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On the mixed problem for hyperbolic partial differential-functional equations of the first order. (English) Zbl 1010.35021
Summary: We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order \[ D_xz(x,y) = f(x,y,z_{(x,y)}, D_yz(x,y)), \] where \(z_{(x,y)} : [-\tau ,0] \times [0,h] \rightarrow \mathbb{R}\) is a function defined by \(z_{(x,y)}(t,s) = z(x+t, y+s)\), \((t,s) \in [-\tau ,0] \times [0,h]\). Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.

MSC:
35D05 Existence of generalized solutions of PDE (MSC2000)
35R10 Functional partial differential equations
35L60 First-order nonlinear hyperbolic equations
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