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We generalize Picard-Lindelof theorem/ the method of characteristics to thefollowing system of PDE: $C_{il}(x,y) {\partial y_i / \partial x_l} + {\partialy_i / \partial x_m} = D_i(x,y)$. With a Lipschitz or $C^r$ $C_{il},D_i: [-a,a]^{m} \times [-b, b]^{n} \rightarrow \mathbb{R}$ and initial condition $I_i:[-\bar{a}, \bar{a}]^{m-1} \rightarrow (-b,b)$, $\bar{a} \leq a$, we obtain alocal unique Lipschitz or $C^r$ solution $f$, respectively that satisfies theinitial condition, $f_i (v, 0 ) = I_i(v)$, $v \in [-\bar{a}, \bar{a}]^{m-1}$.To construct the solution we set bounds on the value of the solution bydiscretizing the domain of the solution along the direction perpendicular tothe initial condition hyperplane. As the number of discretization hyperplanesis taken to infinity the upper and lower bounds of the solution approach eachother, hence this gives a unique function for the solution ($Ufs$). A localitycondition is derived based on the constants of the problem. The dependence of$C_{il}$, $D_i$ and $I_i$ on parameters, the generalization to nonlinearsystems of PDE and the application to hyperbolic quasilinear systems of firstorder PDE in two independent variables is discussed.

A generalization of Picard-Lindelof theorem/ the method of characteristics to systems of PDE