{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T12:17:37Z","timestamp":1767183457467},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T00:00:00Z","timestamp":1663286400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T00:00:00Z","timestamp":1663286400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2022,10]]},"DOI":"10.1007\/s10957-022-02087-y","type":"journal-article","created":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T07:34:04Z","timestamp":1663313644000},"page":"347-373","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["General Versions of the Ekeland Variational Principle: Ekeland Points and Stop and Go Dynamics"],"prefix":"10.1007","volume":"195","author":[{"given":"Le Phuoc","family":"Hai","sequence":"first","affiliation":[]},{"given":"Phan Quoc","family":"Khanh","sequence":"additional","affiliation":[]},{"given":"Antoine","family":"Soubeyran","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,16]]},"reference":[{"issue":"12","key":"2087_CR1","doi-asserted-by":"publisher","first-page":"3577","DOI":"10.1090\/S0002-9939-06-08744-2","volume":"134","author":"D Az\u00e9","year":"2006","unstructured":"Az\u00e9, D., Corvellec, J.-N.: A variational method in fixed point results with inwardness conditions. Proc. Am. Math. Soc. 134(12), 3577\u20133583 (2006)","journal-title":"Proc. Am. Math. Soc."},{"issue":"1\u20132","key":"2087_CR2","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/s10479-016-2368-0","volume":"269","author":"TQ Bao","year":"2018","unstructured":"Bao, T.Q., Cobzas, S., Soubeyran, A.: Variational principles, completeness and the existence of traps in behavioral sciences. Ann. Oper. Res. 269(1\u20132), 53\u201379 (2018)","journal-title":"Ann. Oper. Res."},{"issue":"12","key":"2087_CR3","doi-asserted-by":"publisher","first-page":"2049","DOI":"10.1080\/02331934.2016.1228062","volume":"65","author":"TQ Bao","year":"2016","unstructured":"Bao, T.Q., Khanh, P.Q., Soubeyran, A.: Variational principles with generalized distances and the modelization of organizational change. Optimization 65(12), 2049\u20132066 (2016)","journal-title":"Optimization"},{"issue":"2","key":"2087_CR4","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s10107-008-0249-2","volume":"122","author":"TQ Bao","year":"2010","unstructured":"Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. A 122(2), 301\u2013347 (2010)","journal-title":"Math. Program. A"},{"issue":"1","key":"2087_CR5","doi-asserted-by":"publisher","first-page":"290","DOI":"10.1007\/s10957-014-0569-8","volume":"164","author":"TQ Bao","year":"2015","unstructured":"Bao, T.Q., Mordukhovich, B.S., Soubeyran, A.: Variational analysis in psychological modeling. J. Optim. Theory Appl. 164(1), 290\u2013315 (2015)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"2087_CR6","doi-asserted-by":"publisher","first-page":"458","DOI":"10.1007\/s10957-016-0933-y","volume":"170","author":"TQ Bao","year":"2016","unstructured":"Bao, T.Q., Soubeyran, A.: Variational analysis in cone pseudo-quasimetric spaces and applications to group dynamics. J. Optim. Theory Appl. 170(2), 458\u2013475 (2016)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"2087_CR7","doi-asserted-by":"publisher","first-page":"517","DOI":"10.1090\/S0002-9947-1987-0902782-7","volume":"303","author":"JM Borwein","year":"1987","unstructured":"Borwein, J.M., Preiss, D.: A smooth variational principle with applications to subdifferentiability and differentiability of convex functions. Trans. Am. Math. Soc. 303(2), 517\u2013527 (1987)","journal-title":"Trans. Am. Math. Soc."},{"key":"2087_CR8","volume-title":"Techniques of Variational Analysis","author":"JM Borwein","year":"2005","unstructured":"Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, Berlin (2005)"},{"key":"2087_CR9","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1090\/S0002-9947-1976-0394329-4","volume":"215","author":"J Caristi","year":"1976","unstructured":"Caristi, J.: Fixed point theorem for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241\u2013251 (1976)","journal-title":"Trans. Am. Math. Soc."},{"key":"2087_CR10","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1007\/s00224-010-9310-7","volume":"50","author":"MA Cerd\u00e0-Uguet","year":"2012","unstructured":"Cerd\u00e0-Uguet, M.A., Schellekens, M.P., Valero, O.: The Baire partial quasi-metric space: a mathematical tool for asymptotic complexity analysis in computer science. Theory Comput. Syst. 50, 387\u2013399 (2012)","journal-title":"Theory Comput. Syst."