Operator Equations of the Second Kind: Theorems on the Existence and Uniqueness of the Solution and on the Preservation of Solvability
This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general operator acting on an arbitrary Banach space , new theorems on the existence and uniqueness of a fixed po...
Saved in:
| Published in: | Differential equations Vol. 58; no. 5; pp. 649 - 661 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Moscow
Pleiades Publishing
01.05.2022
Springer Springer Nature B.V |
| Subjects: | |
| ISSN: | 0012-2661, 1608-3083 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general operator
acting on an arbitrary Banach space
, new theorems on the existence and uniqueness of a fixed point are obtained. Here the well-known concept of the cone norm is used:
, where
is, generally speaking, another Banach space semi-ordered by the cone
. These theorems are based on the assumption that the operator analog of the local Lipschitz condition with respect to the cone norm
is satisfied and generalize the result by A.V. Kalinin and S.F. Morozov (
,
). The role of an analog of the Lipschitz constant on a given bounded set
is played by a bounded linear operator
, depending on this set, with spectral radius
. In addition, M.A. Krasnosel’skii’s lemmas on the equivalent norm are used. Based on the statements obtained, we prove theorems on local and total (over the set of admissible controls) preservation of the solvability of the controlled operator equation
,
, where
is the control parameter from, generally speaking, an arbitrary set
. The abstract theory is illustrated by examples of a controlled nonlinear operator differential equation in a Banach space as well as a strongly nonlinear pseudoparabolic equation. |
|---|---|
| AbstractList | This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general operator
acting on an arbitrary Banach space
, new theorems on the existence and uniqueness of a fixed point are obtained. Here the well-known concept of the cone norm is used:
, where
is, generally speaking, another Banach space semi-ordered by the cone
. These theorems are based on the assumption that the operator analog of the local Lipschitz condition with respect to the cone norm
is satisfied and generalize the result by A.V. Kalinin and S.F. Morozov (
,
). The role of an analog of the Lipschitz constant on a given bounded set
is played by a bounded linear operator
, depending on this set, with spectral radius
. In addition, M.A. Krasnosel’skii’s lemmas on the equivalent norm are used. Based on the statements obtained, we prove theorems on local and total (over the set of admissible controls) preservation of the solvability of the controlled operator equation
,
, where
is the control parameter from, generally speaking, an arbitrary set
. The abstract theory is illustrated by examples of a controlled nonlinear operator differential equation in a Banach space as well as a strongly nonlinear pseudoparabolic equation. This paper continues the author's research on the problem of preserving the solvability ofcontrolled operator equations. As a preliminary result (which is of independent interest) for ageneral operator [Formula omitted] acting on an arbitrary Banach space [Formula omitted], new theorems on the existence and uniqueness of afixed point are obtained. Here the well-known concept of the cone norm is used: [Formula omitted],where [Formula omitted] is, generally speaking, another Banach spacesemi-ordered by the cone [Formula omitted].These theorems are based on the assumption that the operator analog of the local Lipschitzcondition with respect to the cone norm [Formula omitted] is satisfied andgeneralize the result by A.V. Kalinin and S.F. Morozov ( [Formula omitted], [Formula omitted]).The role of an analog of the Lipschitz constant on a given bounded set [Formula omitted] is played by a bounded linear operator [Formula omitted], depending on this set,with spectral radius [Formula omitted]. Inaddition, M.A. Krasnosel'skii's lemmas on the equivalent norm are used. Based on the statementsobtained, we prove theorems on