On the shrinking projection method for nonexpansive mappings endowed with graphs

Recently, the convergence of a shrinking projection method for a Hadamard space satisfying some properties endowed with a directed graph defined on a nonempty closed convex subset of this space has been studied by many authors. In this work, we define a new graph and consider a Hadamard space endowe...

Full description

Saved in:
Bibliographic Details
Published inFixed point theory and algorithms for sciences and engineering Vol. 2025; no. 1; pp. 9 - 12
Main Authors Kimura, Yasunori, Phothi, Supaluk, Tontan, Kittisak
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2025
Springer Nature B.V
SpringerOpen
Subjects
Algorithms
Analysis
Applications of Mathematics
Approximation
Convergence
Differential Geometry
Directed graphs
G-nonexpansive mappings
Graph theory
Graphs
Hilbert space
Iterative methods
Mathematical and Computational Biology
Mathematics
Mathematics and Statistics
Methods
Shrinking projection method
Topology
Shrinking projection method
47H09
47H10
Directed graphs
nonexpansive mappings
Online AccessGet full text
ISSN2730-5422
2730-5422
DOI10.1186/s13663-025-00791-8

Cover

Abstract Recently, the convergence of a shrinking projection method for a Hadamard space satisfying some properties endowed with a directed graph defined on a nonempty closed convex subset of this space has been studied by many authors. In this work, we define a new graph and consider a Hadamard space endowed with our modified graph, we present a theorem on the strong convergence of an iterative sequence generated by the shrinking projection method. In particular, we generalize a result in (Khatoon et al. in Proc. Est. Acad. Sci 71(3):275, 2022 ) to more general setting. The similar result is also deduces to a Hilbert space.
AbstractList Abstract Recently, the convergence of a shrinking projection method for a Hadamard space satisfying some properties endowed with a directed graph defined on a nonempty closed convex subset of this space has been studied by many authors. In this work, we define a new graph and consider a Hadamard space endowed with our modified graph, we present a theorem on the strong convergence of an iterative sequence generated by the shrinking projection method. In particular, we generalize a result in (Khatoon et al. in Proc. Est. Acad. Sci 71(3):275, 2022) to more general setting. The similar result is also deduces to a Hilbert space.
Recently, the convergence of a shrinking projection method for a Hadamard space satisfying some properties endowed with a directed graph defined on a nonempty closed convex subset of this space has been studied by many authors. In this work, we define a new graph and consider a Hadamard space endowed with our modified graph, we present a theorem on the strong convergence of an iterative sequence generated by the shrinking projection method. In particular, we generalize a result in (Khatoon et al. in Proc. Est. Acad. Sci 71(3):275, 2022 ) to more general setting. The similar result is also deduces to a Hilbert space.
Recently, the convergence of a shrinking projection method for a Hadamard space satisfying some properties endowed with a directed graph defined on a nonempty closed convex subset of this space has been studied by many authors. In this work, we define a new graph and consider a Hadamard space endowed with our modified graph, we present a theorem on the strong convergence of an iterative sequence generated by the shrinking projection method. In particular, we generalize a result in (Khatoon et al. in Proc. Est. Acad. Sci 71(3):275, 2022) to more general setting. The similar result is also deduces to a Hilbert space.
