Geodesic
On the mixed problem for hyperbolic partial differential-functional equations of the first order
Czechoslovak Mathematical Journal, Tome 49 (1999) no. 4, pp. 791-809.

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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.
Classification : 35A30, 35D05, 35L60, 35R10
Keywords: partial differential-functional equations; mixed problem; generalized solutions; local existence; bicharacteristics; successive approximations 
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     author = {Cz{\l}api\'nski, Tomasz},
     title = {On the mixed problem for hyperbolic partial differential-functional equations of the first order},
     journal = {Czechoslovak Mathematical Journal},
     pages = {791--809},
     publisher = {mathdoc},
     volume = {49},
     number = {4},
     year = {1999},
     mrnumber = {1746704},
     zbl = {1010.35021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_1999__49_4_a9/}
}
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Człapiński, Tomasz. On the mixed problem for hyperbolic partial differential-functional equations of the first order. Czechoslovak Mathematical Journal, Tome 49 (1999) no. 4, pp. 791-809. http://geodesic.mathdoc.fr/item/CMJ_1999__49_4_a9/