local and total (over the set of admissible controls) preservation ofthe solvability of the controlled operator equation [Formula omitted], [Formula omitted], where [Formula omitted] is the control parameter from, generally speaking,an arbitrary set [Formula omitted]. The abstract theory is illustrated by examples of acontrolled nonlinear operator differential equation in a Banach space as well as a stronglynonlinear pseudoparabolic equation. This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of independent interest) for a general operator acting on an arbitrary Banach space , new theorems on the existence and uniqueness of a fixed point are obtained. Here the well-known concept of the cone norm is used: , where is, generally speaking, another Banach space semi-ordered by the cone . These theorems are based on the assumption that the operator analog of the local Lipschitz condition with respect to the cone norm is satisfied and generalize the result by A.V. Kalinin and S.F. Morozov (, ). The role of an analog of the Lipschitz constant on a given bounded set is played by a bounded linear operator , depending on this set, with spectral radius . In addition, M.A. Krasnosel’skii’s lemmas on the equivalent norm are used. Based on the statements obtained, we prove theorems on local and total (over the set of admissible controls) preservation of the solvability of the controlled operator equation , , where is the control parameter from, generally speaking, an arbitrary set . The abstract theory is illustrated by examples of a controlled nonlinear operator differential equation in a Banach space as well as a strongly nonlinear pseudoparabolic equation. |
| Audience | Academic |
| Author | Chernov, A. V. |
| Author_xml | – sequence: 1 givenname: A. V. surname: Chernov fullname: Chernov, A. V. email: chavnn@mail.ru organization: Lobachevsky State University of Nizhny Novgorod, Alekseev Nizhny Novgorod State Technical University |
| BookMark | eNp9kdtqGzEQhkVJoM7hAXK30OtNR5KllXsXgnuggQTiXC-SdtZWWEuOJIfmBfLc0dqGQEuDQIL5_29GM3NCjnzwSMgFhUtK-fTrPQBlTMpygQAQ8hOZUAmq5qD4EZmMcj3qn8lJSo8AMGuomJDX2w1GnUOs5k9bnV3wqQp9lVdY3aMNvqt-O999qxYrDBHXRfQ7cf7HpYzeYqWL58G7py16TO9wGLZjtp18YO4iJozPuyqjr3ietXGDyy9n5LjXQ8Lzw3tKHr7PF9c_65vbH7-ur25qy5TItWJopp3qtTWcNwK1MXRmpxQ0a6Zd1_C-k1QKZjgDKGFhwDasNwhGKSUMPyVf9nk3MZQfp9w-hm30pWTLGlBMgZrNiuty71rqAVvn-5CjtuV0uHZlKNi7Er8qA5RcCCkL0OwBG0NKEfvWurzrs4BuaCm045baf7ZUSPoXuYlurePLhwzbM6l4_RLjexP_h94A-POlEQ |
| CitedBy_id | crossref_primary_10_1134_S0965542524700362 crossref_primary_10_1134_S1064562423600690 crossref_primary_10_1134_S0965542523070035 |
| Cites_doi | 10.20537/vm150207 10.1002/mana.19750672207 10.1007/BF02355077 10.1007/s10625-005-0206-2 10.1007/BF01109723 10.1134/S0965542518120096 10.35634/vm200107 10.3103/S1066369X1706010X 10.1016/0041-5553(90)90002-A 10.1016/j.ifacol.2018.11.454 10.1007/BF01445103 10.3103/S1066369X18110063 10.1016/0362-546X(94)00190-S 10.1134/S096554251312004X 10.1016/j.jde.2017.09.045 10.1090/gsm/112/07 |
| ContentType | Journal Article |
| Copyright | Pleiades Publishing, Ltd. 2022 COPYRIGHT 2022 Springer Pleiades Publishing, Ltd. 2022. |
| Copyright_xml | – notice: Pleiades Publishing, Ltd. 2022 – notice: COPYRIGHT 2022 Springer – notice: Pleiades Publishing, Ltd. 2022. |
| DBID | AAYXX CITATION 7SC 7TB 8FD FR3 JQ2 KR7 L7M L~C L~D |
| DOI | 10.1134/S0012266122050056 |
| DatabaseName | CrossRef Computer and Information Systems Abstracts Mechanical & Transportation Engineering Abstracts Technology Research Database Engineering Research Database ProQuest Computer Science Collection Civil Engineering Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional |
| DatabaseTitle | CrossRef Civil Engineering Abstracts Technology Research Database Computer and Information Systems Abstracts – Academic Mechanical & Transportation Engineering Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Engineering Research Database Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional |
| DatabaseTitleList | Civil Engineering Abstracts |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Sciences (General) Mathematics |
