Functional differential equations of mixed type in Banach spaces

Cui Baotong

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We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay $D_{t} z (t, x) = \sum_{i = 1}^{n} f_{i} (t, x, z_{(t, x)}) D_{x_{i}} z (t, x) + h (t, x, z_{(t, x)})$ , where $z_{(t, x)} \in X ̶_{0}$ is defined by $z_{(t, x)} (τ, s) = z (t + τ, x + s)$ , $(τ, s) \in (- \infty, 0] \times [0, r]$ , and the phase space $X ̶_{0}$ satisfies suitable axioms. Using the method of bicharacteristics and the fixed-point method we prove a theorem on the local existence and uniqueness of Carathéodory solutions of the mixed problem.