Existence of BV solutions for $2\times 2$ hyperbolic balance laws for $L^{\mathrm{\infty }}$ initial data

We prove the existence of BV solutions for $2\times 2$ system of hyperbolic balance laws in one space dimension. The flux is assumed to have two genuinely nonlinear characteristic fields. We consider a general force which may possibly depend on time and space variable as well. To prove the existence, we assume the initial data to be small in $L^{\mathrm{\infty }}$. Furthermore, we also study qualitative behavior for entropy solutions to hyperbolic system of balance laws.