| EISSN | 1608-3083 |
| EndPage | 661 |
| ExternalDocumentID | A715635566 10_1134_S0012266122050056 |
| GroupedDBID | --Z -5D -5G -BR -EM -Y2 -~C -~X .86 .VR 04Q 04W 06D 0R~ 0VY 1N0 29G 2J2 2JN 2JY 2KG 2KM 2LR 2P1 2VQ 2~H 30V 3V. 4.4 408 409 40D 40E 5GY 5VS 67Z 6NX 6TJ 78A 7WY 8FE 8FG 8FL 8G5 8TC 8UJ 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBBX ABBXA ABDBF ABDZT ABECU ABEFU ABFTV ABHQN ABJCF ABJNI ABJOX ABKCH ABKTR ABLLD ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABUWG ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACNCT ACOKC ACOMO ACPIV ACUHS ACZOJ ADHHG ADHIR ADINQ ADKNI ADKPE ADRFC ADTPH ADURQ ADYFF ADZKW AEBTG AEFQL AEGAL AEGNC AEJHL AEJRE AEMSY AENEX AEOHA AEPYU AETLH AEVLU AEXYK AFBBN AFFNX AFGCZ AFKRA AFLOW AFQWF AFWTZ AFZKB AGAYW AGDGC AGJBK AGMZJ AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AI. AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARAPS ARMRJ ASPBG AVWKF AXYYD AZFZN AZQEC B-. B0M BA0 BAPOH BDATZ BENPR BEZIV BGLVJ BGNMA BPHCQ BSONS CAG CCPQU COF CS3 CSCUP DDRTE DL5 DNIVK DPUIP DU5 DWQXO EAD EAP EBLON EBS EIOEI EJD EMK EPL ESBYG ESX FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRNLG FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNUQQ GNWQR GQ6 GQ7 GQ8 GROUPED_ABI_INFORM_COMPLETE GUQSH GXS H13 HCIFZ HF~ HG6 HMJXF HQYDN HRMNR HVGLF HZ~ IAO IHE IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C JBSCW JCJTX JZLTJ K60 K6V K6~ K7- KDC KOV L6V LAK LLZTM M0C M0N M2O M4Y M7S MA- N2Q NB0 NPVJJ NQJWS NU0 O9- O93 O9J OAM OHT OVD P2P P62 P9R PADUT PF0 PKN PQBIZ PQBZA PQQKQ PROAC PT4 PTHSS Q2X QOS R89 R9I RNI RNS ROL RPX RSV RZC RZE S16 S1Z S27 S3B SAP SDH SHX SISQX SJYHP SMT SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 TEORI TN5 TSG TSK TSV TUC TUS TWZ U2A UG4 UOJIU UPT UTJUX UZXMN VC2 VFIZW VH1 W23 W48 WH7 WK8 XU3 YLTOR ~8M ~A9 AAPKM AAYXX ABDBE ABFSG ABRTQ ACSTC ADHKG AEZWR AFDZB AFFHD AFHIU AFOHR AGQPQ AHPBZ AHWEU AIXLP AMVHM ATHPR CITATION M2P PHGZM PHGZT PQGLB 7SC 7TB 8FD FR3 JQ2 KR7 L7M L~C L~D |
| ID | FETCH-LOGICAL-c285t-82eb4d8facb3375eabb19c410a274dd73fd61652b32004105b0c72fbe0b8885b3 |
| IEDL.DBID | RSV |
| ISICitedReferencesCount | 4 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000847695400005&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0012-2661 |
| IngestDate | Wed Sep 17 23:55:16 EDT 2025 Sat Nov 29 10:10:34 EST 2025 Sat Nov 29 01:43:31 EST 2025 Tue Nov 18 22:26:18 EST 2025 Fri Feb 21 02:45:00 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 5 |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c285t-82eb4d8facb3375eabb19c410a274dd73fd61652b32004105b0c72fbe0b8885b3 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| PQID | 2708280899 |
| PQPubID | 54032 |
| PageCount | 13 |
| ParticipantIDs | proquest_journals_2708280899 gale_infotracacademiconefile_A715635566 crossref_citationtrail_10_1134_S0012266122050056 crossref_primary_10_1134_S0012266122050056 springer_journals_10_1134_S0012266122050056 |
| PublicationCentury | 2000 |
| PublicationDate | 2022-05-01 |
| PublicationDateYYYYMMDD | 2022-05-01 |
| PublicationDate_xml | – month: 05 year: 2022 text: 2022-05-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Moscow |
| PublicationPlace_xml | – name: Moscow – name: New York |
| PublicationTitle | Differential equations |
| PublicationTitleAbbrev | Diff Equat |
| PublicationYear | 2022 |
| Publisher | Pleiades Publishing Springer Springer Nature B.V |
| Publisher_xml | – name: Pleiades Publishing – name: Springer – name: Springer Nature B.V |
