{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T06:03:18Z","timestamp":1780293798578,"version":"3.54.0"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T00:00:00Z","timestamp":1768521600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T00:00:00Z","timestamp":1768521600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1007\/s40314-025-03614-z","type":"journal-article","created":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T08:22:45Z","timestamp":1768551765000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Convergence analysis of a modified Dai-Liao-type conjugate gradient projection algorithm without Lipschitz continuity for constrained nonlinear equations with applications"],"prefix":"10.1007","volume":"45","author":[{"given":"Dandan","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yong","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8024-5056","authenticated-orcid":false,"given":"Songhua","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,1,16]]},"reference":[{"issue":"8","key":"3614_CR1","doi-asserted-by":"publisher","first-page":"693","DOI":"10.3390\/math7080693","volume":"7","author":"AB Abubakar","year":"2019","unstructured":"Abubakar AB, Kumam P, Awwal AM et al (2019) A modified self-adaptive conjugate gradient method for solving convex constrained monotone nonlinear equations for signal recovery problems. Mathematics 7(8):693","journal-title":"Mathematics"},{"issue":"2","key":"3614_CR2","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1007\/s10957-023-02281-6","volume":"200","author":"K Ahmed","year":"2024","unstructured":"Ahmed K, Waziri MY, Murtala S et al (2024) On a scaled symmetric dai-liao-type scheme for constrained system of nonlinear equations with applications. J Optim Theory Appl 200(2):669\u2013702","journal-title":"J Optim Theory Appl"},{"issue":"6","key":"3614_CR3","doi-asserted-by":"publisher","first-page":"1949","DOI":"10.1080\/02331934.2023.2187255","volume":"73","author":"K Ahmed","year":"2024","unstructured":"Ahmed K, Yusuf Waziri M, Sani Halilu A et al (2024) Sparse signal reconstruction via hager-zhang-type schemes for constrained system of nonlinear equations. Optimization 73(6):1949\u20131980","journal-title":"Optimization"},{"issue":"1","key":"3614_CR4","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2583","volume":"32","author":"K Ahmed","year":"2025","unstructured":"Ahmed K, Waziri MY, Halilu AS et al (2025) Signal and image reconstruction with a double parameter hager-zhang-type conjugate gradient method for system of nonlinear equations. Numerical Linear Algebra with Applications 32(1):e2583","journal-title":"Numerical Linear Algebra with Applications"},{"issue":"3","key":"3614_CR5","doi-asserted-by":"publisher","first-page":"1683","DOI":"10.11591\/ijeecs.v21.i3.pp1683-1691","volume":"21","author":"HS Chilmeran","year":"2021","unstructured":"Chilmeran HS, Ahmed HI, Hamed ET et al (2021) A new modification of the quasi-Newton method for unconstrained optimization. Indonesian Journal of Electrical Engineering and Computer Science 21(3):1683\u20131691","journal-title":"Indonesian Journal of Electrical Engineering and Computer Science"},{"key":"3614_CR6","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1007\/s002450010019","volume":"43","author":"L Dai","year":"2001","unstructured":"Dai L, Liao L (2001) New conjugacy conditions and related nonlinear conjugate gradient methods. Appl Math Optim 43:87\u2013101","journal-title":"Appl Math Optim"},{"issue":"14","key":"3614_CR7","doi-asserted-by":"publisher","first-page":"7421","DOI":"10.1016\/j.amc.2011.12.091","volume":"218","author":"Z Dai","year":"2012","unstructured":"Dai Z, Wen F (2012) Another improved wei-yao-liu nonlinear conjugate gradient method with sufficient descent property. Appl Math Comput 218(14):7421\u20137430","journal-title":"Appl Math Comput"},{"key":"3614_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.icheatmasstransfer.2020.104595","volume":"114","author":"S Deniz","year":"2020","unstructured":"Deniz S, Sezer M (2020) Rational Chebyshev collocation method for solving nonlinear heat transfer equations. Int Commun Heat Mass Transfer 114:104595","journal-title":"Int Commun Heat Mass Transfer"},{"key":"3614_CR9","doi-asserted-by":"publisher","first-page":"214","DOI":"10.1007\/s10473-019-0117-6","volume":"39","author":"SS Djordjevi\u0107","year":"2019","unstructured":"Djordjevi\u0107 SS (2019) New hybrid conjugate gradient method as a convex combination of LS and FR methods. Acta Mathematica Scientia 39:214\u2013228","journal-title":"Acta Mathematica Scientia"},{"key":"3614_CR10","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"ED Dolan","year":"2002","unstructured":"Dolan ED, Mor\u00e9 JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91:201\u2013213","journal-title":"Math Program"},{"key":"3614_CR11","unstructured":"Fan Jy (2003) A modified Levenberg\u2013Marquardt algorithm for singular system of nonlinear equations. Journal of Computational Mathematics pp 625\u2013636"},{"issue":"5","key":"3614_CR12","doi-asserted-by":"publisher","first-page":"1802","DOI":"10.1108\/EC-06-2021-0317","volume":"39","author":"AS Halilu","year":"2021","unstructured":"Halilu AS, Majumder A, Waziri MY et al (2021) Motion control of the two joint planar robotic manipulators through accelerated dai-liao method for solving system of nonlinear equations. Eng Comput 39(5):1802\u20131840","journal-title":"Eng Comput"},{"issue":"4","key":"3614_CR13","doi-asserted-by":"publisher","first-page":"602","DOI":"10.3390\/math8040602","volume":"8","author":"A Hassan Ibrahim","year":"2020","unstructured":"Hassan Ibrahim A, Kumam P, Abubakar AB et al (2020) Least-square-based three-term conjugate gradient projection method for $$l_1$$-norm problems with application to compressed sensing. Mathematics 8(4):602","journal-title":"Mathematics"},{"key":"3614_CR14","unstructured":"Ibrahim AH, Garba A, Usman H, et\u00a0al (2019) Derivative-free RMIL conjugate gradient algorithm for convex constrained equations. Thai Journal of Mathematics 18(1)"},{"key":"3614_CR15","doi-asserted-by":"crossref","unstructured":"Ibrahim AH, Kumam P, Abubakar AB, et\u00a0al (2020) A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing. Heliyon 6(3)","DOI":"10.1016\/j.heliyon.2020.e03466"},{"key":"3614_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2024.116234","volume":"456","author":"J Jian","year":"2025","unstructured":"Jian J, Jin J, Ma G (2025) An IDFPM-based algorithm without Lipschitz continuity to constrained nonlinear equations for sparse signal and blurred image restoration problems. J Comput Appl Math 456:116234","journal-title":"J Comput Appl Math"},{"key":"3614_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2022.115006","volume":"424","author":"D Li","year":"2023","unstructured":"Li D, Wu J, Li Y et al (2023) A modified spectral gradient projection-based algorithm for large-scale constrained nonlinear equations with applications in compressive sensing. J Comput Appl Math 424:115006","journal-title":"J Comput Appl Math"},{"issue":"16","key":"3614_CR18","doi-asserted-by":"publisher","first-page":"2556","DOI":"10.3390\/math12162556","volume":"12","author":"D Li","year":"2024","unstructured":"Li D, Li Y, Wang S (2024) An improved three-term conjugate gradient algorithm for constrained nonlinear equations under non-Lipschitz conditions and its applications. Mathematics 12(16):2556","journal-title":"Mathematics"},{"key":"3614_CR19","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/j.apnum.2023.09.009","volume":"195","author":"D Li","year":"2024","unstructured":"Li D, Wang S, Li Y et al (2024) A projection-based hybrid PRP-DY type conjugate gradient algorithm for constrained nonlinear equations with applications. Appl Numer Math 195:105\u2013125","journal-title":"Appl Numer Math"},{"issue":"1","key":"3614_CR20","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1007\/s40314-024-03008-7","volume":"44","author":"D Li","year":"2025","unstructured":"Li D, Li Y, Li Y et al (2025) A modified PRP-type derivative-free projection algorithm for constrained nonlinear equations with applications. Comput Appl Math 44(1):48","journal-title":"Comput Appl Math"},{"key":"3614_CR21","doi-asserted-by":"crossref","unstructured":"Liu J, Du S (2019) Modified three-term conjugate gradient method and its applications. mathematical problems in engineering 2019(1):5976595","DOI":"10.1155\/2019\/5976595"},{"key":"3614_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2021.113822","volume":"403","author":"J Liu","year":"2022","unstructured":"Liu J, Lu Z, Xu J et al (2022) An efficient projection-based algorithm without Lipschitz continuity for large-scale nonlinear pseudo-monotone equations. J Comput Appl Math 403:113822","journal-title":"J Comput Appl Math"},{"key":"3614_CR23","doi-asserted-by":"publisher","first-page":"98","DOI":"10.1016\/j.apnum.2022.02.001","volume":"175","author":"P Liu","year":"2022","unstructured":"Liu P, Shao H, Wang Y et al (2022) A three-term CGPM-based algorithm without Lipschitz continuity for constrained nonlinear monotone equations with applications. Appl Numer Math 175:98\u2013107","journal-title":"Appl Numer Math"},{"issue":"5","key":"3614_CR24","doi-asserted-by":"publisher","first-page":"1610","DOI":"10.1016\/j.jedc.2006.05.007","volume":"31","author":"A Ludwig","year":"2007","unstructured":"Ludwig A (2007) The Gauss-Seidel-quasi-Newton method: A hybrid algorithm for solving dynamic economic models. J Econ Dyn Control 