Conditions for the invertibility of the nonlinear difference operator $(\mathscr Rx)(n)=H(x(n),x(n+1))$, $n\in\mathbb Z$, in the space of bounded number sequences

Necessary and sufficient conditions are found for the invertibility of the nonlinear difference operator
$$ (\mathscr Rx)(n)=H(x(n),x(n+1)),\qquad n\in\mathbb Z, $$
in the space of bounded two-sided number sequences. Here $H\colon \mathbb R^2\to \mathbb R $ is a continuous function.
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