Geodesic
Homogeneous differential-operator equations in locally convex spaces
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2017), pp. 26-43.

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

We describe a general method that allows to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is interpreted not in terms of null right-hand side of an equation but in terms that the left-hand side is homogeneous function of operators appearing in an equation. Solutions are presented by a uniformly convergent functional vector-valued series, generated by a set of solutions to some ordinary differential equation of $k$th degree, roots of characteristical polynomial, and some set of elements of locally convex space. We find sufficient conditions of continuous dependence of solution on generating set, and a solution to Cauchy's problem for considered equations. We specify conditions of its existence and uniqueness. Besides, under certain conditions we find a general solution to considered equation (a function of most general form from which any particular solution can be found). The investigation is realized by means of characteristics (order and type) of operator and operator characteristics (operator order and operator type) of vector relative to an operator. Also we use a convergence of operator series relative to equicontinuous bornology.
Keywords: locally convex space, order and type of operators, differential-operator equation, equicontinuous bornology, convergence by bornology, vector-valued function. 
@article{IVM_2017_1_a3,
     author = {S. N. Mishin},
     title = {Homogeneous differential-operator equations in locally convex spaces},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {26--43},
     publisher = {mathdoc},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2017_1_a3/}
}
TY  - JOUR
AU  - S. N. Mishin
TI  - Homogeneous differential-operator equations in locally convex spaces
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2017
SP  - 26
EP  - 43
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2017_1_a3/
LA  - ru
ID  - IVM_2017_1_a3
ER  - 
%0 Journal Article
%A S. N. Mishin
%T Homogeneous differential-operator equations in locally convex spaces
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2017
%P 26-43
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2017_1_a3/
%G ru
%F IVM_2017_1_a3
S. N. Mishin. Homogeneous differential-operator equations in locally convex spaces. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2017), pp. 26-43. http://geodesic.mathdoc.fr/item/IVM_2017_1_a3/

[1] Gorbachuk V. I., Gorbachuk M. L., Granichnye zadachi dlya differentsialno-operatornykh uravnenii, Nauk. dumka, Kiev, 1984

[2] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967

[3] Trenogin V. A., “Kraevye zadachi dlya abstraktnykh ellipticheskikh uravnenii”, DAN SSSR, 170:5 (1966), 1026–1031

[4] Millionschikov V. M., “K teorii differentsialnykh uravnenii v lokalno vypuklykh prostranstvakh”, Matem. sb., 57:4 (1962), 385–406 | MR | Zbl

[5] Yosida K., “Time dependent evolution equations in locally convex space”, Math. Ann., 162:1 (1965), 83–86 | DOI | MR | Zbl

[6] Godunov A. N., “O lineinykh differentsialnykh uravneniyakh v lokalno vypuklykh prostranstvakh”, Vestn. MGU. Ser. 1. Matem. Mekhanika, 1974, no. 5, 31–39

[7] Radyno Ya. V., “Lineinye differentsialnye uravneniya v lokalno vypuklykh prostranstvakh. II: Svoistva reshenii”, Differents. uravneniya, 13:9 (1977), 1615–1624 | MR | Zbl

[8] Radyno Ya. V., Lineinye uravneniya i bornologiya, Izd-vo BGU, Minsk, 1982

[9] Lobanov S. G., “O razreshimosti lineinykh obyknovennykh differentsialnykh uravnenii v lokalno vypuklykh prostranstvakh”, Vestn. MGU. Ser. 1. Matem. Mekhanika, 1980, no. 2, 3–7 | MR

[10] Shkarin S. A., “Neskolko rezultatov o razreshimosti obyknovennykh lineinykh differentsialnykh uravnenii v lokalno vypuklykh prostranstvakh”, Matem. sb., 181:9 (1990), 1183–1195 | Zbl

[11] Gromov V. P., “Analiticheskie resheniya differentsialno-operatornykh uravnenii v lokalno vypuklykh prostranstvakh”, Dokl. RAN, 394:3 (2004), 305–307 | MR

[12] Aksenov H. A., Primenenie teorii poryadka i tipa operatora v lokalno vypuklykh prostranstvakh k issledovaniyu analiticheskikh zadach dlya differentsialno-operatornykh uravnenii, Diss. ... kand. fiz.-matem. nauk, Orlovskii gos. un-t, Orel, 2011

[13] Gelfond A. O., Leontev A. F., “Ob odnom obobschenii ryada Fure”, Matem. sb., 29:3 (1951), 477–507

[14] Mishin S. H., “Ob odnom vide differentsialno-operatornykh uravnenii s peremennymi koeffitsientami”, Vestn. RUDN. Ser. Matematika, informatika, fizika, 2015, no. 1, 3–14

[15] Gromov V. P., Mishin S. N., Panyushkin S. V., Operatory konechnogo poryadka i differentsialno-operatornye uravneniya, OGU, Orel, 2009

[16] Mishin S. H., “O poryadke i tipe operatora”, Dokl. RAN, 381:3 (2001), 309–312 | MR

[17] Mishin S. H., “Svyaz kharakteristik posledovatelnosti operatorov s bornologicheskoi skhodimostyu”, Vestn. RUDN. Ser. Matematika, informatika, fizika, 2010, no. 4, 26–34

[18] Mishin S. H., “O kharakteristikakh rosta operatornoznachnykh funktsii”, Ufimsk. matem. zhurn., 5:1 (2013), 112–124 | MR

[19] Pontryagin L. S., Obyknovennye differentsialnye uravneniya, Nauka, M., 1974

[20] Kurosh A. G., Kurs vysshei algebry, Nauka, M., 1971

[21] Kantopovich L. V., Akilov G. P., Funktsionalnyi analiz v nopmipovannykh ppostpanstvakh, Fizmatgiz, M., 1959

[22] Robertson A., Robertson V., Topologicheskie vektornye prostranstva, Mir, M., 1967

[23] Krasichkov I. F., “Polnota v prostranstvakh kompleksnoznachnykh funktsii, opisyvaemykh povedeniem modulya”, Matem. sb., 68:1 (1965), 26–57 | MR | Zbl

[24] Krasichkov I. F., “O zamknutykh idealakh v lokalno-vypuklykh algebrakh tselykh funktsii”, Izv. AN SSSR. Ser. Matem., 31:1 (1967), 37–60 | Zbl

[25] Panyushkin S. V., Obobschennoe preobrazovanie Fure i ego primeneniya, Diss. ... kand. fiz.-matem. nauk, Orlovsk. gos. un-t, Orel, 2005