{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T01:17:01Z","timestamp":1783127821926,"version":"3.54.6"},"reference-count":21,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100007129","name":"Shandong Province Natural Science Foundation","doi-asserted-by":"publisher","award":["ZR2024QA053"],"award-info":[{"award-number":["ZR2024QA053"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100010018","name":"Liaoning Province Doctoral Start-up Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100010018","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005047","name":"Natural Science Foundation of Liaoning Province","doi-asserted-by":"publisher","award":["2026-BS-0669"],"award-info":[{"award-number":["2026-BS-0669"]}],"id":[{"id":"10.13039\/501100005047","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100014761","name":"Natural Science Foundation of Qingdao","doi-asserted-by":"publisher","award":["24-4-4-zrjj-6-jch"],"award-info":[{"award-number":["24-4-4-zrjj-6-jch"]}],"id":[{"id":"10.13039\/501100014761","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12401601"],"award-info":[{"award-number":["12401601"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.cnsns.2026.110237","type":"journal-article","created":{"date-parts":[[2026,5,30]],"date-time":"2026-05-30T05:30:53Z","timestamp":1780119053000},"page":"110237","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"P1","title":["Static deformation of flexible beams: A unified variational framework for impulsive problems with nonlocal constraints"],"prefix":"10.1016","volume":"162","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6415-9219","authenticated-orcid":false,"given":"Yongfang","family":"Wei","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu","family":"Cheng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lin","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cnsns.2026.110237_bib0001","first-page":"1","article-title":"Applications of morse theory to fourth-order impulsive differential equations","author":"Gao","year":"2022","journal-title":"Math Probl Eng"},{"key":"10.1016\/j.cnsns.2026.110237_bib0002","first-page":"1","article-title":"Applications of variational methods to some three-point boundary value problems with instantaneous and noninstantaneous impulses","volume":"27","author":"Wei","year":"2022","journal-title":"Nonlinear Anal-Model"},{"key":"10.1016\/j.cnsns.2026.110237_bib0003","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1016\/j.na.2010.04.016","article-title":"On superlinear problems without the Ambrosetti and Rabinowitz condition","volume":"73","author":"Liu","year":"2010","journal-title":"Nonlinear Anal"},{"key":"10.1016\/j.cnsns.2026.110237_bib0004","series-title":"Differential and difference equations with applications","article-title":"Triple solutions for elastic beam equations of the fourth-order with boundary conditions subjected to an elastic device","volume":"vol. 333","author":"Bonanno","year":"2020"},{"key":"10.1016\/j.cnsns.2026.110237_bib0005","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.aml.2018.02.006","article-title":"Existence of infinitely many solutions for fourth-order impulsive differential equations","volume":"81","author":"Yue","year":"2018","journal-title":"Appl Math Lett"},{"key":"10.1016\/j.cnsns.2026.110237_bib0006","doi-asserted-by":"crossref","DOI":"10.1155\/2014\/690381","article-title":"The solutions of Sturm-Liouville boundary-value problem for fourth-order impulsive differential equation via variational methods","volume":"2014","author":"Tian","year":"2014","journal-title":"Abstr Appl Anal"},{"key":"10.1016\/j.cnsns.2026.110237_bib0007","first-page":"1","article-title":"Three solutions for fourth-order impulsive differential inclusions via nonsmooth critical point theory","author":"Gao","year":"2018","journal-title":"J Funct Space"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110237_bib0008","doi-asserted-by":"crossref","DOI":"10.1007\/s10915-023-02430-z","article-title":"Numerical methods for fourth-order pdes on overlapping grids with application to Kirchhoff\u2013love plates","volume":"98","author":"Li","year":"2024","journal-title":"J Sci Comput"},{"issue":"10","key":"10.1016\/j.cnsns.2026.110237_bib0009","doi-asserted-by":"crossref","first-page":"6324","DOI":"10.1002\/mma.6373","article-title":"Solvability of the boundary value problems for the elastic beam equations with impulsive effects crossing resonance points","volume":"43","author":"Chao","year":"2020","journal-title":"Math Method Appl Sci"},{"key":"10.1016\/j.cnsns.2026.110237_bib0010","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2025.116471","article-title":"Solution of non-autonomous impulsive evolution equation in reflexive banach spaces","volume":"197","author":"Ma","year":"2025","journal-title":"Chaos Solit Fract"},{"key":"10.1016\/j.cnsns.2026.110237_bib0011","article-title":"Multiple solutions for some nonlinear impulsive differential equations with three-point boundary conditions via variational approach","volume":"11","author":"Wei","year":"2021","journal-title":"J Appl Anal Comput"},{"key":"10.1016\/j.cnsns.2026.110237_bib0012","series-title":"Minimax theorem","author":"Willem","year":"1996"},{"key":"10.1016\/j.cnsns.2026.110237_bib0013","doi-asserted-by":"crossref","DOI":"10.1016\/j.aml.2020.106283","article-title":"Existence of solution of a three-point boundary value problem via variational approach","volume":"104","author":"Lian","year":"2020","journal-title":"Appl Math Lett"},{"key":"10.1016\/j.cnsns.2026.110237_bib0014","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/s10483-022-2832-9","article-title":"Solutions for a class of Hamiltonian systems on time scales with non-local boundary conditions","volume":"43","author":"Wei","year":"2022","journal-title":"Appl Math Mech"},{"issue":"6","key":"10.1016\/j.cnsns.2026.110237_bib0015","first-page":"1009","article-title":"Superlinear damped vibration problems on time scales with nonlocal boundary conditions","volume":"27","author":"Wei","year":"2022","journal-title":"Nonlinear Anal-Model"},{"issue":"6","key":"10.1016\/j.cnsns.2026.110237_bib0016","doi-asserted-by":"crossref","DOI":"10.1007\/s40314-024-02883-4","article-title":"A new robust compact difference scheme on graded meshes for the time-fractional nonlinear Kuramoto\u2013Sivashinsky equation","volume":"43","author":"Wang","year":"2024","journal-title":"Comput Appl Math"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110237_bib0017","article-title":"Nonlinear perturbations of a periodic Kirchhoff\u2013Boussinesq-type problems in RN","volume":"75","author":"Carlos","year":"2024","journal-title":"Z Angew Math Phy"},{"key":"10.1016\/j.cnsns.2026.110237_bib0018","series-title":"Critical point theory and Hamiltonian systems","author":"Mawhin","year":"1989"},{"issue":"10","key":"10.1016\/j.cnsns.2026.110237_bib0019","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1016\/0362-546X(93)90151-H","article-title":"Infinitely many solutions of a symmetric Dirichlet problem","volume":"20","author":"Bartsch","year":"1993","journal-title":"Nonlinear Anal-Theor"},{"key":"10.1016\/j.cnsns.2026.110237_bib0020","first-page":"332","article-title":"An existence criterion for the critical points on unbounded manifolds","volume":"112","author":"Cerami","year":"1978","journal-title":"Ist Lomb Accad Sci Lett Rend Sci Mat Appl A"},{"key":"10.1016\/j.cnsns.2026.110237_bib0021","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s12190-009-0359-x","article-title":"Variational methods to fourth-order impulsive differential equations","volume":"35","author":"Sun","year":"2011","journal-title":"J Appl Math Comput"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426005940?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426005940?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T00:32:22Z","timestamp":1783125142000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570426005940"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":21,"alternative-id":["S1007570426005940"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110237","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Static deformation of flexible beams: A unified variational framework for impulsive problems with nonlocal constraints","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110237","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110237"}}