| References | KorpusovM.O.SveshnikovA.G.Blow-up of solutions of strongly nonlinear equations of pseudoparabolic typeSovrem. Mat. Pril.2006403138 Chernov, A.V., On overcoming the singularity of distributed control systems, Tr. Tret’ei Vseross. nauchn. konf. “Matematicheskoe modelirovanie i kraevye zadachi” (Proc. Third All-Russ. Sci. Conf. “Mathematical Modeling and Boundary Value Problems”), Samara, 2006, Part 2, pp. 171–174. TersenovA.The Dirichlet problem for second order semilinear elliptic and parabolic equationsDiffer. Equat. Appl.20091339341125549751177.35100 ChernovA.V.Preservation of the solvability of a semilinear global electric circuit equationComput. Math. Math. Phys.2018581220182030390893810.1134/S0965542518120096 ChernovA.V.On the totally global solvability of a controlled Hammerstein type equation with a varied linear operatorVestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki201525223024310.20537/vm150207 SuminV.I.Controlled Volterra functional equations and contraction mapping principleTr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk201925. N 12622783935655 SuminV.I.Conditions of stability for the existence of global solutions of controlled boundary value problems for nonlinear parabolic equationsVestn. Tambov. Gos. Univ. Ser. Estestv. Tekh. Nauki200054493495 Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz, Moscow: Nauka, 1984. Translated under the title: Functional Analysis, Oxford: Pergamon, 1982. Sumin, V.I., Funktsional’nye vol’terrovy uravneniya v teorii optimal’nogo upravleniya raspredelennymi sistemami. Ch. I (Functional Volterra Equations in the Theory of Optimal Control of Distributed Systems. Part I), Nizhny Novgorod: Izd. Nizhegorod. Gos. Univ., 1992. SaitoH.Global solvability of the Navier–Stokes equations with a free surface in the maximal regularity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}—\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q $$\end{document} classJ. Differ. Equat.201826431475152010.1016/j.jde.2017.09.045 SuminV.I.Volterra functional-operator equations in the theory of optimal control of distributed systemsIFAC PapersOnLine2018513275976410.1016/j.ifacol.2018.11.454 Sumin, V.I. and Chernov, A.V., Volterra functional-operator equations in the theory of optimization of distributed systems, Tr. Mezhdunar. konf. “Dinamika sistem i protsessy upravleniya,” posvyashch. 90-letiyu so dnya rozhd. akad. N.N. Krasovskogo (Proc. Int. Conf. “Dynamics of Systems and Control Processes” Dedicated 90th Anniv. Acad. N.N. Krasovskii), (Yekaterinburg, September 15–20, 2014), Yekaterinburg, 2015, pp. 293–300. Krasnosel’skii, M.A., Vainikko, G.M., Zabreiko, P.P., Rutitskii, Ya.B., and Stetsenko, V.Ya., Priblizhennoe reshenie operatornykh uravnenii (Approximate Solution of Operator Equations), Moscow: Nauka, 1969. Sumin, V.I., On the problem of singularity of distributed control systems. II, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 2001, no. 1 (23), pp. 198–204. ChernovA.V.On total preservation of solvability of controlled Hammerstein-type equation with non-isotone and non-majorizable operatorRuss. Math.2017616728110.3103/S1066369X1706010X KorpusovM.O.Global solvability conditions for an initial–boundary value problem for a nonlinear equation of pseudoparabolic typeDiffer. Equations2005415712720220068010.1007/s10625-005-0206-2 KalantarovV.K.LadyzhenskayaO.A.The occurrence of collapse for quasilinear equations of parabolic and hyperbolic typesJ. Math. Sci.1978101537010.1007/BF01109723 SuminV.I.The features of gradient methods for distributed optimal-control problemsUSSR Comput. Math. Math. Phys.199030111510.1016/0041-5553(90)90002-A Sumin, V.I., On the problem of singularity of distributed control systems. I, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 1999, no. 2 (21), pp. 145–155. Tröltzsch, F., Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Grad. Stud. Math., Providence, 2010, vol. 112. Sumin, V.I., Functional Volterra equations in the mathematical theory of optimal control of distributed systems, Doctoral (Phys.-Math.) Dissertation, Nizhny Novgorod, 1998. Gaewski, H., Gröger, K, and Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Berlin: Akademie-Verlag, 1974. SuminV.I.ChernovA.V.Operators in spaces of measurable functions: Volterra and quasi-nilpotencyDiffer. Equations199834101403141117130110958.47014 Chernov, A.V., Volterra operator equations and their application in the theory of optimization of hyperbolic systems, Cand. Sci. (Phys.