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Moreover, we present some fixed circle and fixed disc results of integral type contractive self-mappings, which generalize many results of invariance and transformations in the literature.<\/jats:p>","DOI":"10.3390\/sym12111825","type":"journal-article","created":{"date-parts":[[2020,11,5]],"date-time":"2020-11-05T00:00:37Z","timestamp":1604534437000},"page":"1825","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Fixed Circle and Fixed Disc Results for New Types of \u0398c-Contractive Mappings in Metric Spaces"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8343-4036","authenticated-orcid":false,"given":"Hayel N.","family":"Saleh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"},{"name":"Department of Mathematics, Taiz University, Taiz 6803, Yemen"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4303-2884","authenticated-orcid":false,"given":"Salvatore","family":"Sessa","sequence":"additional","affiliation":[{"name":"Dipartimento di Architettura, Universit\u00e0 degli Studi di Napoli Federico II, Via Toledo 402, 80134 Napoli, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6281-1229","authenticated-orcid":false,"given":"Waleed M.","family":"Alfaqih","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hajjah University, Hajjah 1729, Yemen"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3270-1365","authenticated-orcid":false,"given":"Mohammad","family":"Imdad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"additional","affiliation":[{"name":"Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"133","DOI":"10.4064\/fm-3-1-133-181","article-title":"Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales","volume":"3","author":"Banach","year":"1922","journal-title":"Fund. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1186\/1687-1812-2012-94","article-title":"Fixed point theory of a new type of contractive mappings in complete metric spaces","volume":"2012","author":"Wardowski","year":"2012","journal-title":"Fixed Point Theory Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1186\/1029-242X-2014-439","article-title":"Further generalizations of the Banach contraction principle","volume":"2014","author":"Jleli","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1861","DOI":"10.1090\/S0002-9939-07-09055-7","article-title":"A generalized Banach contraction principle that characterizes metric completeness","volume":"136","author":"Suzuki","year":"2008","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Imdad, M., Khan, A.R., Saleh, H.N., and Alfaqih, W.M. (2019). Some \u03c6-fixed point results for (F, \u03c6, \u03b1-\u03c8)-contractive type mappings with applications. Mathematics, 7.","DOI":"10.3390\/math7020122"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/0022-247X(69)90031-6","article-title":"A theorem on contraction mappings","volume":"28","author":"Meir","year":"1969","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"121","DOI":"10.4153\/CMB-1971-024-9","article-title":"Some remarks concerning contraction mappings","volume":"14","author":"Reich","year":"1971","journal-title":"Can. Math. Bull."},{"key":"ref_8","first-page":"19","article-title":"Generalized contractions and fixed-point theorems","volume":"12","author":"Ciric","year":"1971","journal-title":"Publ. Inst. Math."},{"key":"ref_9","first-page":"43","article-title":"Approximating fixed points of weak contractions using the picard iteration","volume":"9","author":"Berinde","year":"2004","journal-title":"Nonlinear Anal. Forum"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1007\/BF01304884","article-title":"Fix point theorems in metric spaces","volume":"23","author":"Zamfirescu","year":"1972","journal-title":"Arch. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1186\/1029-242X-2014-38","article-title":"A new generalization of the Banach contraction principle","volume":"2014","author":"Jleli","year":"2014","journal-title":"J. Inequal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1109\/TNN.2008.2005605","article-title":"The graph neural network model","volume":"20","author":"Scarselli","year":"2008","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_13","unstructured":"Mandic, D.P. (2000, January 11\u201313). The use of Mobius transformations in neural networks and signal processing. Proceedings of the 2000 IEEE Signal Processing Society Workshop (Cat. No.00TH8501), Neural Networks for Signal Processing X, Sydney, NSW, Australia."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4698","DOI":"10.1016\/j.cnsns.2011.03.005","article-title":"Complex valued neural network with m\u00f6bius activation function","volume":"16","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1007\/s40840-017-0555-z","article-title":"Some fixed circle theorems on metric spaces","volume":"42","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Aydi, H., Ta\u015f, N., \u00d6zg\u00fcr, N.Y., and Mlaiki, N. (2019). Fixed-discs in rectangular metric spaces. Symmetry, 11.","DOI":"10.3390\/sym11020294"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2794","DOI":"10.3906\/mat-1812-44","article-title":"Fixed-disc results via simulation functions","volume":"43","year":"2019","journal-title":"Turk. J. Math."},{"key":"ref_18","first-page":"15","article-title":"Fixed Point to Fixed Circle and Activation Function in Partial Metric Space","volume":"1","author":"Tomar","year":"2020","journal-title":"J. Appl. Anal."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ta\u015f, N., \u00d6zg\u00fcr, N., and Mlaiki, M. (2018). New types of Fc-contractions and the fixed circle problem. Mathematics, 6.","DOI":"10.3390\/math6100188"},{"key":"ref_20","first-page":"10","article-title":"New fixed circle results on S-metric spaces","volume":"9","author":"Tas","year":"2017","journal-title":"Bull. Math. Anal. Appl."},{"key":"ref_21","first-page":"459","article-title":"Fixed-circle problem on S-metric spaces with a geometric viewpoint","volume":"34","year":"2019","journal-title":"Facta Univ. Ser. Math. Inform."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"020048","DOI":"10.1063\/1.5020497","article-title":"Some fixed circle theorems and discontinuity at fixed circle","volume":"1926","year":"2018","journal-title":"AIP Conf. Proc."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Mlaiki, N., Ta\u015f, N., and \u00d6zg\u00fcr, N. (2018). On the fixed circle problem and Khan type contractions. Axioms, 7.","DOI":"10.3390\/axioms7040080"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s40840-018-0698-6","article-title":"On discontinuity problem at fixed point","volume":"43","author":"Pant","year":"2020","journal-title":"Bull. Malays. Math. Sci. 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