{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T11:54:18Z","timestamp":1767182058787,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T00:00:00Z","timestamp":1611100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The purpose of this paper is to prove fixed point theorems for cyclic-type operators in extended b-metric spaces. The well-posedness of the fixed point problem and limit shadowing property are also discussed. Some examples are given in order to support our results, and the last part of the paper considers some applications of the main results. The first part of this section is devoted to the study of the existence of a solution to the boundary value problem. In the second part of this section, we study the existence of solutions to fractional boundary value problems with integral-type boundary conditions in the frame of some Caputo-type fractional operators.<\/jats:p>","DOI":"10.3390\/sym13020158","type":"journal-article","created":{"date-parts":[[2021,1,21]],"date-time":"2021-01-21T02:36:05Z","timestamp":1611196565000},"page":"158","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Existence of the Solutions of Nonlinear Fractional Differential Equations Using the Fixed Point Technique in Extended b-Metric Spaces"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8304-1574","authenticated-orcid":false,"given":"Liliana","family":"Guran","sequence":"first","affiliation":[{"name":"Department of Pharmaceutical Sciences, \u201cVasile Goldi\u015f\u201d Western University of Arad, L. Rebreanu Street, No. 86, 310048 Arad, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5513-0801","authenticated-orcid":false,"given":"Monica-Felicia","family":"Bota","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Babe\u015f-Bolyai University, Kog\u0103lniceanu Street No. 1, 400084 Cluj-Napoca, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les operations dans les ensembles abstraits et leur applications aux equations integrales","volume":"3","author":"Banach","year":"1922","journal-title":"Fundam. Math."},{"key":"ref_2","first-page":"79","article-title":"Fixed points for mappings satisfying cyclical contractive conditions","volume":"4","author":"Kirk","year":"2003","journal-title":"Fixed Point Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1016\/j.jmaa.2005.10.081","article-title":"Existence and convergence of best proximity points","volume":"323","author":"Eldred","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_4","first-page":"5","article-title":"Fixed point theorems for cyclic contractions in b-metric spaces","volume":"2015","author":"George","year":"2015","journal-title":"J. Nonlinear Funct. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"867216","DOI":"10.1155\/2012\/867216","article-title":"Fixed point theorems for various classes of cyclic mappings","volume":"2012","author":"Aydi","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"931","DOI":"10.18514\/MMN.2017.1966","article-title":"A Perov type theorem for cyclic contractions and applications to systems of integral equations","volume":"17","author":"Magdas","year":"2017","journal-title":"Miskolc Math. Notes"},{"key":"ref_7","first-page":"1637","article-title":"Fixed points, coupled fixed points and best proximity points for cyclic operators","volume":"20","author":"Petrusel","year":"2019","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6160","DOI":"10.1016\/j.na.2012.06.021","article-title":"Cyclic generalized \u03a8-weakly contractive mappings and fixed point results with applications to integral equations","volume":"75","author":"Nashine","year":"2012","journal-title":"Nonlinear Anal. Methods Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1016\/j.neucom.2016.08.025","article-title":"A Combinatorial Necessary and Sufficient Condition for Cluster Consensus","volume":"216","author":"Shang","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_10","first-page":"26","article-title":"The contraction mapping principle in quasimetric spaces","volume":"30","author":"Bakhtin","year":"1989","journal-title":"Funct. Anal. Unianowsk Gos. Ped. Inst."},{"key":"ref_11","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_12","first-page":"337","article-title":"A common fixed point for weak \u03d5-contractions on b-metric spaces","volume":"13","author":"Aydi","year":"2012","journal-title":"Fixed Point Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"321","DOI":"10.24193\/fpt-ro.2018.1.25","article-title":"Fixed point theorems for various type of F-contractions in complete b-metric spaces","volume":"19","author":"Lukacs","year":"2018","journal-title":"Fixed Point Theory"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Saleem, N., Vujakovr\u00edc, J., Baloch, W.U., and Radenovi\u0107, S. (2019). Coincidence point results for multivalued Suzuki type mappings using \u03b8-contraction in b-metric spaces. Mathematics, 7.","DOI":"10.3390\/math7111017"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.1007\/s11784-016-0400-2","article-title":"New fixed point theorems for set-valued contractions in b-metric spaces","volume":"19","author":"Miculescu","year":"2017","journal-title":"J. Fixed Point Theory Appl."},{"key":"ref_16","first-page":"101","article-title":"Coincience points and common fixed points for expansive type mappings in b-metric spaces","volume":"11","author":"Mohanta","year":"2016","journal-title":"Iran J. Math. Sci. Inform."},{"key":"ref_17","first-page":"263","article-title":"Nonlinear set-valued contraction mappings in b-metric spaces","volume":"46","author":"Czerwik","year":"1998","journal-title":"Atti Sem. Mat. Univ. Modena"},{"key":"ref_18","first-page":"3","article-title":"Generalized contractions in quasimetric spaces","volume":"3","author":"Berinde","year":"1993","journal-title":"Semin. Fixed Point Theory"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Abdeljawad, T., Agarwal, R.P., Karap\u0131nar, E., and Kumari, P.S. (2019). Solutions of the Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space. Symmetry, 11.","DOI":"10.3390\/sym11050686"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Kamran, T., Samreen, M., and Ain, Q.U.L. (2017). A generalization of b-metric space and some fixed point theorems. Mathematics, 5.","DOI":"10.3390\/math5020019"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s11083-005-9018-5","article-title":"Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations","volume":"22","author":"Nieto","year":"2005","journal-title":"Order"},{"key":"ref_22","unstructured":"Ladde, G.S., Lakshmikantham, V., and Vatsala, A.S. (1985). Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman."},{"key":"ref_23","first-page":"1","article-title":"On new evolution of Ri\u2019s result via w-distances and the study on the solution for nonlinear integral equations and fractional differential equations","volume":"132","author":"Aydi","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_24","unstructured":"Kilbas, A.A., Srivastava, H.H., and Trujillo, J.J. (2006). Theory and applications of fractional differential equations. North-Holland Mathematics Studies, Elsevier."},{"key":"ref_25","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integral and Derivative, Gordon Breach."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"20120144","DOI":"10.1098\/rsta.2012.0144","article-title":"Some existence results on nonlinear fractional differential equations","volume":"371","author":"Baleanu","year":"2013","journal-title":"Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/158\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:13:00Z","timestamp":1760159580000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/158"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,20]]},"references-count":26,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["sym13020158"],"URL":"https:\/\/doi.org\/10.3390\/sym13020158","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,1,20]]}}}