-Math.) Dissertation, Nizhny Novgorod, 2000. ChernovA.V.Smooth finite-dimensional approximations of distributed optimization problems via control discretizationComput. Math. Math. Phys.2013531218391852314657110.1134/S096554251312004X KalininA.V.MorozovS.F.The Cauchy problem for a nonlinear integro-differential transport equationMath. Notes1997615566573162011310.1007/BF02355077 Krasnosel’skii, M.A., Polozhitel’nye resheniya operatornykh uravnenii. Glavy nelineinogo analiza (Positive Solutions of Operator Equations. Chapters of Nonlinear Analysis), Moscow: Fizmatgiz, 1962. KobayashiT.PecherH.ShibataY.On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosityMath. Ann.19932962215234121990010.1007/BF01445103 LionsJ.-L.Controle de systèmes distribués singuliers1983ParisGauthier-Villars0514.93001 Sumin, V.I., On the problem of singularity of distributed control systems. III, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 2002, no. 1 (25), pp. 164–174. Sumin, V.I., The problem of stability of the existence of global solutions of controlled boundary value problems and Volterra functional equations, Vestn. Nizhegorod. Gos. Univ. Mat., 2003, no. 1, pp. 91–107. ChernovA.V.On totally global solvability of controlled second kind operator equationVestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki202030192111411950210.35634/vm200107 ChernovA.V.The total preservation of unique global solvability of the first kind operator equation with additional controlled nonlinearityRuss. Math.20186211536610.3103/S1066369X18110063 LuG.Global existence and blow-up for a class of semilinear parabolic systems: a Cauchy problemNonlinear Anal. Theory Methods Appl.199524811931206132564210.1016/0362-546X(94)00190-S FursikovA.V.Optimal Control of Distributed Systems. Theory and Applications2000Providence, R.I.Am. Math. Soc.1027.93500 2370_CR7 2370_CR25 2370_CR9 A.V. Chernov (2370_CR22) 2017; 61 2370_CR8 A.V. Chernov (2370_CR23) 2020; 30 2370_CR27 V.I. Sumin (2370_CR21) 2019; 25. N 1 A.V. Kalinin (2370_CR2) 1997; 61 H. Saito (2370_CR31) 2018; 264 A. Tersenov (2370_CR30) 2009; 1 A.V. Chernov (2370_CR20) 2018; 58 A.V. Chernov (2370_CR32) 2018; 62 A.V. Fursikov (2370_CR6) 2000 2370_CR10 V.I. Sumin (2370_CR11) 2000; 5 2370_CR13 2370_CR12 2370_CR34 2370_CR33 G. Lu (2370_CR29) 1995; 24 2370_CR17 V.I. Sumin (2370_CR19) 2018; 51 2370_CR16 A.V. Chernov (2370_CR18) 2015; 25 M.O. Korpusov (2370_CR26) 2006; 40 V.K. Kalantarov (2370_CR24) 1978; 10 T. Kobayashi (2370_CR28) 1993; 296 M.O. Korpusov (2370_CR4) 2005; 41 V.I. Sumin (2370_CR14) 1990; 30 V.I. Sumin (2370_CR35) 1998; 34 2370_CR3 2370_CR1 J.-L. Lions (2370_CR5) 1983 A.V. Chernov (2370_CR15) 2013; 53 |
| References_xml | – reference: Sumin, V.I., Functional Volterra equations in the mathematical theory of optimal control of distributed systems, Doctoral (Phys.-Math.) Dissertation, Nizhny Novgorod, 1998. – reference: ChernovA.V.On totally global solvability of controlled second kind operator equationVestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki202030192111411950210.35634/vm200107 – reference: Sumin, V.I., On the problem of singularity of distributed control systems. II, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 2001, no. 1 (23), pp. 198–204. – reference: KorpusovM.O.SveshnikovA.G.Blow-up of solutions of strongly nonlinear equations of pseudoparabolic typeSovrem. Mat. Pril.2006403138 – reference: KorpusovM.O.Global solvability conditions for an initial–boundary value problem for a nonlinear equation of pseudoparabolic typeDiffer. Equations2005415712720220068010.1007/s10625-005-0206-2 – reference: Sumin, V.I., The problem of stability of the existence of global solutions of controlled boundary value problems and Volterra functional equations, Vestn. Nizhegorod. Gos. Univ. Mat., 2003, no. 1, pp. 91–107. – reference: Sumin, V.I. and Chernov, A.V., Volterra functional-operator equations in the theory of optimization of distributed systems, Tr. Mezhdunar. konf. “Dinamika sistem i protsessy upravleniya,” posvyashch. 