Carathéodory solutions of hyperbolic functional differential inequalities with first order derivatives

Adrian Karpowicz. "Carathéodory solutions of hyperbolic functional differential inequalities with first order derivatives." Annales Polonici Mathematici 94.1 (2008): 53-78. <http://eudml.org/doc/281110>.

@article{AdrianKarpowicz2008,
abstract = {We consider the Darboux problem for a functional differential equation: $(∂²u)/(∂x∂y) (x,y) = f(x,y,u_\{(x,y)\},u(x,y),∂u/∂x (x,y),∂u/∂y (x,y))$ 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 give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.},
author = {Adrian Karpowicz},
journal = {Annales Polonici Mathematici},
keywords = {functional differential inequalities; hyperbolic equations; Darboux problem; solutions in the sense of Carathéodory},
language = {eng},
number = {1},
pages = {53-78},
title = {Carathéodory solutions of hyperbolic functional differential inequalities with first order derivatives},
url = {http://eudml.org/doc/281110},
volume = {94},
year = {2008},
}

TY - JOUR
AU - Adrian Karpowicz
TI - Carathéodory solutions of hyperbolic functional differential inequalities with first order derivatives
JO - Annales Polonici Mathematici
PY - 2008
VL - 94
IS - 1
SP - 53
EP - 78
AB - We consider the Darboux problem for a functional differential equation: $(∂²u)/(∂x∂y) (x,y) = f(x,y,u_{(x,y)},u(x,y),∂u/∂x (x,y),∂u/∂y (x,y))$ 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 give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.
LA - eng
KW - functional differential inequalities; hyperbolic equations; Darboux problem; solutions in the sense of Carathéodory
UR - http://eudml.org/doc/281110
ER -