},{"issue":"3","key":"2087_CR11","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s00186-007-0205-6","volume":"67","author":"GY Chen","year":"2008","unstructured":"Chen, G.Y., Yang, X.Q., Yu, H.: Vector Ekeland\u2019s variational principle in an F-type topological space. Math. Methods Oper. Res. 67(3), 471\u2013478 (2008)","journal-title":"Math. Methods Oper. Res."},{"issue":"2","key":"2087_CR12","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1287\/moor.1.2.165","volume":"1","author":"FH Clarke","year":"1976","unstructured":"Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Res. 1(2), 165\u2013174 (1976)","journal-title":"Math. Oper. Res."},{"issue":"10\u201311","key":"2087_CR13","doi-asserted-by":"publisher","first-page":"2558","DOI":"10.1016\/j.topol.2012.04.015","volume":"159","author":"S Cobza\u015f","year":"2012","unstructured":"Cobza\u015f, S.: Ekeland variational principle in asymmetric locally convex spaces. Topol. Appl. 159(10\u201311), 2558\u20132569 (2012)","journal-title":"Topol. Appl."},{"issue":"4","key":"2087_CR14","first-page":"369","volume":"6","author":"J Danes\u0306","year":"1972","unstructured":"Danes\u0306, J.: A geometric theorem useful in nonlinear analysis. Bull. U.M.I. 6(4), 369\u2013375 (1972)","journal-title":"Bull. U.M.I."},{"issue":"1\u20132","key":"2087_CR15","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1007\/s10107-007-0127-3","volume":"116","author":"A Daniilidis","year":"2009","unstructured":"Daniilidis, A., Jules, F., Lassonde, M.: Subdifferential characterizations of approximate convexity: the lower semicontinuous case. Math. Program. B 116(1\u20132), 115\u2013127 (2009)","journal-title":"Math. Program. B"},{"issue":"1\u20133","key":"2087_CR16","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/S0166-8641(97)00140-5","volume":"85","author":"JW de Bakker","year":"1998","unstructured":"de Bakker, J.W., de Vink, E.P.: Denotational models for programming languages: applications of Banach\u2019s fixed point theorem. Topol. Appl. 85(1\u20133), 35\u201352 (1998)","journal-title":"Topol. Appl."},{"issue":"1","key":"2087_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s002080200327","volume":"324","author":"M del Pino","year":"2002","unstructured":"del Pino, M., Felmer, P.: Semi-classical states of nonlinear Schr\u00f6dinger equations: a variational reduction method. Math. Ann. 324(1), 1\u201332 (2002)","journal-title":"Math. Ann."},{"issue":"2","key":"2087_CR18","doi-asserted-by":"publisher","first-page":"324","DOI":"10.1016\/0022-247X(74)90025-0","volume":"47","author":"I Ekeland","year":"1974","unstructured":"Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47(2), 324\u2013353 (1974)","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"2087_CR19","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1090\/S0273-0979-1979-14595-6","volume":"1","author":"I Ekeland","year":"1979","unstructured":"Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. 1(3), 443\u2013474 (1979)","journal-title":"Bull. Am. Math. Soc."},{"key":"2087_CR20","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/s10898-021-01061-8","volume":"82","author":"M Fakhar","year":"2022","unstructured":"Fakhar, M., Khodakhah, M., Mazyaki, A., Soubeyran, A., Zafarani, J.: Variational rationality, variational principles and the existence of traps in a changing environment. J. Glob. Optim. 82, 161\u2013177 (2022)","journal-title":"J. Glob. Optim."},{"issue":"7\u20138","key":"2087_CR21","doi-asserted-by":"publisher","first-page":"1481","DOI":"10.1080\/02331934.2019.1589468","volume":"69","author":"LP Hai","year":"2020","unstructured":"Hai, L.P., Khanh, P.Q.: An induction theorem and Ekeland\u2019s variational principle in partial metric spaces with applications. Optimization 69(7\u20138), 1481\u20131511 (2020)","journal-title":"Optimization"},{"issue":"10","key":"2087_CR22","doi-asserted-by":"publisher","first-page":"3025","DOI":"10.1090\/S0002-9939-03-07066-7","volume":"131","author":"AH Hamel","year":"2003","unstructured":"Hamel, A.H.: Phelps\u2019 lemma, Danes\u2019 drop theorem, and Ekeland\u2019s principle in locally convex spaces. Proc. Am. Math. Soc. 131(10), 3025\u20133038 (2003)","journal-title":"Proc. Am. Math. Soc."