{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T16:32:38Z","timestamp":1768840358252,"version":"3.49.0"},"reference-count":31,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,7]],"date-time":"2022-11-07T00:00:00Z","timestamp":1667779200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific Research Deanship at the University of Ha\u2019il\u2014Saudi Arabia","award":["RG-21 159"],"award-info":[{"award-number":["RG-21 159"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many techniques have been recently used by various researchers to solve some types of symmetrical fractional differential equations. In this article, we show the existence and uniqueness to the solution of \u03c2-Caputo stochastic fractional differential equations (CSFDE) using the Banach fixed point technique (BFPT). We analyze the Hyers\u2013Ulam stability of CSFDE using the stochastic calculus techniques. We illustrate our results with three examples.<\/jats:p>","DOI":"10.3390\/sym14112336","type":"journal-article","created":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T10:54:40Z","timestamp":1667904880000},"page":"2336","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Some Existence and Uniqueness Results for a Class of Fractional Stochastic Differential Equations"],"prefix":"10.3390","volume":"14","author":[{"given":"Omar","family":"Kahouli","sequence":"first","affiliation":[{"name":"Department of Electronics Engineering, Community College, University of Ha\u2019il, Ha\u2019il P.O. Box 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7142-7026","authenticated-orcid":false,"given":"Abdellatif","family":"Ben Makhlouf","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lassaad","family":"Mchiri","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Sciences of Sfax, University of Sfax, Route Soukra, BP 1171, Sfax 3000, Tunisia"},{"name":"ENSIIE, University of Evry-Val-d\u2019Essonne, 1 Square de la R\u00e9sistance, CEDEX, 91025 \u00c9vry-Courcouronnes, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7755-2837","authenticated-orcid":false,"given":"Pushpendra","family":"Kumar","sequence":"additional","affiliation":[{"name":"Institute for the Future of Knowledge, University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6739-679X","authenticated-orcid":false,"given":"Naim","family":"Ben Ali","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, College of Engineering, University of Ha\u2019il, Ha\u2019il P.O. Box 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ali","family":"Aloui","sequence":"additional","affiliation":[{"name":"Department of Electronics Engineering, Community College, University of Ha\u2019il, Ha\u2019il P.O. Box 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,7]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_2","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Machado, J.A., and Luo, A.C. (2011). Fractional Dynamics and Control, Springer Science and Business Media.","DOI":"10.1007\/978-1-4614-0457-6"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1115\/1.3167616","article-title":"Applications of fractional calculus to the theory of viscoelasticity","volume":"51","author":"Koeller","year":"1984","journal-title":"ASME J. Appl. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Li, C.P., and Zeng, F.H. (2015). Numerical Methods for Fractional Calculus, Chapman and Hall\/CRC Press.","DOI":"10.1201\/b18503"},{"key":"ref_6","first-page":"7667370","article-title":"Existence and Uniqueness of the Solution for an Inverse Problem of a Fractional Diffusion Equation with Integral Condition","volume":"2020","author":"Oussaeif","year":"2020","journal-title":"J. Funct. Spaces"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2581","DOI":"10.1108\/EC-07-2021-0393","article-title":"Existence and uniqueness of solutions for generalized Sturm\u2013Liouville and Langevin equations via Caputo\u2013Hadamard fractional-order operator","volume":"39","author":"Batiha","year":"2022","journal-title":"Eng. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.neucom.2012.11.034","article-title":"Dynamic analysis of a class of fractional-order neural networks with delay","volume":"111","author":"Chen","year":"2013","journal-title":"Neurocomputing"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.chemolab.2015.03.009","article-title":"A method of approximate fractional order differentiation with noise immunity","volume":"144","author":"Li","year":"2015","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1016\/j.camwa.2009.07.050","article-title":"On the fractional Adams method","volume":"58","author":"Li","year":"2009","journal-title":"Comput. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1002\/mma.4617","article-title":"Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications","volume":"41","author":"Almeida","year":"2018","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_12","first-page":"1","article-title":"Caputo-type modification of the Hadamard fractional derivatives","volume":"142","author":"Jarad","year":"2012","journal-title":"Adv. Differ. Equ."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s12044-019-0514-8","article-title":"Fractional boundary value problem with \u03c8-Caputo fractional derivative","volume":"129","author":"Abdo","year":"2019","journal-title":"Proc. Indian Acad. Sci. Math. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2459","DOI":"10.1216\/RMJ-2019-49-8-2459","article-title":"Further properties of Osler\u2019s generalized fractional integrals and derivatives with respect to another function","volume":"49","author":"Almeida","year":"2019","journal-title":"Rocky Mt. J. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1873","DOI":"10.1007\/s13398-018-0590-0","article-title":"A numerical study of fractional relaxation-oscillation equations involving \u03c8-Caputo fractional derivative","volume":"113","author":"Almeida","year":"2019","journal-title":"Rev. R. Acad. Cienc. Exactas F\u00eds Nat. Ser. Mat."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4866","DOI":"10.1080\/00036811.2021.1873300","article-title":"Ulam stability for nonlinear-Langevin fractional differential equations involving two fractional orders in the \u03c8-Caputo sense","volume":"101","author":"Baitiche","year":"2022","journal-title":"Appl. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Derbazi, C., Baitiche, Z., Benchohra, M., and Cabada, A. (2020). Initial value problem for nonlinear fractional differential equations with \u03c8-Caputo derivative via monotone iterative technique. Axioms, 9.","DOI":"10.3390\/axioms9020057"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.chaos.2017.03.010","article-title":"A survey on Hadamard and Hilfer fractional differential equations: Analysis and stability","volume":"102","author":"Abbas","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","first-page":"1","article-title":"The existence and Hyers\u2013Ulam stability of solution for an impulsive Riemann\u2013Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1 < \u03b2 < 2","volume":"59","author":"Guo","year":"2019","journal-title":"Bound. Value Probl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1073\/pnas.27.4.222","article-title":"On the stability of the linear functional equation","volume":"27","author":"Hyers","year":"1941","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1080\/07362994.2020.1824677","article-title":"The existence and Hyers-Ulam stability of solution for almost periodical fractional stochastic differential equation with fBm","volume":"39","author":"Guo","year":"2021","journal-title":"Stoch. Anal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1080\/17442508.2018.1551400","article-title":"Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays","volume":"91","author":"Li","year":"2019","journal-title":"Stochastics"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1007\/s40306-017-0207-2","article-title":"Ulam stability for fractional partial integrodi differential equation with uncertainty","volume":"42","author":"Long","year":"2017","journal-title":"Acta Math. Vietnam."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5937\/MatMor2101001D","article-title":"Existence, uniqueness, approximation of solutions and Ealpha-Ulam stability results for a class of nonlinear fractional differential equations involving psi-Caputo derivative with initial conditions","volume":"25","author":"Derbazi","year":"2021","journal-title":"Math. Moravica"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2380","DOI":"10.3906\/mat-2010-9","article-title":"Existence results for \u03c8-Caputo fractional neutral functional integro-differential equations with finite delay","volume":"44","author":"Boutiara","year":"2020","journal-title":"Turk. J. Math."},{"key":"ref_26","first-page":"887","article-title":"Uniqueness and Ulam\u2013Hyers\u2013Mittag\u2013Leffler stability results for the delayed fractional multiterm differential equation involving the \u03d5-Caputo fractional derivative","volume":"52","author":"Derbazi","year":"2020","journal-title":"Rocky Mt. J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"108949","DOI":"10.1016\/j.spl.2020.108949","article-title":"Ulam-Hyers stability of Caputo type fractional stochastic neutral differential equations","volume":"168","author":"Ahmadova","year":"2021","journal-title":"Stat. Probab. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"111757","DOI":"10.1016\/j.chaos.2021.111757","article-title":"Some results on the study of Caputo-Hadamard fractional stochastic differential equations","volume":"155","author":"Mchiri","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_29","first-page":"1","article-title":"Asymptotic separation between solutions of Caputo fractional stochastic differential equations","volume":"36","author":"Doan","year":"2018","journal-title":"Stoch. Anal. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1090\/S0002-9904-1968-11933-0","article-title":"A fixed point theorem of the alternative, for contractions on a generalized complete metric space","volume":"74","author":"Diaz","year":"1968","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_31","unstructured":"Vanterler da Sousa, J., and Capelas de Oliveira, E. (2017). A Gronwall inequality and the Cauchy-type problem by means of \u03c8-Hilfer operator. arXiv."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2336\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:11:43Z","timestamp":1760145103000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/11\/2336"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,7]]},"references-count":31,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["sym14112336"],"URL":"https:\/\/doi.org\/10.3390\/sym14112336","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,7]]}}}