Geodesic
On the Cauchy Problem for Differential Equations in a~Banach Space over the Field of $p$-Adic Numbers
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 99-106.

Voir la notice de l'article provenant de la source Math-Net.Ru

For an operator-differential equation of the form $y^{(m)}(z) = Ay(z)$, where $A$ is a closed linear operator on a Banach space over the field of $p$-adic numbers, conditions on the initial data are given that are necessary and sufficient for the Cauchy problem to be well-posed in the class of locally analytic vector-valued functions. The result is illustrated by $p$-adic partial differential equations. 
@article{TM_2004_245_a10,
     author = {M. L. Gorbachuk and V. I. Gorbachuk},
     title = {On the {Cauchy} {Problem} for {Differential} {Equations} in {a~Banach} {Space} over the {Field} of $p${-Adic} {Numbers}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {99--106},
     publisher = {mathdoc},
     volume = {245},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_245_a10/}
}
TY  - JOUR
AU  - M. L. Gorbachuk
AU  - V. I. Gorbachuk
TI  - On the Cauchy Problem for Differential Equations in a~Banach Space over the Field of $p$-Adic Numbers
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2004
SP  - 99
EP  - 106
VL  - 245
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2004_245_a10/
LA  - ru
ID  - TM_2004_245_a10
ER  - 
%0 Journal Article
%A M. L. Gorbachuk
%A V. I. Gorbachuk
%T On the Cauchy Problem for Differential Equations in a~Banach Space over the Field of $p$-Adic Numbers
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2004
%P 99-106
%V 245
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2004_245_a10/
%G ru
%F TM_2004_245_a10
M. L. Gorbachuk; V. I. Gorbachuk. On the Cauchy Problem for Differential Equations in a~Banach Space over the Field of $p$-Adic Numbers. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 99-106. http://geodesic.mathdoc.fr/item/TM_2004_245_a10/

[1] Koblits N., $p$-Adicheskie chisla, $p$-adicheskii analiz i dzeta-funktsii, Mir, M., 1982 | MR

[2] Dwork B., Gerotto G., Sullivan F. J., An introduction to $G$-functions, Princeton Univ. Press, Princeton, 1994 | MR | Zbl

[3] Schikhof W. H., Ultrametric calculus. An introduction to $p$-adic analysis, Cambridge Univ. Press, Cambridge, 1984 | MR | Zbl

[4] Khrennikov A. Yu., Nearkhimedov analiz i ego prilozheniya, Fizmatlit, M., 2003 | MR | Zbl

[5] Gorbachuk M. L., Gorbachuk V. I., “On the Cauchy problem for differential equations in a Banach space over the field of $p$-adic numbers”, Meth. Funct. Anal. and Topology, 9:3 (2003), 207–212 | MR | Zbl

[6] Gorbachuk M. L., “O korrektnosti zadachi Koshi dlya differentsialno-operatornykh uravnenii v klassakh analiticheskikh vektor-funktsii”, Dokl. RAN, 374:1 (2000), 7–9 | MR | Zbl

[7] Khrennikov A. Yu., “Matematicheskie metody nearkhimedovoi fiziki”, UMN, 45:4 (1990), 79–110 | MR | Zbl