Geodesic
On some differential-operator equations of arbitrary order
Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 1-21 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

On the half-line $(0,+\infty)$ we investigate the following equation in a Banach space: \begin{equation} \sum^s_{j=0}Aj\frac{d^ju(t)}{dt^j}=h(t),\quad s\geqslant1, \end{equation} where $A_0,\dots,A_s$ are closed operators which commute with $\frac d{dt}$. We consider the following classes of equations: parabolic, inverse parabolic, hyperbolic, quasi-elliptic, and quasi-hyperbolic. We present boundary value problems for these classes and prove that they are well-posed. The proofs are based on a solvability theorem for the operator equation $\sum^s_{j=0}A_jB^ju=h $, where $B$ is a closed operator. Bibliography: 20 titles. 
@article{SM_1973_19_1_a0,
     author = {Yu. A. Dubinskii},
     title = {On some differential-operator equations of arbitrary order},
     journal = {Sbornik. Mathematics},
     pages = {1--21},
     year = {1973},
     volume = {19},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1973_19_1_a0/}
}
TY  - JOUR
AU  - Yu. A. Dubinskii
TI  - On some differential-operator equations of arbitrary order
JO  - Sbornik. Mathematics
PY  - 1973
SP  - 1
EP  - 21
VL  - 19
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1973_19_1_a0/
LA  - en
ID  - SM_1973_19_1_a0
ER  - 
%0 Journal Article
%A Yu. A. Dubinskii
%T On some differential-operator equations of arbitrary order
%J Sbornik. Mathematics
%D 1973
%P 1-21
%V 19
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1973_19_1_a0/
%G en
%F SM_1973_19_1_a0
Yu. A. Dubinskii. On some differential-operator equations of arbitrary order. Sbornik. Mathematics, Tome 19 (1973) no. 1, pp. 1-21. http://geodesic.mathdoc.fr/item/SM_1973_19_1_a0/

[1] E. Khille, R. Fillips, Funktsionalnyi analiz i polugruppy, IL, Moskva, 1962

[2] T. Lions, Equations operationelles-differentielles et problemes aux limites, Springer-Verlag, Berlin, 1962 | Zbl

[3] S. G. Krein, Lineinye differentsialnye uravneniya v banakhovom prostranstve, izd-vo «Nauka», Moskva, 1967 | MR

[4] Zh. Lions, E. Madzhenes, Neodnorodnye granichnye zadachi, izd-vo «Nauka», Moskva, 1971

[5] P. Crisvard, “Equations operationelles abstraites et problemes aux limites”, Ann. sc. norm. super, Pisa, XXI:3 (1967), 308–347

[6] V. K. Romanko, “K teorii operatorov $u^{(m)}-A$”, Diff. ur-niya, 3:11 (1967), 1957–1970 | MR | Zbl

[7] A. A. Dezin, “Teoremy suschestvovaniya i edinstvennosti reshenii granichnykh zadach”, Uspekhi matem. nauk, XIV:3 (87) (1959), 21–73 | MR

[8] V. P. Mikhailov, “O pervoi kraevoi zadache dlya odnogo klassa gipoellipticheskikh uravnenii”, Matem. sb., 63 (105) (1964), 238–264

[9] Yu. A. Dubinskii, “Zadacha Koshi dlya operatorno-differentsialnykh uravnenii”, DAN SSSR, 181:5 (1968), 1046–1049 | MR | Zbl

[10] Yu. A. Dubinskii, “Ob odnoi abstraktnoi teoreme i ee prilozheniyakh k kraevym zadacham dlya neklassicheskikh uravnenii”, Matem. sb., 79 (121) (1969), 91–117 | MR | Zbl

[11] Yu. A. Dubinskii, “Kraevye zadachi dlya nekotorykh klassov differentsialno-operatornykh uravnenii vysokogo poryadka”, DAN SSSR, 196:1 (1971), 32–35 | MR

[12] A. Friedman, “Singular perturbations for the Cauchy problems and for boundary value problems”, J. diff. equat., 5:2 (1969), 570–579 | DOI | MR

[13] R. Hersh, “Explicit solution of a class of higher order abstract Cauchy problems”, J. diff. equat., 8:3 (1970), 570–579 | DOI | MR | Zbl

[14] S. Ya. Yakubov, “Reshenie smeshannykh zadach dlya differentsialnykh uravnenii v chastnykh proizvodnykh operatornym metodom”, DAN SSSR, 196:3 (1971), 545–548 | Zbl

[15] V. G. Mazya, B. A. Plamenevskii, “Ob asimptotike reshenii differentsialnykh uravnenii s operatornymi koeffitsientami”, DAN SSSR, 196:3 (1971), 512–515

[16] M. G. Gasymov, “K teorii polinomialnykh operatornykh puchkov”, DAN SSSR, 199:4 (1971), 747–750 | Zbl

[17] M. G. Gasymov, “K teorii evolyutsionnykh uravnenii regulyarnogo tipa”, DAN SSSR, 200:1 (1971), 13–16 | MR | Zbl

[18] J. Lions, E. Magenes, “Priblemi ai limiti non omogenei, III”, Ann. sc. norm. super, Pisa, VX:1 (1961), 39–71

[19] J. Chazarain, “Problemes de Cauchy abstraits et applications a quelques problemes mixtes”, J. funct. anal., 7:3 (1971), 386–446 | DOI | MR | Zbl

[20] Yu. A. Dubinskii, “O nekotorykh differentsialno-operatornykh uravneniyakh obschego tipa”, DAN SSSR, 201:5 (1971), 1033–1036 | MR | Zbl