{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T09:56:53Z","timestamp":1766138213248,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T00:00:00Z","timestamp":1675036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we introduce the concept of orthogonal convex structure contraction mapping and prove some fixed point theorems on orthogonal \u266d-metric spaces. We adopt an example to highlight the utility of our main result. Finally, we apply our result to examine the existence and uniqueness of the solution for the spring-mass system via an integral equation with a numerical example.<\/jats:p>","DOI":"10.3390\/axioms12020143","type":"journal-article","created":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T08:28:23Z","timestamp":1675067303000},"page":"143","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Fixed Point Theorems via Orthogonal Convex Contraction in Orthogonal \u266d-Metric Spaces and Applications"],"prefix":"10.3390","volume":"12","author":[{"given":"Gunasekaran","family":"Nallaselli","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kanchipuram 603203, India"}]},{"given":"Amani S.","family":"Baazeem","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, IMSIU (Imam Mohammed Ibn Saud Islamic University), P.O. Box 90950, Riyadh 11623, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6346-605X","authenticated-orcid":false,"given":"Arul Joseph","family":"Gnanaprakasam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kanchipuram 603203, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4191-3338","authenticated-orcid":false,"given":"Gunaseelan","family":"Mani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, India"}]},{"given":"Khalil","family":"Javed","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, International Islamic University Islamabad, Islamabad 04436, Pakistan"}]},{"given":"Eskandar","family":"Ameer","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Taiz University, Taiz P.O. Box 6803, Yemen"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Agarwal, R.P., Meehan, M., and O\u2019regan, D. (2001). Fixed Point Theory and Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511543005"},{"key":"ref_2","first-page":"1","article-title":"Fixed point and coupled fixed point theorems on b-metric-like spaces","volume":"2013","author":"Alghamdi","year":"2013","journal-title":"J. Inequalities Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1711","DOI":"10.2478\/s11533-013-0272-2","article-title":"A note on \u201cSome results on multi-valued weakly Jungck mappings in b-metric space\u201d","volume":"11","author":"Bota","year":"2013","journal-title":"Open Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2015\/969726","article-title":"Fixed points of generalized-Suzuki type contraction in complete-metric spaces","volume":"2015","author":"Alsulami","year":"2015","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_5","first-page":"108","article-title":"Notes on some recent papers concerning F-contractions in b-metric spaces","volume":"1","author":"Kadelburg","year":"2018","journal-title":"Constr. Math. Anal."},{"key":"ref_6","first-page":"1","article-title":"A note on the results of Suzuki, Miculescu and Mihail","volume":"21","year":"2019","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Popescu, O., and Stan, G. (2020). Two fixed point theorems concerning F-contraction in complete metric spaces. Symmetry, 12.","DOI":"10.3390\/sym12010058"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s11784-020-00815-3","article-title":"Coincidence and fixed points of multivalued F-contractions in generalized metric space with application","volume":"22","author":"Saleem","year":"2020","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_9","first-page":"5","article-title":"Contraction mappings in b-metric spaces","volume":"1","author":"Czerwik","year":"1993","journal-title":"Acta Math. Inform. Univ. Ostrav."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Debnath, P., Konwar, N., and Radenovi\u0107, S. (2021). Metric Fixed Point Theory: Applications in Science, Engineering and Behavioural Sciences, Springer.","DOI":"10.1007\/978-981-16-4896-0"},{"key":"ref_11","first-page":"123","article-title":"Fixed points of generalized multi-valued contractive mappings in metric type spaces","volume":"6","author":"Latif","year":"2022","journal-title":"J. Nonlinear Var. Anal."},{"key":"ref_12","first-page":"27","article-title":"An example of a metric space with finite decomposition complexity","volume":"2022","author":"Jiang","year":"2022","journal-title":"J. Nonlinear Funct. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1687-1812-2012-106","article-title":"Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces","volume":"2012","author":"Shatanawi","year":"2012","journal-title":"Fixed Point Theory Appl."},{"key":"ref_14","first-page":"142","article-title":"A convexity in metric space and nonexpansive mappings. I","volume":"22","author":"Takahashi","year":"1970","journal-title":"Inkodai Math. Semin. Rep."