ArticleNumber 9
Author Kimura, Yasunori
Tontan, Kittisak
Phothi, Supaluk
Author_xml – sequence: 1
  givenname: Yasunori
  surname: Kimura
  fullname: Kimura, Yasunori
  organization: Toho University
– sequence: 2
  givenname: Supaluk
  surname: Phothi
  fullname: Phothi, Supaluk
  organization: Chiang Mai University
– sequence: 3
  givenname: Kittisak
  surname: Tontan
  fullname: Tontan, Kittisak
  email: kittisak_tontan@cmu.ac.th
  organization: Chiang Mai University
BookMark eNp9kUtPxSAQhYnRxOcfcEXiujpAS-nSGB83MdGFrgmF6W2vXqhQX_9etEZduYKQc74zw9klmz54JOSQwTFjSp4kJqQUBfCqAKgbVqgNssNrAUVVcr75575NDlJaAWQpE7yudsjtjadTjzT1cfAPg1_SMYYV2mkInq5x6oOjXYj0M_JtND4NL0jXZhyzNFH0Lryio6_D1NNlNGOf9slWZx4THnyfe-T-4vzu7Kq4vrlcnJ1eF5Y3tSochxrRAOctA8dabNrW1RadksoC77ApTWmUKgGkq4W0Vimr2ryuaKAqG7FHFjPXBbPSYxzWJr7rYAb99RDiUps4DfYRtVQd65rW2QwpW2FasAxdxR3IsoSuzqyjmZV3f3rGNOlVeI4-j68FZzLHM8mzis8qG0NKEbufVAb6swg9F6Hz7-qvIrTKJjGbUhb7JcZf9D-uD04hjSM
Cites_doi 10.1515/9783110361629
10.1090/S0002-9939-07-09110-1
10.3176/proc.2022.3.07
10.1155/2010/582475
10.1016/j.jmaa.2007.09.062
10.18514/MMN.2019.2954
10.1186/s13663-015-0436-9
10.1007/BF01305855
ContentType Journal Article
Copyright The Author(s) 2025
Copyright Springer Nature B.V. Dec 2025
Copyright_xml – notice: The Author(s) 2025
– notice: Copyright Springer Nature B.V. Dec 2025
DBID C6C
AAYXX
CITATION
3V.
7SC
7TB
7XB
8AL
8FD
8FE
8FG
8FK
ABJCF
ABUWG
AFKRA
ARAPS
AZQEC
BENPR
BGLVJ
CCPQU
DWQXO
FR3
GNUQQ
HCIFZ
JQ2
K7-
KR7
L6V
L7M
L~C
L~D
M0N
M7S
P5Z
P62
PHGZM
PHGZT
PIMPY
PKEHL
PQEST
PQGLB
PQQKQ
PQUKI
PRINS
PTHSS
Q9U
DOA
DOI 10.1186/s13663-025-00791-8
DatabaseName Springer Nature OA Free Journals
CrossRef
ProQuest Central (Corporate)
Computer and Information Systems Abstracts
Mechanical & Transportation Engineering Abstracts
ProQuest Central (purchase pre-March 2016)
Computing Database (Alumni Edition)
Technology Research Database
ProQuest SciTech Collection
ProQuest Technology Collection
ProQuest Central (Alumni) (purchase pre-March 2016)
Materials Science & Engineering Collection
ProQuest Central (Alumni)
ProQuest Central UK/Ireland
Advanced Technologies & Aerospace Collection
ProQuest Central Essentials
ProQuest Central
Technology Collection
ProQuest One Community College
ProQuest Central
Engineering Research Database
ProQuest Central Student
SciTech Premium Collection
ProQuest Computer Science Collection
Computer Science Database
Civil Engineering Abstracts
ProQuest Engineering Collection
Advanced Technologies Database with Aerospace
Computer and Information Systems Abstracts – Academic
Computer and Information Systems Abstracts Professional
Computing Database
Engineering Database
Advanced Technologies & Aerospace Database
ProQuest Advanced Technologies & Aerospace Collection
Proquest Central Premium
ProQuest One Academic (New)
Publicly Available Content Database
ProQuest One Academic Middle East (New)
ProQuest One Academic Eastern Edition (DO NOT USE)
ProQuest One Applied & Life Sciences
ProQuest One Academic