31(5):1610\u20131632","journal-title":"J Econ Dyn Control"},{"issue":"2","key":"3614_CR25","doi-asserted-by":"publisher","first-page":"1032","DOI":"10.1016\/j.amc.2006.07.004","volume":"184","author":"C Ma","year":"2007","unstructured":"Ma C, Jiang L (2007) Some research on Levenberg-Marquardt method for the nonlinear equations. Appl Math Comput 184(2):1032\u20131040","journal-title":"Appl Math Comput"},{"key":"3614_CR26","volume":"1","author":"A Ramli","year":"2010","unstructured":"Ramli A, Abdullah ML (2010) Mamat M (2010) Broyden\u2019s method for solving fuzzy nonlinear equations. Adv Fuzzy Syst 1:763270","journal-title":"Adv Fuzzy Syst"},{"issue":"22","key":"3614_CR27","doi-asserted-by":"publisher","first-page":"11323","DOI":"10.1016\/j.amc.2012.05.030","volume":"218","author":"M Rivaie","year":"2012","unstructured":"Rivaie M, Mamat M, June LW et al (2012) A new class of nonlinear conjugate gradient coefficients with global convergence properties. Appl Math Comput 218(22):11323\u201311332","journal-title":"Appl Math Comput"},{"key":"3614_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2020.124800","volume":"556","author":"KK Sabelfeld","year":"2020","unstructured":"Sabelfeld KK, Kireeva A (2020) Stochastic simulation algorithms for solving a nonlinear system of drift-diffusion-Poisson equations of semiconductors. Physica A 556:124800","journal-title":"Physica A"},{"key":"3614_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13660-020-2301-6","volume":"2020","author":"C Tang","year":"2020","unstructured":"Tang C, Li S, Cui Z (2020) Least-squares-based three-term conjugate gradient methods. Journal of Inequalities and Applications 2020:1\u201322","journal-title":"Journal of Inequalities and Applications"},{"key":"3614_CR30","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/s40324-020-00228-9","volume":"78","author":"M Waziri","year":"2021","unstructured":"Waziri M, Ahmed K, Sabiu J et al (2021) Enhanced dai-liao conjugate gradient methods for systems of monotone nonlinear equations. SeMA Journal 78:15\u201351","journal-title":"SeMA Journal"},{"key":"3614_CR31","unstructured":"Wei Z, Huang H, Tao Y (2010) A modified hestenes stiefel conjugate gradient method and its convergence. In: Journal of Mathematical Research and Exposition, pp 297\u2013308"},{"issue":"1","key":"3614_CR32","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1016\/j.jmaa.2013.04.017","volume":"405","author":"Y Xiao","year":"2013","unstructured":"Xiao Y, Zhu H (2013) A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J Math Anal Appl 405(1):310\u2013319","journal-title":"J Math Anal Appl"},{"issue":"4","key":"3614_CR33","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1080\/10556780701223293","volume":"22","author":"L Zhang","year":"2007","unstructured":"Zhang L, Zhou W, Li D (2007) Some descent three-term conjugate gradient methods and their global convergence. Optimisation Methods and Software 22(4):697\u2013711","journal-title":"Optimisation Methods and Software"},{"issue":"3","key":"3614_CR34","doi-asserted-by":"publisher","first-page":"2427","DOI":"10.1007\/s12190-024-02062-y","volume":"70","author":"H Zheng","year":"2024","unstructured":"Zheng H, Li J, Liu P et al (2024) An inertial Fletcher-Reeves-type conjugate gradient projection-based method and its spectral extension for constrained nonlinear equations. J Appl Math Comput 70(3):2427\u20132452","journal-title":"J Appl Math Comput"},{"key":"3614_CR35","doi-asserted-by":"publisher","first-page":"502","DOI":"10.1007\/s10957-017-1140-1","volume":"175","author":"Y Zheng","year":"2017","unstructured":"Zheng Y, Zheng B (2017) Two new Dai-Liao-type conjugate gradient methods for unconstrained optimization problems. J Optim Theory Appl 175:502\u2013509","journal-title":"J Optim Theory Appl"},{"key":"3614_CR36","unstructured":"Zhou W, Li D (2007) Limited memory BFGS method for nonlinear monotone equations. Journal of Computational Mathematics pp 89\u201396"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03614-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03614-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03614-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T05:39:31Z","timestamp":1780292371000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03614-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,16]]},"references-count":36,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["3614"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03614-z","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,16]]},"assertion":[{"value":"14 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 September 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 January 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"204"}}