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1090\/conm\/021\/729507","article-title":"Iteration processes for nonexpansive mappings","volume":"21","author":"Goebel","year":"1983","journal-title":"J. Contemp. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/0022-247X(88)90047-9","article-title":"Iteration processes for nonlinear mappings in convex metric spaces","volume":"132","author":"Ding","year":"1988","journal-title":"J. Math. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1016\/0362-546X(90)90058-O","article-title":"Nonexpansive iterations in hyperbolic spaces","volume":"19","author":"Reich","year":"1990","journal-title":"Nonlinear Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"365","DOI":"10.37193\/CJM.2015.03.13","article-title":"On the theory of fixed point theorems for convex contraction mappings","volume":"31","year":"2015","journal-title":"Carpathian J. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"315","DOI":"10.11650\/tjm.19.2015.4746","article-title":"Approximate fixed point theorems for partial generalized convex contraction mappings in \u03b1-complete metric spaces","volume":"19","author":"Latif","year":"2015","journal-title":"Taiwan. J. Math."},{"key":"ref_20","first-page":"119","article-title":"Iterated function systems consisting of generalized convex contractions in the framework of complete strong b-metric spaces","volume":"55","author":"Georgescu","year":"2017","journal-title":"Ann. West Univ. Timis. Math. Comput. Sci."},{"key":"ref_21","first-page":"29","article-title":"Fixed point theorem for Reich contraction mapping in convex b-metric spaces","volume":"40","author":"Karaca","year":"2021","journal-title":"Aligarh Bull. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1359","DOI":"10.1080\/02331934.2021.1887180","article-title":"Fixed point theorems for the Mann\u2019s iteration scheme in convex graphical rectangular b-metric spaces","volume":"70","author":"Chen","year":"2021","journal-title":"Optimization"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"67","DOI":"10.5937\/univtho9-21890","article-title":"Some new fixed point results for convex contractions in b-metric spaces","volume":"9","year":"2019","journal-title":"Univ. Thought Publ. Nat. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6929","DOI":"10.3934\/math.2020444","article-title":"3D analysis of modified F-contractions in convex b-metric spaces with application to Fredholm integral equations","volume":"5","author":"Asif","year":"2020","journal-title":"AIMS Math."},{"key":"ref_25","first-page":"11","article-title":"Approximating fixed points of almost convex contractions in metric spaces","volume":"12","author":"Berinde","year":"2020","journal-title":"Ann. Acad. Rom. Sci. Ser. Math. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s12215-018-0386-2","article-title":"Fixed points of convex and generalized convex contractions","volume":"69","author":"Bisht","year":"2020","journal-title":"Rend. Circ. Mat. Palermo Ser."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Chen, L., Li, C., Kaczmarek, R., and Zhao, Y. (2020). Several fixed point theorems in convex b-metric spaces and applications. Mathematics, 8.","DOI":"10.3390\/math8020242"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2902","DOI":"10.22436\/jnsa.010.06.05","article-title":"On generalized convex contractions of type-2 in b-metric and 2-metric spaces","volume":"10","author":"Khan","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Nallaselli, G., Gnanaprakasam, A.J., Mani, G., Mitrovi\u0107, Z.D., Aloqaily, A., and Mlaiki, N. (2023). Integral equation via fixed point theorems on a new type of convex contraction in b-metric and 2-metric spaces. Mathematics, 11.","DOI":"10.3390\/math11020344"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"569","DOI":"10.24193\/fpt-ro.2017.2.45","article-title":"On orthogonal sets and Banach fixed point theorem","volume":"18","author":"Gordji","year":"2017","journal-title":"Fixed Point Theory"},{"key":"ref_31","first-page":"251","article-title":"Fixed point theory in generalized orthogonal metric space","volume":"6","author":"Gordji","year":"2017","journal-title":"J. Linear Topol. Algebra"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11784-019-0737-4","article-title":"Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces","volume":"22","author":"Sawangsup","year":"2020","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1859","DOI":"10.3390\/sym14091859","article-title":"Common fixed-points technique for the existence of a solution to fractional integro-differential equations via orthogonal Branciari metric spaces","volume":"14","author":"Gunasekaran","year":"2022","journal-title":"Symmetry"},{"key":"ref_34","first-page":"1","article-title":"Fixed point of orthogonal F-Suzuki contraction mapping on O-complete b-metric spaces with applications","volume":"2021","author":"Beg","year":"2021","journal-title":"J. Funct. 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