ProQuest One Academic UKI Edition
ProQuest Central China
Engineering Collection
ProQuest Central Basic
DOAJ Open Access Full Text
DatabaseTitle CrossRef
Publicly Available Content Database
Computer Science Database
ProQuest Central Student
Technology Collection
Technology Research Database
Computer and Information Systems Abstracts – Academic
ProQuest One Academic Middle East (New)
Mechanical & Transportation Engineering Abstracts
ProQuest Advanced Technologies & Aerospace Collection
ProQuest Central Essentials
ProQuest Computer Science Collection
Computer and Information Systems Abstracts
ProQuest Central (Alumni Edition)
SciTech Premium Collection
ProQuest One Community College
ProQuest Central China
ProQuest Central
ProQuest One Applied & Life Sciences
ProQuest Engineering Collection
ProQuest Central Korea
ProQuest Central (New)
Advanced Technologies Database with Aerospace
Engineering Collection
Advanced Technologies & Aerospace Collection
Civil Engineering Abstracts
ProQuest Computing
Engineering Database
ProQuest Central Basic
ProQuest Computing (Alumni Edition)
ProQuest One Academic Eastern Edition
ProQuest Technology Collection
ProQuest SciTech Collection
Computer and Information Systems Abstracts Professional
Advanced Technologies & Aerospace Database
ProQuest One Academic UKI Edition
Materials Science & Engineering Collection
Engineering Research Database
ProQuest One Academic
ProQuest One Academic (New)
ProQuest Central (Alumni)
DatabaseTitleList

Publicly Available Content Database
Database_xml – sequence: 1
  dbid: C6C
  name: Springer Nature OA Free Journals
  url: http://www.springeropen.com/
  sourceTypes: Publisher
– sequence: 2
  dbid: DOA
  name: DOAJ Open Access Full Text
  url: https://www.doaj.org/
  sourceTypes: Open Website
– sequence: 3
  dbid: 8FG
  name: ProQuest Technology Collection
  url: https://search.proquest.com/technologycollection1
  sourceTypes: Aggregation Database
DeliveryMethod fulltext_linktorsrc
Discipline Mathematics
EISSN 2730-5422
EndPage 12
ExternalDocumentID oai_doaj_org_article_68f1f9bdccc84b3ab0c1ed52d06440f7
10_1186_s13663_025_00791_8
GrantInformation_xml – fundername: Japan Society for the Promotion of Science
  grantid: JP21K03316
  funderid: http://dx.doi.org/10.13039/501100001691
GroupedDBID 0R~
AAFWJ
AAJSJ
AAKKN
AASML
ABDBF
ABEEZ
ABJCF
ABUWG
ACACY
ACUHS
ACULB
AFGXO
AFKRA
AFPKN
ALMA_UNASSIGNED_HOLDINGS
ARAPS
AZQEC
BENPR
BGLVJ
C24
C6C
CCPQU
DWQXO
EBLON
EBS
ESX
GNUQQ
GROUPED_DOAJ
HCIFZ
I-F
K7-
M7S
M~E
PHGZM
PHGZT
PIMPY
PMFND
PTHSS
SOJ
TUS
~8M
AAYXX
CITATION
PQGLB
3V.
7SC
7TB
7XB
8AL
8FD
8FE
8FG
8FK
FR3
JQ2
KR7
L6V
L7M
L~C
L~D
M0N
P62
PKEHL
PQEST
PQQKQ
PQUKI
PRINS
Q9U
PUEGO
ID FETCH-LOGICAL-c2978-d207eea022b10d1be9bbd7ced868c02fe94a4a884006d736cc88c8b1863905493
IEDL.DBID DOA
ISSN 2730-5422
IngestDate Wed Aug 27 01:29:03 EDT 2025
Fri Jul 25 09:24:37 EDT 2025
Tue Aug 12 03:10:09 EDT 2025
Mon Jun 09 02:42:42 EDT 2025
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 1
Keywords Shrinking projection method
