The existence of Carathéodory solutions of hyperbolic functional differential equations

Adrian Karpowicz

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

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

Abstract

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We consider the following Darboux problem for the functional differential equation ∂ ² u / ∂ 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 ) : [ - a ₀ , 0 ] × [ - b ₀ , 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.

How to cite

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Adrian 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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  10. [10] B. Rzepecki, On the existence of solutions of the Darboux problem for the hyperbolic partial differential equations in Banach space, Rend. Semin. Mat.Univ. Padova 76 (1986), 201-206. Zbl0656.35087
  11. [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. [12] J. Straburzyński, The existence of solutions of some functional-differential equations of hyperbolic type, Demonstr. Math. 12 (1979), 105-121. Zbl0434.35062
  13. [13] J. Straburzyński, Existence of solutions of the Goursa problem for some functional-differential equations, Demonstr. Math. 15 (1982), 883-897. Zbl0536.35013
  14. [14] W. Walter, Ordinary functional differential equations and inequalities in the sense of Carathéodory, Appl. Anal. 70 (1998), 85-95. Zbl1031.34068
  15. [15] W. Walter, Differential and Integral Inequalities, Springer, 1970. doi:10.1007/978-3-642-86405-6 

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