Geodesic
On total preservation of solvability for a controlled Hammerstein type equation with non-isotone and non-majorized operator
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2017), pp. 83-94.

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

We prove some nontrivial corollaries of the Schauder theorem. We use these corollaries to prove a theorem concerning the total preservation of solvability for a controlled functional operator equation of the Hammerstein type with non-isotone and non-majorized operator component of the right-hand side. We illustrate an application of the abstract theory by the example of the Dirichlet problem associated with a semilinear elliptic equation of the stationary diffusion-reaction type.
Keywords: fixed point, equation of the Hammerstein type, semilinear elliptic equation of the diffusion-reaction type. 
@article{IVM_2017_6_a9,
     author = {A. V. Chernov},
     title = {On total preservation of solvability for a controlled {Hammerstein} type equation with non-isotone and non-majorized operator},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {83--94},
     publisher = {mathdoc},
     number = {6},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2017_6_a9/}
}
TY  - JOUR
AU  - A. V. Chernov
TI  - On total preservation of solvability for a controlled Hammerstein type equation with non-isotone and non-majorized operator
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2017
SP  - 83
EP  - 94
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2017_6_a9/
LA  - ru
ID  - IVM_2017_6_a9
ER  - 
%0 Journal Article
%A A. V. Chernov
%T On total preservation of solvability for a controlled Hammerstein type equation with non-isotone and non-majorized operator
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2017
%P 83-94
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2017_6_a9/
%G ru
%F IVM_2017_6_a9
A. V. Chernov. On total preservation of solvability for a controlled Hammerstein type equation with non-isotone and non-majorized operator. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2017), pp. 83-94. http://geodesic.mathdoc.fr/item/IVM_2017_6_a9/

[1] Chernov A. V., “Ob odnom mazhorantnom priznake totalnogo sokhraneniya globalnoi razreshimosti upravlyaemogo funktsionalno-operatornogo uravneniya”, Izv. vuzov. Matem., 2011, no. 3, 95–107

[2] Kalantarov V. K., Ladyzhenskaya O. A., “O vozniknovenii kollapsov dlya kvazilineinykh uravnenii parabolicheskogo i giperbolicheskogo tipov”, Zap. nauchn. semin. LOMI, 69, 1977, 77–102 | Zbl

[3] Sumin V. I., “Ob obosnovanii gradientnykh metodov dlya raspredelennykh zadach optimalnogo upravleniya”, Zhurn. vychisl. matem. i matem. fiz., 30:1 (1990), 3–21 | MR | Zbl

[4] Lions Zh.-L., Upravlenie singulyarnymi raspredelennymi sistemami, Nauka, M., 1987 | MR

[5] Fursikov A. V., Optimalnoe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya, Nauchnaya kniga, Novosibirsk, 1999

[6] Chernov A. V., “O gladkikh konechnomernykh approksimatsiyakh raspredelennykh optimizatsionnykh zadach s pomoschyu diskretizatsii upravleniya”, Zhurn. vychisl. matem. i matem. fiz., 53:12 (2013), 2029–2043 | DOI | MR | Zbl

[7] Chernov A. V., “O primenimosti tekhniki parametrizatsii upravleniya k resheniyu raspredelennykh zadach optimizatsii”, Vestn. Udmurtsk. un-ta. Matem. Mekhan. Kompyuternye nauki, 24:1 (2014), 102–117

[8] Sergeev Ya. D., Kvasov D. E., Diagonalnye metody globalnoi optimizatsii, Fizmatlit, M., 2008

[9] Chernov A. V., “O kusochno postoyannoi approksimatsii v raspredelennykh zadachakh optimizatsii”, Tr. IMM UrO RAN, 21, no. 1, 2015, 305–321

[10] Chernov A. V., “O dostatochnykh usloviyakh upravlyaemosti nelineinykh raspredelennykh sistem”, Zhurn. vychisl. matem. i matem. fiz., 52:8 (2012), 1400–1414 | MR | Zbl

[11] Petrosyan L. A., Zenkevich N. A., Semina E. A., Teoriya igr, Vyssh. shkola, M., 1998

[12] Chernov A. V., “O suschestvovanii ravnovesiya po Neshu v differentsialnoi igre, svyazannoi s ellipticheskimi uravneniyami: monotonnyi sluchai”, Matem. teoriya igr i ee prilozh., 7:3 (2015), 48–78

[13] Casas E., “Boundary control of semilinear elliptic equations with pointwise state constraints”, SIAM J. Control and Optim., 31 (1993), 993–1006 | DOI | MR | Zbl

[14] Chernov A. V., “O volterrovom obobschenii metoda monotonizatsii dlya nelineinykh funktsionalno-operatornykh uravnenii”, Vestn. Udmurtsk. un-ta. Matem. Mekhan. Kompyuternye nauki, 22:2 (2012), 84–99

[15] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977

[16] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1984

[17] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973

[18] Karchevskii M. M., Pavlova M. F., Uravneniya matematicheskoi fiziki. Dopolnitelnye glavy, KGU, Kazan, 2012

[19] Vorobev A. Kh., Diffuzionnye zadachi v khimicheskoi kinetike, MGU, M., 2003

[20] Tröltzsch F., Optimal control of partial differential equations: theory, methods and applications, Graduate Stud. in Math., 112, American mathematical society, Providence, R.I., 2010 | DOI | MR | Zbl

[21] Gilbarg D., Trudinger N., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989

[22] Chernov A. V., “O totalnom sokhranenii razreshimosti upravlyaemoi zadachi Dirikhle dlya ellipticheskogo uravneniya”, Dinamika sistem i protsessy upravleniya, Tr. Mezhdunarodn. konf., posvyasch. 90-letiyu so dnya rozhdeniya akad. N. N. Krasovskogo, IMM UrO RAN, Ekaterinburg, 2015, 359–366