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Tytuł artykułu

The existence of Carathéodory solutions of hyperbolic functional differential equations

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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.
Twórcy
  • Institute of Mathematics, University of Gdańsk, Wit Stwosz St. 57, 80-952 Gdańsk, Poland
Bibliografia
  • [1] A. Alexiewicz and W. Orlicz, Some remarks on the existence and uniqueness of solutions of the hyperbolic equation ∂²z/∂x∂y = f(x,y,z,∂z/∂x,∂z/∂y), Stud. Math. 15 (1956), 201-215.
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  • [6] K. Deimling, A Carathéodory theory for systems of integral equations, Ann. Mat. Pura Appl. 86 (1970), 217-260.
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  • [8] A. Pelczar, Some functional differential equations, Diss. Math. 100 (1973), 3-74.
  • [9] R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic Publisher, cop. 2002.
  • [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.
  • [11] J. Simon, Compact sets in the Space $L^{p}(0,T;B)$, Annali di Matematica Pura ed Applicate 146 (1986), 65-96.
  • [12] J. Straburzyński, The existence of solutions of some functional-differential equations of hyperbolic type, Demonstr. Math. 12 (1979), 105-121.
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  • [14] W. Walter, Ordinary functional differential equations and inequalities in the sense of Carathéodory, Appl. Anal. 70 (1998), 85-95.
  • [15] W. Walter, Differential and Integral Inequalities, Springer, 1970. doi:10.1007/978-3-642-86405-6
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1115
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