47H09
47H10
Directed graphs
nonexpansive mappings
Language English
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c2978-d207eea022b10d1be9bbd7ced868c02fe94a4a884006d736cc88c8b1863905493
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
OpenAccessLink https://doaj.org/article/68f1f9bdccc84b3ab0c1ed52d06440f7
PQID 3216736162
PQPubID 237292
PageCount 12
ParticipantIDs doaj_primary_oai_doaj_org_article_68f1f9bdccc84b3ab0c1ed52d06440f7
proquest_journals_3216736162
crossref_primary_10_1186_s13663_025_00791_8
springer_journals_10_1186_s13663_025_00791_8
PublicationCentury 2000
PublicationDate 2025-12-01
PublicationDateYYYYMMDD 2025-12-01
PublicationDate_xml – month: 12
  year: 2025
  text: 2025-12-01
  day: 01
PublicationDecade 2020
PublicationPlace Cham
PublicationPlace_xml – name: Cham
– name: New York
PublicationTitle Fixed point theory and algorithms for sciences and engineering
PublicationTitleAbbrev Fixed Point Theory Algorithms Sci Eng
PublicationYear 2025
Publisher Springer International Publishing
Springer Nature B.V
SpringerOpen
Publisher_xml – name: Springer International Publishing
– name: Springer Nature B.V
– name: SpringerOpen
References E. Blum (791_CR2) 1994; 63
Y. Kimura (791_CR7) 2010; 2010
S. Khatoon (791_CR6) 2022; 71
W. Takahashi (791_CR9) 2008; 341
J. Tiammee (791_CR10) 2015; 2015
K. Goebel (791_CR3) 1984
791_CR4
M. Bačák (791_CR1) 2014
J. Jachymski (791_CR5) 2008; 136
P.L. Papini (791_CR8) 1979; 88
References_xml – volume-title: Convex Analysis and Optimization in Hadamard Spaces
  year: 2014
  ident: 791_CR1
  doi: 10.1515/9783110361629
– volume: 136
  start-page: 1359
  issue: 4
  year: 2008
  ident: 791_CR5
  publication-title: Proc. Am. Math. Soc.
  doi: 10.1090/S0002-9939-07-09110-1
– volume: 71
  start-page: 275
  issue: 3
  year: 2022
  ident: 791_CR6
  publication-title: Proc. Est. Acad. Sci.
  doi: 10.3176/proc.2022.3.07
– volume-title: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings
  year: 1984
  ident: 791_CR3
– volume: 2010
  year: 2010
  ident: 791_CR7
  publication-title: Abstr. Appl. Anal.
  doi: 10.1155/2010/582475
– volume: 341
  start-page: 276
  year: 2008
  ident: 791_CR9
  publication-title: J. Math. Anal. Appl.
  doi: 10.1016/j.jmaa.2007.09.062
– volume: 63
  start-page: 123
  year: 1994
  ident: 791_CR2
  publication-title: Math. Stud.
– ident: 791_CR4
  doi: 10.18514/MMN.2019.2954
– volume: 2015
  year: 2015
  ident: 791_CR10
  publication-title: Fixed Point Theory Appl.
  doi: 10.1186/s13663-015-0436-9
– volume: 88
  start-page: 27
  year: 1979
  ident: 791_CR8
  publication-title: Monatshefte Math.
  doi: 10.1007/BF01305855
SSID ssj0002513275
Score 2.4072862
Snippet Recently, the convergence of a shrinking projection method for a Hadamard space satisfying some properties endowed with a directed graph defined on a nonempty...
Abstract Recently, the convergence of a shrinking projection method for a Hadamard space satisfying some properties endowed with a directed graph defined on a...
