# The existence of Carathéodory solutions of hyperbolic functional differential equations

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2010)

- Volume: 30, Issue: 1, page 121-140
- ISSN: 1509-9407

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topAdrian Karpowicz. "The existence of Carathéodory solutions of hyperbolic functional differential equations." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 30.1 (2010): 121-140. <http://eudml.org/doc/271181>.

@article{AdrianKarpowicz2010,

abstract = {We consider the following Darboux problem for the functional differential equation
$∂²u/∂x∂y(x,y) = f(x,y,u_\{(x,y)\},∂u/∂x(x,y),∂u/∂y(x,y))$ a.e. in [0,a]×[0,b],
u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]$0,a]×(0,b],
$where the function $u_\{(x,y)\}:[-a₀,0]×[-b₀,0] → ℝ^\{k\}$ is defined by $u_\{(x,y)\}(s,t) = u(s+x,t+y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.},

author = {Adrian Karpowicz},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {existence theorem; functional differential equation; hyperbolic equation; Darboux problem; solution in the sense of Carathéodory},

language = {eng},

number = {1},

pages = {121-140},

title = {The existence of Carathéodory solutions of hyperbolic functional differential equations},

url = {http://eudml.org/doc/271181},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Adrian Karpowicz

TI - The existence of Carathéodory solutions of hyperbolic functional differential equations

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2010

VL - 30

IS - 1

SP - 121

EP - 140

AB - We consider the following Darboux problem for the functional differential equation
$∂²u/∂x∂y(x,y) = f(x,y,u_{(x,y)},∂u/∂x(x,y),∂u/∂y(x,y))$ a.e. in [0,a]×[0,b],
u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]$0,a]×(0,b],
$where the function $u_{(x,y)}:[-a₀,0]×[-b₀,0] → ℝ^{k}$ is defined by $u_{(x,y)}(s,t) = u(s+x,t+y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We prove a theorem on existence of the Carathéodory solutions of the above problem.

LA - eng

KW - existence theorem; functional differential equation; hyperbolic equation; Darboux problem; solution in the sense of Carathéodory

UR - http://eudml.org/doc/271181

ER -

## References

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- [11] J. Simon, Compact sets in the Space ${L}^{p}(0,T;B)$, Annali di Matematica Pura ed Applicate 146 (1986), 65-96. Zbl0629.46031
- [12] J. Straburzyński, The existence of solutions of some functional-differential equations of hyperbolic type, Demonstr. Math. 12 (1979), 105-121. Zbl0434.35062
- [13] J. Straburzyński, Existence of solutions of the Goursa problem for some functional-differential equations, Demonstr. Math. 15 (1982), 883-897. Zbl0536.35013
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