90-letiyu so dnya rozhd. akad. N.N. Krasovskogo (Proc. Int. Conf. “Dynamics of Systems and Control Processes” Dedicated 90th Anniv. Acad. N.N. Krasovskii), (Yekaterinburg, September 15–20, 2014), Yekaterinburg, 2015, pp. 293–300. – reference: Tröltzsch, F., Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Grad. Stud. Math., Providence, 2010, vol. 112. – reference: Gaewski, H., Gröger, K, and Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Berlin: Akademie-Verlag, 1974. – reference: Krasnosel’skii, M.A., Polozhitel’nye resheniya operatornykh uravnenii. Glavy nelineinogo analiza (Positive Solutions of Operator Equations. Chapters of Nonlinear Analysis), Moscow: Fizmatgiz, 1962. – reference: Sumin, V.I., On the problem of singularity of distributed control systems. III, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 2002, no. 1 (25), pp. 164–174. – reference: Chernov, A.V., Volterra operator equations and their application in the theory of optimization of hyperbolic systems, Cand. Sci. (Phys.-Math.) Dissertation, Nizhny Novgorod, 2000. – reference: Chernov, A.V., On overcoming the singularity of distributed control systems, Tr. Tret’ei Vseross. nauchn. konf. “Matematicheskoe modelirovanie i kraevye zadachi” (Proc. Third All-Russ. Sci. Conf. “Mathematical Modeling and Boundary Value Problems”), Samara, 2006, Part 2, pp. 171–174. – reference: Sumin, V.I., Funktsional’nye vol’terrovy uravneniya v teorii optimal’nogo upravleniya raspredelennymi sistemami. Ch. I (Functional Volterra Equations in the Theory of Optimal Control of Distributed Systems. Part I), Nizhny Novgorod: Izd. Nizhegorod. Gos. Univ., 1992. – reference: Sumin, V.I., On the problem of singularity of distributed control systems. I, Vestn. Nizhegorod. Gos. Univ. Mat. Model. Optim. Upr., 1999, no. 2 (21), pp. 145–155. – reference: KobayashiT.PecherH.ShibataY.On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosityMath. Ann.19932962215234121990010.1007/BF01445103 – reference: LuG.Global existence and blow-up for a class of semilinear parabolic systems: a Cauchy problemNonlinear Anal. Theory Methods Appl.199524811931206132564210.1016/0362-546X(94)00190-S – reference: ChernovA.V.Preservation of the solvability of a semilinear global electric circuit equationComput. Math. Math. Phys.2018581220182030390893810.1134/S0965542518120096 – reference: SuminV.I.Volterra functional-operator equations in the theory of optimal control of distributed systemsIFAC PapersOnLine2018513275976410.1016/j.ifacol.2018.11.454 – reference: ChernovA.V.Smooth finite-dimensional approximations of distributed optimization problems via control discretizationComput. Math. Math. Phys.2013531218391852314657110.1134/S096554251312004X – reference: Krasnosel’skii, M.A., Vainikko, G.M., Zabreiko, P.P., Rutitskii, Ya.B., and Stetsenko, V.Ya., Priblizhennoe reshenie operatornykh uravnenii (Approximate Solution of Operator Equations), Moscow: Nauka, 1969. – reference: LionsJ.-L.Controle de systèmes distribués singuliers1983ParisGauthier-Villars0514.93001 – reference: SaitoH.Global solvability of the Navier–Stokes equations with a free surface in the maximal regularity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}—\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q $$\end{document} classJ. Differ. Equat.201826431475152010.1016/j.jde.2017.09.045 – reference: ChernovA.V.On total preservation of solvability of controlled Hammerstein-type equation with non-isotone and non-majorizable operatorRuss. Math.2017616728110.3103/S1066369X1706010X – reference: KalantarovV.K.LadyzhenskayaO.A.The occurrence of collapse for quasilinear equations of parabolic and hyperbolic typesJ. Math. Sci.1978101537010.1007/BF01109723 – reference: FursikovA.V.Optimal Control of Distributed Systems. Theory and Applications2000Providence, R.I.Am. Math. Soc.1027.93500 – reference: ChernovA.V.On the totally global solvability of a controlled Hammerstein type equation with a varied linear operatorVestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki201525223024310.20537/vm150207 – reference: KalininA.V.MorozovS.F.The Cauchy problem for a nonlinear integro-differential transport equationMath. Notes1997615566573162011310.1007/BF02355077 – reference: ChernovA.V.The total preservation of unique global solvability of the first kind operator equation with additional controlled