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

Człapiński, Tomasz

Annales Polonici Mathematici, Tome 72 (1999), p. 87-98 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider the mixed problem for the quasilinear partial functional differential equation with unbounded delay $D_{t} z (t, x) = \sum_{i = 1}^{n} f_{i} (t, x, z_{(t, x)}) D_{x_{i}} z (t, x) + h (t, x, z_{(t, x)})$ , where $z_{(t, x)} \in X ̶_{0}$ is defined by $z_{(t, x)} (τ, s) = z (t + τ, x + s)$ , $(τ, s) \in (- \infty, 0] \times [0, r]$ , and the phase space $X ̶_{0}$ 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.

Publié le : 1999-01-01

Zbl 0947.35170

EUDML-ID : urn:eudml:doc:262827

@article{bwmeta1.element.bwnjournal-article-apmv72z1p87bwm,
     author = {Cz\l api\'nski, Tomasz},
     title = {On the mixed problem for quasilinear partial functional differential equations with unbounded delay},
     journal = {Annales Polonici Mathematici},
     volume = {72},
     year = {1999},
     pages = {87-98},
     zbl = {0947.35170},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv72z1p87bwm}
}

Człapiński, Tomasz. On the mixed problem for quasilinear partial functional differential equations with unbounded delay. Annales Polonici Mathematici, Tome 72 (1999) pp. 87-98. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv72z1p87bwm/

Bibliographie

[00000] [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. | Zbl 0737.35133

[00001] [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. | Zbl 0859.35131

[00002] [3] T. Człapiński, On the mixed problem for quasilinear differential-functional equations of the first order, Z. Anal. Anwendungen 16 (1997), 463-478. | Zbl 0878.35027

[00003] [4] T. Człapiński, On the Chaplyghin method for partial differential-functional equations of the first order, Univ. Iagel. Acta Math. 35 (1997), 137-149. | Zbl 0942.35044

[00004] [5] T. Człapiński, On the mixed problem for hyperbolic partial differential-functional equations of the first order, Czechoslovak Math. J., to appear. | Zbl 1010.35021

[00005] [6] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer, 1991. | Zbl 0732.34051

[00006] [7] G. A. Kamenskii and A. D. Myshkis, On the mixed type functional-differential equations, Nonlinear Anal. 30 (1997), 2577-2584. | Zbl 0896.34058

[00007] [8] Z. Kamont, Hyperbolic functional differential equations with unbounded delay, Z. Anal. Anwendungen 18 (1999), 97-109. | Zbl 0924.35187

[00008] [9] Z. Kamont and K. Topolski, Mixed problems for quasilinear hyperbolic differential-functional systems, Math. Balkanica 6 (1992), 313-324. | Zbl 0832.35143

[00009] [10] V. Lakshmikantham, L. Wen and B. Zhang, Theory of Differential Equations with Unbounded Delay, Kluwer, 1994. | Zbl 0823.34069

[00010] [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. | Zbl 0685.35065