{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:33:53Z","timestamp":1760060033818,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,1]],"date-time":"2025-08-01T00:00:00Z","timestamp":1754006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11871302","71773067","ZR2022MA009","ZR2023MA011"],"award-info":[{"award-number":["11871302","71773067","ZR2022MA009","ZR2023MA011"]}]},{"name":"Natural Science Foundation of Shandong Province of China","award":["11871302","71773067","ZR2022MA009","ZR2023MA011"],"award-info":[{"award-number":["11871302","71773067","ZR2022MA009","ZR2023MA011"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Since the monotonicity of the best approximant is crucial to establish partial ordering methods, in this paper, we, respectively, characterize the best approximants in Banach function spaces and Lorentz spaces \u0393p,w, in which we especially focus on the monotonicity characterizations. We first study monotonicity characterizations of the metric projection operator onto sublattices in general Banach function spaces by the property Hg. The sufficient and necessary conditions for monotonicity of the metric projection onto cones and sublattices are then, respectively, established in \u0393p,w. The Lorentz spaces \u0393p,w are also shown to be reflexive under the condition RBp, which is the basis for the existence of the best approximant. As applications, by establishing the partial ordering methods based on the obtained monotonicity characterizations, the solvability and approximation theorems for best proximity points are deduced without imposing any contractive and compact conditions in \u0393p,w. Our results extend and improve many previous results in the field of the approximation and partial ordering theory.<\/jats:p>","DOI":"10.3390\/axioms14080600","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T08:46:55Z","timestamp":1754383615000},"page":"600","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Characterization of the Best Approximation and Establishment of the Best Proximity Point Theorems in Lorentz Spaces"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8399-7763","authenticated-orcid":false,"given":"Dezhou","family":"Kong","sequence":"first","affiliation":[{"name":"College of Information Science and Engineering, Shandong Agricultural University, Tai\u2019an 271018, China"}]},{"given":"Zhihao","family":"Xu","sequence":"additional","affiliation":[{"name":"College of Information Science and Engineering, Shandong Agricultural University, Tai\u2019an 271018, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5993-114X","authenticated-orcid":false,"given":"Yun","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Information Science and Engineering, Shandong Agricultural University, Tai\u2019an 271018, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3589-4898","authenticated-orcid":false,"given":"Li","family":"Sun","sequence":"additional","affiliation":[{"name":"College of Information Science and Engineering, Shandong Agricultural University, Tai\u2019an 271018, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1007\/BF01195027","article-title":"Monotonicity of metric projections onto positive cones of ordered Euclidean spaces","volume":"46","author":"Isac","year":"1986","journal-title":"Arch. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1287\/moor.1120.0553","article-title":"Solvability of variational inequalities on Hilbert lattices","volume":"37","author":"Nishimura","year":"2012","journal-title":"Math. Oper. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.na.2013.11.013","article-title":"A duality between the metric projection onto a convex cone and the metric projection onto its dual in Hilbert spaces","volume":"97","year":"2014","journal-title":"Nonlinear Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2815","DOI":"10.1016\/j.laa.2013.08.032","article-title":"Lattice-like operations and isotone projection sets","volume":"439","year":"2013","journal-title":"Linear Algebra Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1157","DOI":"10.1016\/j.jmaa.2011.11.009","article-title":"Optimal solutions to variational inequalities on Banach lattices","volume":"388","author":"Li","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1080\/02331934.2018.1445741","article-title":"Isotone cones in Banach spaces and applications to best approximations of operators without continuity conditions","volume":"67","author":"Li","year":"2018","journal-title":"Optimization"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"391952","DOI":"10.1155\/2014\/391952","article-title":"Generalized contraction and invariant approximation results on nonconvex subsets of normed spaces","volume":"2014","author":"Abbas","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/0021-9045(86)90048-1","article-title":"Best monotone approximations in L1[0, 1]","volume":"47","author":"Huotari","year":"1986","journal-title":"J. Approx. Theory"},{"key":"ref_9","first-page":"177","article-title":"L\u03d5-approximation by nondecreasing function on the interval","volume":"13","author":"Marano","year":"1997","journal-title":"Constr. Approx."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/BF00536190","article-title":"Best approximants in L\u03d5 spaces","volume":"51","author":"Landers","year":"1980","journal-title":"Z. Wahrsch. Verw. Gabiete"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"29","DOI":"10.14321\/realanalexch.30.1.0029","article-title":"A Lebesgue type differentiation theorem for best approximations by constants in Orlicz spaces","volume":"30","author":"Favier","year":"2004","journal-title":"Real Anal. Exch."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jat.2015.12.001","article-title":"Characterizing best isotone approximations in Lp spaces, 1 \u2264 p < \u221e","volume":"203","author":"Deutsch","year":"2016","journal-title":"J. Approx. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Dong, J., and Cui, Y. (2024). Monotonicities of quasi-normed Orlicz spaces. Axioms, 13.","DOI":"10.3390\/axioms13100696"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Liu, Y., Xue, Y., and Cui, Y. (2024). Lower local uniform monotonicity in F-normed Musielak\u2013Orlicz spaces. Axioms, 13.","DOI":"10.3390\/axioms13040243"},{"key":"ref_15","unstructured":"Luxemburg, W.A.J. (1955). Banach Function Spaces. [Ph.D. Thesis, Technische Hogeschool te Delft]."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1007\/s12220-024-01673-y","article-title":"On the properties of quasi-Banach function spaces","volume":"34","author":"Nekvinda","year":"2024","journal-title":"J. Geom. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"411","DOI":"10.2140\/pjm.1951.1.411","article-title":"On the theory of spaces \u039b","volume":"1","author":"Lorentz","year":"1951","journal-title":"Pac. J. