Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates

We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$((A(x\mathrm{/}\epsilon )+B(x\mathrm{/}\epsilon ))u^{\mathrm{\prime }}(x)+c(x,u(x)){)}^{\mathrm{\prime }}=d(x,u(x))forx\in (0,1),u(0)=u(1)=0.$$ For small $\epsilon >0$ we show existence of weak solutions $u=u_{\epsilon }$ as well as their local uniqueness for $\mathrm{\parallel }u-u_0{\mathrm{\parallel }}_{\mathrm{\infty }}\approx 0$, where $u=u_0$ is a given solution to the homogenized problem $$(A_0{u}^{\mathrm{\prime }}+c(x,u(x)){)}^{\mathrm{\prime }}=d(x,u(x))forx\in (0,1),u(0)=u(1)=0,A_0:={({\int }_0^{1}A(y{)}^{-1}dy)}^{-1}$$ such that the linearized problem $$(A_0{u}^{\mathrm{\prime }}+{\mathrm{\partial }}_{u}c(x,u_0(x))u(x){)}^{\mathrm{\prime }}={\mathrm{\partial }}_{u}d(x,u_0(x))u(x)forx\in (0,1),u(0)=u(1)=0$$ does not have weak solutions $u≠0$. Further, we prove that $\mathrm{\parallel }{u}_{\epsilon }-u_0{\mathrm{\parallel }}_{\mathrm{\infty }}\to 0$ and, if $c(\cdot ,u)\in W^{1,\mathrm{\infty }}((0,1);{\mathbb{R}}^n)$, that $\mathrm{\parallel }{u}_{\epsilon }-u_0{\mathrm{\parallel }}_{\mathrm{\infty }}=O(\epsilon )$ for $\epsilon \to 0$. Moreover, all these statements are true, roughly speaking, uniformly with respect to the localized defects $B$. We assume that $A\in L^{\mathrm{\infty }}(\mathbb{R};{\mathbb{M}}_n)$ is 1-periodic, $B\in L^{\mathrm{\infty }}(\mathbb{R};{\mathbb{M}}_n)\cap L^1(\mathbb{R};{\mathbb{M}}_n)$, $A(y)$ and $A(y)+B(y)$ are positive definite uniformly with respect to $y$, $c(x,\cdot ),d(x,\cdot )\in C^1({\mathbb{R}}^n;{\mathbb{R}}^n)$ and $c(\cdot ,u),d(\cdot ,u)\in L^{\mathrm{\infty }}((0,1);{\mathbb{R}}^n)$. The main tool of the proofs is an abstract result of implicit function theorem type which has been tailored for applications to nonlinear singular perturbation and homogenization problems.