y,\cdot,\cdot)$ are$C^1$-smooth, the maps $a(x,\cdot,u,u')$ and $f(x,\cdot,u,u')$ are 1-periodic,and the maps $a(x,y,u,\cdot)$ are strongly monotone and Lipschitz continuousuniformly with respect to $x$, $y$ and bounded $u$. Neither global solutionuniqueness is supposed nor $W^{2,2}$-regularity of $u_0$.  The main tool of the proofs is an abstract result of implicit functiontheorem type which in the past has been applied to singularly perturbednonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits acommon approach to existence and local uniqueness results for singularlyperturbed problems and and for homogenization problems. %Hence, the presentpaper is a first step (on the ODE level) to create a common approach toexistence, local uniqueness and error estimates for nonlinear singularlyperturbed problems and for nonlinear homogenization problems.

A common approach to singular perturbation and homogenization I: Quasilinear ODE systems