# On the mixed problem for quasilinear partial functional differential equations with unbounded delay

Annales Polonici Mathematici (1999)

- Volume: 72, Issue: 1, page 87-98
- ISSN: 0066-2216

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topCzłapiński, Tomasz. "On the mixed problem for quasilinear partial functional differential equations with unbounded delay." Annales Polonici Mathematici 72.1 (1999): 87-98. <http://eudml.org/doc/262827>.

@article{Człapiński1999,

abstract = {We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay
$D_tz(t,x) = ∑_\{i=1\}^n f_i(t,x,z_\{(t,x)\})D_\{x_i\}z(t,x) + h(t,x,z_\{(t,x)\})$,
where $z_\{(t,x)\} ∈ X̶_0$ is defined by $z_\{(t,x)\}(τ,s) = z(t+τ,x+s)$, $(τ,s) ∈ (-∞,0]×[0,r]$, and the phase space $X̶_0$ 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.},

author = {Człapiński, Tomasz},

journal = {Annales Polonici Mathematici},

keywords = {Carathéodory solutions; functional differential equation; bicharacteristics; fixed-point theorem; mixed problem; unbounded delay; fixed-point method; local existence and uniqueness of Carathéodory solutions},

language = {eng},

number = {1},

pages = {87-98},

title = {On the mixed problem for quasilinear partial functional differential equations with unbounded delay},

url = {http://eudml.org/doc/262827},

volume = {72},

year = {1999},

}

TY - JOUR

AU - Człapiński, Tomasz

TI - On the mixed problem for quasilinear partial functional differential equations with unbounded delay

JO - Annales Polonici Mathematici

PY - 1999

VL - 72

IS - 1

SP - 87

EP - 98

AB - We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay
$D_tz(t,x) = ∑_{i=1}^n f_i(t,x,z_{(t,x)})D_{x_i}z(t,x) + h(t,x,z_{(t,x)})$,
where $z_{(t,x)} ∈ X̶_0$ is defined by $z_{(t,x)}(τ,s) = z(t+τ,x+s)$, $(τ,s) ∈ (-∞,0]×[0,r]$, and the phase space $X̶_0$ 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.

LA - eng

KW - Carathéodory solutions; functional differential equation; bicharacteristics; fixed-point theorem; mixed problem; unbounded delay; fixed-point method; local existence and uniqueness of Carathéodory solutions

UR - http://eudml.org/doc/262827

ER -

## References

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- [8] Z. Kamont, Hyperbolic functional differential equations with unbounded delay, Z. Anal. Anwendungen 18 (1999), 97-109. Zbl0924.35187
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- [10] V. Lakshmikantham, L. Wen and B. Zhang, Theory of Differential Equations with Unbounded Delay, Kluwer, 1994. Zbl0823.34069
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