},{"issue":"1","key":"2087_CR23","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1112\/jlms\/s2-41.1.175","volume":"41","author":"AD Ioffe","year":"1990","unstructured":"Ioffe, A.D.: Proximal analysis and approximate subdifferentials. J. London Math. Soc. 41(1), 175\u2013192 (1990)","journal-title":"J. London Math. Soc."},{"issue":"4","key":"2087_CR24","doi-asserted-by":"publisher","first-page":"1255","DOI":"10.1090\/S0002-9947-1995-1290719-8","volume":"347","author":"A Jourani","year":"1995","unstructured":"Jourani, A., Thibault, L.: Verifiable conditions for openness and regularity of multivalued mappings in Banach spaces. Trans. Am. Math. Soc. 347(4), 1255\u20131268 (1995)","journal-title":"Trans. Am. Math. Soc."},{"issue":"9\u201310","key":"2087_CR25","doi-asserted-by":"publisher","first-page":"2442","DOI":"10.1016\/j.mcm.2012.06.036","volume":"57","author":"E Karapinar","year":"2013","unstructured":"Karapinar, E., Erhan, I.M., \u00d6zt\u00fcrk, A.: Fixed point theorems on quasi-partial metric spaces. Math. Comput. Model. 57(9\u201310), 2442\u20132448 (2013)","journal-title":"Math. Comput. Model."},{"issue":"2","key":"2087_CR26","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1016\/0022-247X(86)90279-9","volume":"118","author":"PQ Khanh","year":"1986","unstructured":"Khanh, P.Q.: An induction theorem and general open mapping theorems. J. Math. Anal. Appl. 118(2), 519\u2013534 (1986)","journal-title":"J. Math. Anal. Appl."},{"issue":"1\u20136","key":"2087_CR27","first-page":"33","volume":"37","author":"PQ Khanh","year":"1989","unstructured":"Khanh, P.Q.: On Caristi\u2013Kirk\u2019s theorem and Ekeland\u2019s variational principle for Pareto extrema. Bull. Pol. Acad. Sci. Math. 37(1\u20136), 33\u201339 (1989)","journal-title":"Bull. Pol. Acad. Sci. Math."},{"issue":"4","key":"2087_CR28","doi-asserted-by":"publisher","first-page":"2561","DOI":"10.1137\/140991157","volume":"25","author":"PQ Khanh","year":"2015","unstructured":"Khanh, P.Q., Kruger, A., Thao, N.H.: An induction theorem and nonlinear regularity models. SIAM J. Optim. 25(4), 2561\u20132588 (2015)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"2087_CR29","doi-asserted-by":"publisher","first-page":"951","DOI":"10.1007\/s10898-012-9983-3","volume":"57","author":"PQ Khanh","year":"2013","unstructured":"Khanh, P.Q., Quy, D.N.: Versions of Ekeland\u2019s variational principle involving set perturbations. J. Glob. Optim. 57(3), 951\u2013968 (2013)","journal-title":"J. Glob. Optim."},{"key":"2087_CR30","volume-title":"A Dynamic Theory of Personality","author":"K Lewin","year":"1935","unstructured":"Lewin, K.: A Dynamic Theory of Personality. McGraw-Hill, New York (1935)"},{"key":"2087_CR31","doi-asserted-by":"publisher","DOI":"10.1037\/10019-000","volume-title":"Principles of Topological Psychology","author":"K Lewin","year":"1936","unstructured":"Lewin, K.: Principles of Topological Psychology. McGraw-Hill, New York (1936)"},{"key":"2087_CR32","doi-asserted-by":"publisher","DOI":"10.1037\/13613-000","volume-title":"The Conceptual Representation and Measurement of Psychological Forces","author":"K Lewin","year":"1938","unstructured":"Lewin, K.: The Conceptual Representation and Measurement of Psychological Forces. Duke University Press, Durham (1938)"},{"issue":"1","key":"2087_CR33","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1137\/090760660","volume":"21","author":"CG Liu","year":"2011","unstructured":"Liu, C.G., Ng, K.F.: Ekeland\u2019s variational principle for set-valued functions. SIAM J. Optim. 21(1), 41\u201356 (2011)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"2087_CR34","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1111\/j.1749-6632.1994.tb44144.x","volume":"728","author":"SG Matthews","year":"1994","unstructured":"Matthews, S.G.: Partial metric topology. Ann. N. Y. Acad. Sci. 728(1), 183\u2013197 (1994)","journal-title":"Ann. N. Y. Acad. Sci."},{"issue":"1","key":"2087_CR35","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1016\/0304-3975(95)00051-W","volume":"151","author":"SG Matthews","year":"1995","unstructured":"Matthews, S.G.: A extensional treatment of lazy data flow deadlock. Theoret. Comput. Sci. 151(1), 195\u2013205 (1995)","journal-title":"Theoret. Comput. Sci."