SourceID doaj
proquest
crossref
springer
SourceType Open Website
Aggregation Database
Index Database
Publisher
StartPage 9
SubjectTerms Algorithms
Analysis
Applications of Mathematics
Approximation
Convergence
Differential Geometry
Directed graphs
G-nonexpansive mappings
Graph theory
Graphs
Hilbert space
Iterative methods
Mathematical and Computational Biology
Mathematics
Mathematics and Statistics
Methods
Shrinking projection method
Topology
SummonAdditionalLinks – databaseName: ProQuest Central
  dbid: BENPR
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwfV1LT9wwEB7BcimHqg-qbksrH3prI_yIvfahqgoCoUpsUQUSNyv2OHDpLt2A6M_v2EkWgVSOeWo0Y8_b3wB80inUJgc5JlKsSvZYVFanWEXDk9KtQh5yoHgyN8fn9Y8LfbEB8_EsTG6rHHViUdS4jDlHvqekyC1Iwshv13-qPDUqV1fHERrNMFoBvxaIsU3YIpWs-QS29g_np7_WWRey5krO9Hh6xpq9TihTSpm6InPpiNQHFqoA-T_wPh8VTIsdOnoBzwcHkn3vJf4SNtLiFWyfrNFXu9dw-nPB6JJ1V6t-MgIb8i0kA9aPjGbkqzKK_NNf0ga5g539bjJSw2XH0gKXdwlZztCygmfd7cD50eHZwXE1TE6oosxhIUo-S6kh-xwERxGSCwFnMaE1NnLZJlc3dWMpuOMGiasxWhttIH4oRz6cU29gkml4C0y3LrYuoRT0SXSq4cEgmqBci7ThcQqfR2756x4gw5fAwhrf89YTb33hrbdT2M8MXb-Zwa3LjeXq0g97xRvbitYFjERWHVQTeBQJtURyn2rezqawO4rDDzuu8_frYwpfRhHdP_4_Se-e_tt7eCbz4igdLLswuVndpg_kh9yEj8Pi-gc9MdtO
  priority: 102
  providerName: ProQuest
– databaseName: Springer Nature OA Free Journals
  dbid: C6C
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV25TsQwELU4GigQp1hYkAs6iPARO3YJKxBCWqBgpe2s-IKGBREQfD5jJ7scgoIyjh1ZbzyZeT6eEToQwZYykRzpgKtCPKaFEsEVTpLAReSe2EQUh1fyYlRejsW4k8lJZ2G-rt9TJY8bymVeaRQFRDMNX5pHiwJ-vGk0D-RgNp8CcZqzSkzPxfza9FvsyRL93_LKH0uhOcKcr6KVLjXEJ60t19BcmKyj5eFMV7XZQDfXEwyPuLl_bu88wN1MCqCL28ugMWShGDh9eAc_T3vT8UOdNBjuGhwm_vEteJzmXnFWqm420ej87HZwUXR3IhSOJcLnGalCqCHyWko8tUFb6ysXvJLKERaDLuuyVkDbiPQVl84p5ZQFPLiG7EzzLbSQ-rCNsIjaRR08o9DEaV4TK72XluvowZV9Dx1O0TJPrfSFyZRBSdNiawBbk7E1qodOE6Czmkm2OheANU3nBUaqSKO23kG3SstrSxwNXjAPiVFJYtVD_ak5TOdLjeGMps1nVLIeOpqa6PP1313a-V_1XbTE0mDJe1X6aOHl-TXsQcbxYvfzUPsAw3DNrg
  priority: 102
  providerName: Springer Nature
Title On the shrinking projection method for nonexpansive mappings endowed with graphs
URI https://link.springer.com/article/10.1186/s13663-025-00791-8
https://www.proquest.com/docview/3216736162
https://doaj.org/article/68f1f9bdccc84b3ab0c1ed52d06440f7
Volume 2025
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LS8QwEB58XPQgPnF9LDl402IebTY56rKrCKsiCt5C86heXMUqevK3O0m7vkC8eAmkaWH4JunMl0xmAHaKYHMZSY50yFXRHrNMFcFlTtIgikp4aiNRHJ3K46v85Lq4_lLqK8aENemBG-D2papYpa13zqncitJSx4IvuEdbmtMq3SOnmn4hU_EfjFZb8F4xuSWj5H7NhExHlkWGZlGjSN8sUUrY_83L_HEwmuzNcBEWWkeRHDQCLsFUGC_D_Ogjy2q9AudnY4JdUt8-NhUQSLuvgliTpjQ0QZ-UIMMPr7jqY6Q6uStjRoabmoSxv38JnsSdWJLyVtercDUcXPaPs7ZCQuZ4pH-e014IJdphy6hnNmhrfc8Fr6RylFdB52VeKiRxVPqekIihcsoiHkKjr6bFGsxEGdaBFJV2lQ6eM_zEaVFSK72XVujK48L2HdidoGUemkQYJhEIJU2DrUFsTcLWqA4cRkA_3oxJrNMDVK1pVWv-Um0HtibqMO3Kqo3gLIaiMck7sDdR0efw7yJt_IdImzDH4xRK8SxbMPP0-By20St5sl2YVsOjLsweDk7PL7DX53lsZb-bpia2o7fBO4Kt48s