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"113","DOI":"10.4064\/sm-24-2-113-190","article-title":"Intermiediate spaces and interpolation, the complex method","volume":"24","year":"1964","journal-title":"Stud. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.jat.2004.10.001","article-title":"Best constant approximants in Lorentz spaces","volume":"131","author":"Levis","year":"2004","journal-title":"J. Approx. Theory"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1518","DOI":"10.1016\/j.jat.2010.04.002","article-title":"The best constant approximant operators in Lorentz spaces \u0393p,w and their applications","volume":"162","author":"Ciesielski","year":"2010","journal-title":"J. Approx. Theory"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1080\/01630563.2019.1641115","article-title":"Characterization of the Cone and Applications in Banach Spaces","volume":"40","author":"Kong","year":"2019","journal-title":"Numer. Func. Anal. Optim."},{"key":"ref_22","first-page":"59","article-title":"Best constant approximants in Orlicz-Lorentz spaces","volume":"48","author":"Levis","year":"2008","journal-title":"Comment. Math. Prace Mat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/j.jat.2009.04.005","article-title":"Weak inequalities for maximal functions in Orlicz-Lorentz spaces and applications","volume":"162","author":"Levis","year":"2010","journal-title":"J. Approx. Theory"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Sababe, S.H., and Nikoufar, I. (2025). Weighted Lorentz spaces, variable exponent analysis, and operator extensions. Axioms, 14.","DOI":"10.3390\/axioms14080562"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/0021-9045(92)90141-A","article-title":"Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation","volume":"69","author":"Kurc","year":"1992","journal-title":"J. Approx. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2713","DOI":"10.1016\/j.na.2011.11.011","article-title":"Monotonicity and rotundity of Lorentz spaces \u0393p,w","volume":"75","author":"Ciesielski","year":"2012","journal-title":"Nonlinear Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"999","DOI":"10.1007\/s11117-016-0397-1","article-title":"Strong extensions for q-summing operators actingin p-convex Banach function spaces for 1 \u2264 p \u2264 q","volume":"20","author":"Delgado","year":"2016","journal-title":"Positivity"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.jmaa.2015.01.064","article-title":"Local approach to Kadec-Klee properties in symmetric function spaces","volume":"426","author":"Ciesielski","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Rynne, B.P., and Youngson, M.A. (2008). Linear Functional Analysis, Spinger. [2nd ed.].","DOI":"10.1007\/978-1-84800-005-6"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.insmatheco.2017.10.006","article-title":"Optimal reinsurance under risk and uncertainty on Orlicz hearts","volume":"81","author":"Kong","year":"2018","journal-title":"Insur. Math. Econom."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/s10957-017-1162-8","article-title":"Isotonicity of the metric projection and complementarity problems in Hilbert spaces","volume":"175","author":"Kong","year":"2017","journal-title":"J. Optim. Theory Appl."},{"key":"ref_32","unstructured":"Birkhoff, G. (1967). Lattice Theory, American Mathematical Society."},{"key":"ref_33","unstructured":"Guo, D., Cho, Y., and Zhu, J. (2004). Partial Ordering Methods in Nonlinear Problems, Nova Science Publishers Inc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1216\/rmjm\/1021477253","article-title":"Monotonicity and rotundity properties in Banach lattices","volume":"30","author":"Hudzik","year":"2000","journal-title":"Rocky Mountain J. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s10957-017-1084-5","article-title":"Isotonicity of the metric projection by Lorentz cone and variational inequalities","volume":"173","author":"Kong","year":"2017","journal-title":"J. Optim. Theory Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1007\/s11117-016-0430-4","article-title":"Banach envelopes in symmetric spaces of measurable operators","volume":"21","year":"2017","journal-title":"Positivity"},{"key":"ref_37","unstructured":"Bennett, C., and Sharpley, R. (1988). Interpolation of Operators. Pure and Applied Mathematics, Academic Press Inc."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1889","DOI":"10.1007\/s11784-016-0337-5","article-title":"Isotonicity of the metric projection with applications to variational inequalities and fixed point theory in Banach spaces","volume":"19","author":"Kong","year":"2017","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_39","unstructured":"Krein, S.G., Petunin, Y.I., and Semenov, E.M. (1978). Interpolation of Linear Operators, Nauka. (In Russian)."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/BF02786637","article-title":"On Lorentz spaces \u0393p,w","volume":"140","author":"Maligranda","year":"2004","journal-title":"Israel J. Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1242","DOI":"10.1002\/mana.200810798","article-title":"G\u00e2teaux derivatives and their applications to approximation in Lorentz spaces","volume":"282","author":"Ciesielski","year":"2009","journal-title":"Math. Nachr."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.jmaa.2015.04.040","article-title":"On geometric structure of symmetric spaces","volume":"430","author":"Ciesielski","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"145","DOI":"10.4064\/sm-96-2-145-158","article-title":"Boundedness of classical operators on classical Lorentz spaces","volume":"96","author":"Sawyer","year":"1990","journal-title":"Studia Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2581","DOI":"10.1007\/s11425-015-5020-6","article-title":"The best approximation theorems and variational inequalities for discontinuous mappings in Banach spaces","volume":"58","author":"Liu","year":"2015","journal-title":"Sci. China Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1081\/NFA-120026380","article-title":"Proximinal retracts and best proximity pair theorems","volume":"24","author":"Kirk","year":"2003","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1006\/jath.1999.3415","article-title":"Best proximity pair theorems for multifunctions with open fibres","volume":"103","author":"Basha","year":"2000","journal-title":"J. Approx. Theory"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/600\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:21:28Z","timestamp":1760034088000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/8\/600"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,1]]},"references-count":46,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["axioms14080600"],"URL":"https:\/\/doi.org\/10.3390\/axioms14080600","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,8,1]]}}}