This paper is concerned with
the Cauchy problem
$(*) \quad \quad u_t+[F(u)]_x=g(t,x,u),\quad
u(0,x)=\overline{u}(x),$
for a nonlinear $2\times 2$ hyperbolic system of
inhomogeneous balance laws
in one space dimension. As usual,
we assume that the system is strictly hyperbolic and that each
characteristic field is either
linearly degenerate or genuinely nonlinear.
Under suitable assumptions on $g$, we prove that there exists
$T>0$ such that, for every $\overline{u}$ with sufficiently small
total variation, the Cauchy problem ($*$)
has a unique "viscosity solution",
defined for $t\in [0,T]$,
depending continuously on the initial data.
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