},{"key":"2087_CR36","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.topol.2015.12.074","volume":"203","author":"O Mohammad","year":"2016","unstructured":"Mohammad, O.: A new contribution to the fixed point theory in partial quasi-metric spaces and its applications to asymptotic complexity analysis of algorithms. Topol. Appl. 203, 42\u201356 (2016)","journal-title":"Topol. Appl."},{"key":"2087_CR37","doi-asserted-by":"publisher","first-page":"623","DOI":"10.1137\/S0363012900374816","volume":"41","author":"BS Mordukhovich","year":"2002","unstructured":"Mordukhovich, B.S., Wang, B.: Necessary suboptimality and optimality conditions via variational principles. SIAM J. Control Optim. 41, 623\u2013640 (2002)","journal-title":"SIAM J. Control Optim."},{"key":"2087_CR38","unstructured":"O\u2019Neill, S.J.: A fundamental study into the theory and application of the partial metric spaces. PhD Thesis Uni Warwick (1998)"},{"issue":"9","key":"2087_CR39","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1016\/0362-546X(86)90069-6","volume":"10","author":"JP Penot","year":"1986","unstructured":"Penot, J.P.: The drop theorem, the petal theorem and Ekeland\u2019s variational principle. Nonlinear Anal. 10(9), 813\u2013822 (1986)","journal-title":"Nonlinear Anal."},{"issue":"1","key":"2087_CR40","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0001-8708(74)90062-0","volume":"13","author":"RR Phelps","year":"1974","unstructured":"Phelps, R.R.: Support cones in Banach spaces and their applications. Adv. Math. 13(1), 1\u201319 (1974)","journal-title":"Adv. Math."},{"issue":"1","key":"2087_CR41","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1080\/02331934.2017.1387257","volume":"67","author":"JH Qiu","year":"2018","unstructured":"Qiu, J.H., He, F., Soubeyran, A.: Equilibrium versions of variational principles in quasi-metric spaces and the robust trap problem. Optimization 67(1), 25\u201353 (2018)","journal-title":"Optimization"},{"issue":"2","key":"2087_CR42","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1112\/plms\/s3-39.2.331","volume":"39","author":"RT Rockafellar","year":"1979","unstructured":"Rockafellar, R.T.: Directionally Lipschitzian functions and subdifferential calculus. Proc. Lond. Math. Soc. 39(2), 331\u2013355 (1979)","journal-title":"Proc. Lond. Math. Soc."},{"key":"2087_CR43","first-page":"1292","volume":"268","author":"N Shahzad","year":"2015","unstructured":"Shahzad, N., Valero, O., Alghhamdi, M.A., Alghamdi, M.A.: A fixed point theorem in partial quasi-metric spaces and an application to software engineering. Appl. Math. Comput. 268, 1292\u20131301 (2015)","journal-title":"Appl. Math. Comput."},{"key":"2087_CR44","first-page":"397","volume-title":"Fixed Point Theory and Applications","author":"W Takahashi","year":"1991","unstructured":"Takahashi, W.: Existence theorems generalizing fixed point theorems for multivalued mappings. In: Th\u00e9ra, M.A., Baillon, J.B. (eds.) Fixed Point Theory and Applications, pp. 397\u2013406. Longman Scientific and Technical, Essex (1991)"},{"key":"2087_CR45","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s101070100278","volume":"92","author":"Z Wu","year":"2002","unstructured":"Wu, Z., Ye, J.J.: On error bounds for lower semicontinuous functions. Math. Program. A. 92, 301\u2013314 (2002)","journal-title":"Math. Program. A."},{"issue":"3","key":"2087_CR46","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1007\/BF01078126","volume":"5","author":"PP Zabreiko","year":"1971","unstructured":"Zabreiko, P.P., Krasnoselski, M.A.: The solvability of nonlinear operator equations. Funct. Anal. Appl. 5(3), 42\u201344 (1971)","journal-title":"Funct. Anal. Appl."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02087-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-022-02087-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-022-02087-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T16:28:31Z","timestamp":1664382511000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-022-02087-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,16]]},"references-count":46,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["2087"],"URL":"https:\/\/doi.org\/10.1007\/s10957-022-02087-y","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,16]]},"assertion":[{"value":"20 December 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 July 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 September 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}