nonlinearityRuss. Math.20186211536610.3103/S1066369X18110063 – reference: SuminV.I.ChernovA.V.Operators in spaces of measurable functions: Volterra and quasi-nilpotencyDiffer. Equations199834101403141117130110958.47014 – reference: SuminV.I.Controlled Volterra functional equations and contraction mapping principleTr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk201925. N 12622783935655 – reference: Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz, Moscow: Nauka, 1984. Translated under the title: Functional Analysis, Oxford: Pergamon, 1982. – reference: SuminV.I.The features of gradient methods for distributed optimal-control problemsUSSR Comput. Math. Math. Phys.199030111510.1016/0041-5553(90)90002-A – reference: TersenovA.The Dirichlet problem for second order semilinear elliptic and parabolic equationsDiffer. Equat. Appl.20091339341125549751177.35100 – reference: SuminV.I.Conditions of stability for the existence of global solutions of controlled boundary value problems for nonlinear parabolic equationsVestn. Tambov. Gos. Univ. Ser. Estestv. Tekh. Nauki200054493495 – volume: 25 start-page: 230 issue: 2 year: 2015 ident: 2370_CR18 publication-title: Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki doi: 10.20537/vm150207 – volume: 34 start-page: 1403 issue: 10 year: 1998 ident: 2370_CR35 publication-title: Differ. Equations – ident: 2370_CR3 doi: 10.1002/mana.19750672207 – volume: 5 start-page: 493 issue: 4 year: 2000 ident: 2370_CR11 publication-title: Vestn. Tambov. Gos. Univ. Ser. Estestv. Tekh. Nauki – volume: 61 start-page: 566 issue: 5 year: 1997 ident: 2370_CR2 publication-title: Math. Notes doi: 10.1007/BF02355077 – ident: 2370_CR34 – volume: 41 start-page: 712 issue: 5 year: 2005 ident: 2370_CR4 publication-title: Differ. Equations doi: 10.1007/s10625-005-0206-2 – ident: 2370_CR8 – volume: 10 start-page: 53 issue: 1 year: 1978 ident: 2370_CR24 publication-title: J. Math. Sci. doi: 10.1007/BF01109723 – ident: 2370_CR16 – volume: 58 start-page: 2018 issue: 12 year: 2018 ident: 2370_CR20 publication-title: Comput. Math. Math. Phys. doi: 10.1134/S0965542518120096 – volume: 30 start-page: 92 issue: 1 year: 2020 ident: 2370_CR23 publication-title: Vestn. Udmurt. Univ. Mat. Mekh. Komp’yut. Nauki doi: 10.35634/vm200107 – volume: 61 start-page: 72 issue: 6 year: 2017 ident: 2370_CR22 publication-title: Russ. Math. doi: 10.3103/S1066369X1706010X – volume: 30 start-page: 1 issue: 1 year: 1990 ident: 2370_CR14 publication-title: USSR Comput. Math. Math. Phys. doi: 10.1016/0041-5553(90)90002-A – volume: 51 start-page: 759 issue: 32 year: 2018 ident: 2370_CR19 publication-title: IFAC PapersOnLine doi: 10.1016/j.ifacol.2018.11.454 – volume: 296 start-page: 215 issue: 2 year: 1993 ident: 2370_CR28 publication-title: Math. Ann. doi: 10.1007/BF01445103 – volume: 62 start-page: 53 issue: 11 year: 2018 ident: 2370_CR32 publication-title: Russ. Math. doi: 10.3103/S1066369X18110063 – volume-title: Controle de systèmes distribués singuliers year: 1983 ident: 2370_CR5 – volume: 24 start-page: 1193 issue: 8 year: 1995 ident: 2370_CR29 publication-title: Nonlinear Anal. Theory Methods Appl. doi: 10.1016/0362-546X(94)00190-S – ident: 2370_CR25 – volume: 53 start-page: 1839 issue: 12 year: 2013 ident: 2370_CR15 publication-title: Comput. Math. Math. Phys. doi: 10.1134/S096554251312004X – ident: 2370_CR10 – ident: 2370_CR33 – ident: 2370_CR12 – volume: 1 start-page: 393 issue: 3 year: 2009 ident: 2370_CR30 publication-title: Differ. Equat. Appl. – ident: 2370_CR9 – ident: 2370_CR7 – volume: 40 start-page: 3 year: 2006 ident: 2370_CR26 publication-title: Sovrem. Mat. Pril. – ident: 2370_CR1 – ident: 2370_CR13 – ident: 2370_CR17 – volume: 25. N 1 start-page: 262 year: 2019 ident: 2370_CR21 publication-title: Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk – volume-title: Optimal Control of Distributed Systems. Theory and Applications year: 2000 ident: 2370_CR6 – volume: 264 start-page: 1475 issue: 3 year: 2018 ident: 2370_CR31 publication-title: J. Differ. Equat. doi: 10.1016/j.jde.2017.09.045 – ident: 2370_CR27 doi: 10.1090/gsm/112/07 |
| SSID | ssj0009715 |