linkProvider Directory of Open Access Journals
linkToHtml http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1LTxwxDLbQcqA9VPSlbgttDu2pHZFkZrKZA0KlBS2F3aIKJG7pJM7ApbN0hwr4c_y2OpmZRVQqN47zlGU7_mzHsQHe595mKgQ5ylGsSngsEp17lzjFfZpXKXIbAsXJVI2Ps28n-ckS3PRnYUJZZW8To6HGmQs58o1UilCCJJTcOv-dhKlRYXe1H6FRdqMVcDO2GOsOduz760sK4ZrNva8k7w9S7u4cfRkn3ZSBxMkQQqHkI-9LwjIrOArrC2tx5DxqpR2XlS-yMis1BUJcIVHgnNZOW6EJ28nfCc2YCAKWs5BAGcDy9s708Mciy0PeQypHeX9aR6uNRqQqbp3mCcFzQay5g4hxcMAdb_efDdqIe7ur8KRzWNnnVsOewpKvn8HjyaLba_McDr_XjC5ZczZvJzGwLr9DMmftiGpGvjGrZ7W_IusTKubZrzJ0hjhtmK9xdumRhYwwi_2zmxdw_CA8fAmDQMMrYHlVuKrwKAV94oq05FYhKpsWFZKBwSF87LllztuGHCYGMlqZlreGeGsib40ewnZg6OLN0Ew73pjNT023No3SlagKi47IymxaWu6Ex1wiuWsZr0ZDWOvFYboV3phbfRzCp15Et4__T9Lr-__2DlbGR5MDc7A33X8Dj2RQlFg9swaDi_kfv04-0IV92ykag58Prdt_AWsAF6I
linkToPdf http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV3NTxUxEG8IJEYPxM_4ALUHPenmtd3dbvdAjAgvIPJ8MZJwq9tOixf3wVsM8C_6Vzntdh_BRG8c9zOTmenM_KbTGUJel84UMoAcaRGroj_mmSqdzaxkLi99DswEoHg0lfvHxaeT8mSF_B7OwoSyysEmRkMNcxty5ONc8FCCxKUY-1QWMdudvD87z8IEqbDTOozTaNKYBdiO7cbSIY9Dd32JcK7bPthF2b8RYrL37eN-liYOZFYEOAWCVc416NcMZ8CNq42ByjpQUlkmvKuLpmgUgiImAamxVimrDFfo5zH2CY2Z0B2sVej1EQiu7exNZ1-XGR-MJHJRlcPJHSXHHc9l3EYtM3TVNbLplneMQwRuRb5_bdZGHzh5SNZT8Eo_9Nr2iKy49jF5cLTs_No9IbMvLcVL2v1Y9FMZaMr1oPxpP66aYpxM23nrrtAShep5-rMJXSJOO-pamF86oCE7TGMv7e4pOb4THj4jq4GG54SWvra-diA4fmLrvGFGAkiT1x7Q2MCIvB24pc_65hw6gholdc9bjbzVkbdajchOYOjyzdBYO96YL051WqdaKs99bcAiWYXJG8Msd1AKwNCtYL4aka1BHDqt9k7f6OaIvBtEdPP43yRt_P9vr8g91HH9-WB6uEnui6AnsZBmi6xeLH65FxgOXZiXSc8o-X7Xqv0Hf2Eb5g
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=On+the+shrinking+projection+method+for+nonexpansive+mappings+endowed+with+graphs&rft.jtitle=Fixed+point+theory+and+algorithms+for+sciences+and+engineering&rft.au=Yasunori+Kimura&rft.au=Supaluk+Phothi&rft.au=Kittisak+Tontan&rft.date=2025-12-01&rft.pub=SpringerOpen&rft.eissn=2730-5422&rft.volume=2025&rft.issue=1&rft.spage=1&rft.epage=12&rft_id=info:doi/10.1186%2Fs13663-025-00791-8&rft.externalDBID=DOA&rft.externalDocID=oai_doaj_org_article_68f1f9bdccc84b3ab0c1ed52d06440f7
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2730-5422&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2730-5422&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2730-5422&client=summon