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## Annales Polonici Mathematici

1999 | 72 | 1 | 87-98
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
87-98
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-11-18
Twórcy
autor
• Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
Bibliografia
• [1] P. Brandi, Z. Kamont and A. Salvadori, Differential and differential-difference inequalities related to mixed problems for first order partial differential-functional equations, Atti. Sem. Mat. Fis. Univ. Modena 39 (1991), 255-276.
• [2] T. Człapiński, Existence of solutions of the Darboux problem for partial differential-functional equations with infinite delay in a Banach space, Comm. Math. 35 (1995), 111-122.
• [3] T. Człapiński, On the mixed problem for quasilinear differential-functional equations of the first order, Z. Anal. Anwendungen 16 (1997), 463-478.
• [4] T. Człapiński, On the Chaplyghin method for partial differential-functional equations of the first order, Univ. Iagel. Acta Math. 35 (1997), 137-149.
• [5] T. Człapiński, On the mixed problem for hyperbolic partial differential-functional equations of the first order, Czechoslovak Math. J., to appear.
• [6] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer, 1991.
• [7] G. A. Kamenskii and A. D. Myshkis, On the mixed type functional-differential equations, Nonlinear Anal. 30 (1997), 2577-2584.
• [8] Z. Kamont, Hyperbolic functional differential equations with unbounded delay, Z. Anal. Anwendungen 18 (1999), 97-109.
• [9] Z. Kamont and K. Topolski, Mixed problems for quasilinear hyperbolic differential-functional systems, Math. Balkanica 6 (1992), 313-324.
• [10] V. Lakshmikantham, L. Wen and B. Zhang, Theory of Differential Equations with Unbounded Delay, Kluwer, 1994.
• [11] J. Turo, Local generalized solutions of mixed problems for quasilinear hyperbolic systems of functional partial differential equations in two independent variables, Ann. Polon. Math. 49 (1989), 256-278.
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