| Score | 2.2569292 |
| Snippet | This paper continues the author’s research on the problem of preserving the solvability of controlled operator equations. As a preliminary result (which is of... This paper continues the author's research on the problem of preserving the solvability ofcontrolled operator equations. As a preliminary result (which is of... |
| SourceID | proquest gale crossref springer |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 649 |
| SubjectTerms | Banach spaces Difference and Functional Equations Differential equations Existence theorems Linear operators Lipschitz condition Mathematical analysis Mathematics Mathematics and Statistics Nonlinear control Operators (mathematics) Ordinary Differential Equations Partial Differential Equations Uniqueness |
| Title | Operator Equations of the Second Kind: Theorems on the Existence and Uniqueness of the Solution and on the Preservation of Solvability |
| URI | https://link.springer.com/article/10.1134/S0012266122050056 https://www.proquest.com/docview/2708280899 |
| Volume | 58 |
| WOSCitedRecordID | wos000847695400005&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVAVX databaseName: Springer customDbUrl: eissn: 1608-3083 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0009715 issn: 0012-2661 databaseCode: RSV dateStart: 20000101 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1JS8VADA5uBz24i8-NOQhuFNrp7k3kiSAuuOGtdDYQ9D21T9E_4O82mU6fuIKek0yHNplMmuQLwGoUK5FJFXt5YlIvSvzcE8Zwz2iMJgIlpPbrYRPp0VF2dZWfuD7uqql2b1KS9qSu545E1NMbcHIn1BpKCJaDMIzeLiNrPD27fEfaTZuxBdwjdpfK_HaJD87o85H8JTdqXc7exL82Ownj7obJdmqVmIIB3ZmGscM-PGs1DVPOoiu27mCnN2bg9fhO25w7a9_X-N8V6xqGYuyMombFDjB-32a2m1_fIrFjie1n0hRcjZXIc2EBYen87As75bZkJ0OFH83fYOJDnqcaMPxlFi722ue7-56b0uBJnsU9L-NaRCozpRRhmMa6FCLIZRT4JQa8SqWhUUmQxFyEZJB4nRO-TLkR2hcYfccinIOhTrej54HJvEzRW4ZGBzpKhBbKlBgRlolKjPJN1AK_-VyFdBDmNEnjprChTBgVX957Czb7Inc1fsdvzGukAwXZNq4rS9eigLsjlKxiB1WLLmgJci41alI4o68KnhIeIOVRW7DVqMU7-cfHLvyJexFGObVg2KLLJRjqPTzqZRiRT73r6mHF2sIbRSsA5g |
| linkProvider | Springer Nature |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3dT9RAEJ_oYaI-gCCGgxP3wcQP0qTdbretb8QcwQCnETC8Nd2vhAQOvJ4X-Qf4u5nZbo8oSqLPM7PdtDM7O52Z3wC8FplRhTZZVEqXR0LGZaSc45GzGE0kRmkbt8Mm8tGoODkpv4Q-7qardu9Skv6kbueOCOrpTTi5E2oNJQTLh7Ag0GFRHd_Xw2-3SLt5N7aAR8QeUpl_XOIXZ_T7kXwnN-pdzs7Sf232GSyGGybbblViGR7Y8Qo8PZjDszYrsBwsumFvA-z0u-dw_fnS-pw7G35v8b8bduEYirFDipoN28P4_QPz3fz2HIljTxz-JE3B1ViNPMceEJbOz7lwUG5PDjJU-NH9DSY-5Jm1gOFXq3C8Mzz6uBuFKQ2R5kU2jQpulTCFq7VK0zyztVJJqUUS1xjwGpOnzshEZlylZJB4nVOxzrlTNlYYfWcqfQG98cXYrgHTZZ2jt0ydTayQyirjaowIa2mkM7ETfYi7z1XpAGFOkzTOKh_KpKK689778H4uctnid9zH_IZ0oCLbxnV1HVoUcHeEklVto2rRBU0i56BTkyoYfVPxnPAAKY_ah61OLW7Jf33s-j9xv4LHu0cH-9X-p9HeBjzh1I7hCzAH0JtOftiX8EjPpqfNZNPbxQ1VkAPK |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3dS9xAEB_8KEUf_KrS02vdh4JWCSabb9-k3tFiewrW4lvIfoHQ5q7mPOw_0L_bmc3mpNoKxeed2YTszM7-Mju_AXgXxUpkUsVenpjUixI_94Qx3DMa0USghNR-02wiHQyyy8v8zPU5rdvb7m1KsqlpIJamanwwUsb1IImovjfgFFqoTJTYLGdhPqKeQQTXz7_ds-6mbQsD7pG4S2v-dYo_AtPD7flRntSGn_7ys198BZbcyZMdNaayCjO6WoPFL1Pa1noNVp2n12zX0VG_fwW_T0fa5uJZ72fDC16zoWGoxs4JTSt2grj-kNkqf_0DBys72LslC8LZWIkyF5YolvbVqbIzejvsdOhCSPuXmORQZtIQif9ah4t-7-uHj57r3uBJnsVjL-NaRCozpRRhmMa6FCLIJa5SiUBYqTQ0KgmSmIuQHBWPecKXKTdC-wJReSzCDZirhpV-DUzmZYpRNDQ60FEitFCmRKRYJioxyjdRB_x26QrpqM2pw8b3wkKcMCoeffcO7E1VRg2vx1PCO2QPBfk8zitLV7qAb0fsWcURmhkd3BKU7LYmU7jNoC54SjyBlF_twH5rIvfD_3zs5n9Jb8PLs-N-8fnT4GQLFjhVadh7mV2YG1_f6DfwQk7GV_X1W-sid6b7DK4 |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Operator+Equations+of+the+Second+Kind%3A+Theorems+on+the+Existence+and+Uniqueness+of+the+Solution+and+on+the+Preservation+of+Solvability&rft.jtitle=Differential+equations&rft.au=Chernov%2C+A.+V.&rft.date=2022-05-01&rft.issn=0012-2661&rft.eissn=1608-3083&rft.volume=58&rft.issue=5&rft.spage=649&rft.epage=661&rft_id=info:doi/10.1134%2FS0012266122050056&rft.externalDBID=n%2Fa&rft.externalDocID=10_1134_S0012266122050056 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0012-2661&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0012-2661&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0012-2661&client=summon |