{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:27:28Z","timestamp":1760146048651,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T00:00:00Z","timestamp":1727740800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Applied Technology College of Soochow University"},{"name":"Taizhou University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we delve into delayed fractional differential equations with Riemann\u2013Stieltjes integral boundary conditions and fractional impulses. By using differential inequality techniques and some fixed-point theorems, some novel sufficient assessments for convenient verification have been devised to ensure the existence and uniqueness of solutions. We further employ the nonlinear analysis to reveal that this problem is Ulam\u2013Hyers (UH) stable. Finally, some examples and numerical simulations are presented to illustrate the reliability and validity of our main results.<\/jats:p>","DOI":"10.3390\/axioms13100682","type":"journal-article","created":{"date-parts":[[2024,10,1]],"date-time":"2024-10-01T11:08:47Z","timestamp":1727780927000},"page":"682","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Ulam\u2013Hyers Stability and Simulation of a Delayed Fractional Differential Equation with Riemann\u2013Stieltjes Integral Boundary Conditions and Fractional Impulses"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4248-1442","authenticated-orcid":false,"given":"Xiaojun","family":"Lv","sequence":"first","affiliation":[{"name":"Applied Technology College, Soochow University, Suzhou 215325, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2236-3016","authenticated-orcid":false,"given":"Kaihong","family":"Zhao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Electronics & Information Engineering, Taizhou University, Taizhou 318000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haiping","family":"Xie","sequence":"additional","affiliation":[{"name":"Applied Technology College, Soochow University, Suzhou 215325, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,1]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel","volume":"7","author":"Atangana","year":"2015","journal-title":"Adv. Mech. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1186\/s13662-020-2527-0","article-title":"Analyzing transient response of the parallel RCL circuit by using the Caputo-Fabrizio fractional derivative","volume":"2020","author":"Alizadeh","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Jajarmi, A., Mohammadi, H., and Rezapour, S. (2020). A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Soliton. Fract., 134.","DOI":"10.1016\/j.chaos.2020.109705"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Rahman, M., Ahmad, S., Matoog, R., Alshehri, N., and Khan, T. (2021). Study on the mathematical modelling of COVID-19 with Caputo-Fabrizio operator. Chaos Soliton. Fract., 150.","DOI":"10.1016\/j.chaos.2021.111121"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.ecolmodel.2015.06.016","article-title":"Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton system","volume":"318","author":"Javidi","year":"2015","journal-title":"Ecol. Model."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Chatterjee, A., and Ahmad, B. (2021). A fractional-order differential equation model of COVID-19 infection of epithelial cells. Chaos Soliton. Fract., 147.","DOI":"10.1016\/j.chaos.2021.110952"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Zhao, K. (2022). Stability of a nonlinear ML-nonsingular kernel fractional Langevin system with distributed lags and integral control. Axioms, 11.","DOI":"10.3390\/axioms11070350"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Zhao, K. (2022). Stability of a nonlinear Langevin system of ML-type fractional derivative affected by time-varying delays and differential feedback control. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6120725"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1186\/s13662-020-02618-9","article-title":"Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions","volume":"2020","author":"Li","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1263","DOI":"10.1016\/j.mcm.2011.10.006","article-title":"Multiple positive solutions of a singular fractional differential equation with negatively perturbed term","volume":"55","author":"Zhang","year":"2012","journal-title":"Math. Comput. Model."},{"key":"ref_11","first-page":"6950","article-title":"Positive solutions for boundary value problems of nonlinear fractional differential equations","volume":"217","author":"Zhao","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1300","DOI":"10.1016\/j.camwa.2009.06.034","article-title":"Positive solutions to singular boundary value problem for nonlinear fractional differential equation","volume":"59","author":"Zhang","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6434","DOI":"10.1016\/j.na.2011.06.026","article-title":"Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity","volume":"74","author":"Wang","year":"2011","journal-title":"Nonlinear Anal.-Theor."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1515\/ijnsns-2017-0009","article-title":"Solvability of anti-periodic BVPs for impulsive fractional differential systems involving Caputo and Riemann-Liouville fractional derivatives","volume":"19","author":"Liu","year":"2018","journal-title":"Int. J. Nonlinear Sci. Numer. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1016\/j.na.2009.07.012","article-title":"The positive properties of the green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application","volume":"72","author":"Jiang","year":"2010","journal-title":"Nonlinear Anal. TMA"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2094","DOI":"10.22436\/jnsa.010.04.63","article-title":"Positive properties of the green function for two-term fractional differential equations and its application","volume":"10","author":"Wang","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.mcm.2011.07.037","article-title":"Impulsive fractional differential equations with nonlinear boundary conditions","volume":"55","author":"Cao","year":"2012","journal-title":"Math. Comput. Model."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.jmaa.2011.05.082","article-title":"Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative","volume":"384","author":"Bai","year":"2011","journal-title":"J. Math. Anal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1016\/j.na.2010.09.030","article-title":"Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order","volume":"74","author":"Wang","year":"2011","journal-title":"Nonlinear Anal.-Theor."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.2298\/FIL2304053Z","article-title":"Existence and UH-stability of integral boundary problem for a class of nonlinear higher-order Hadamard fractional Langevin equation via Mittag-Leffler functions","volume":"37","author":"Zhao","year":"2023","journal-title":"Filomat"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.aml.2018.04.024","article-title":"Fractional differential equations involving generalized derivative with Stieltjes and fractional integral boundary conditions","volume":"84","author":"Ahmad","year":"2018","journal-title":"Appl. Math. Lett."},{"key":"ref_22","unstructured":"Ulam, S. (1906). A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathmatics, Interscience."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2222","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_24","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/BF01831117","article-title":"Hyers-Ulam stability of functional equations in several variables","volume":"50","author":"Forti","year":"1995","journal-title":"Aeq. Math."},{"key":"ref_25","unstructured":"Brzdek, J., Popa, D., Ra\u015fa, I., and Xu, B. (2018). Ulam Stability of Operators, Academic Press."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Hyers, D., Isac, G., and Rassias, T. (1998). Stability of Functional Equations in Several Variables, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-1790-9"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Czerwik, S. (2002). Functional Equations and Inequalities in Several Variables, World Scientific.","DOI":"10.1142\/9789812778116"},{"key":"ref_28","unstructured":"Aderyani, S., Saadati, R., Li, C., and Allahviranloo, T. (2024). Towards Ulam Type Multi Stability Analysis, Springer."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Zhao, K. (2023). Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with p-Laplacian. Axioms, 12.","DOI":"10.3390\/axioms12080733"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Zhao, K., Liu, J., and Lv, X. (2024). A Unified Approach to Solvability and Stability of Multipoint BVPs for Langevin and Sturm-Liouville Equations with CH-Fractional Derivatives and Impulses via Coincidence Theory. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8020111"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.jmaa.2013.02.034","article-title":"Laplace transform and Hyers-Ulam stability of linear differential equations","volume":"403","author":"Rezaei","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Lv, X., Zhao, K., and Xie, H. (2024). Stability and Numerical Simulation of a Nonlinear Hadamard Fractional Coupling Laplacian System with Symmetric Periodic Boundary Conditions. Symmetry, 16.","DOI":"10.3390\/sym16060774"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1186\/s13662-024-03801-y","article-title":"Study on the stability and its simulation algorithm of a nonlinear impulsive ABC-fractional coupled system with a Laplacian operator via F-contractive mapping","volume":"2024","author":"Zhao","year":"2024","journal-title":"Adv. Contin. Discret. Model."},{"key":"ref_34","first-page":"383","article-title":"Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives","volume":"60","author":"Wang","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_35","first-page":"72","article-title":"Ulam-Hyers stability of fractional Langevin equations","volume":"258","author":"Wang","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Zhao, K. (2022). Existence, stability and simulation of a class of nonlinear fractional Langevin equations involving nonsingular Mittag-Leffler kernel. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6090469"},{"key":"ref_37","unstructured":"Podlubny, I. (1993). Fractional Differential Equations, Academic Press."},{"key":"ref_38","unstructured":"Klbas, A., Srivastava, H., and Trujillo, J. (2006). Theory and Applications of Fractional Differential Equation, Elsevier."},{"key":"ref_39","unstructured":"Guo, D., and Lakshmikantham, V. (1988). Nonlinear Problems in Abstract Cone, Academic Press."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/682\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:08:50Z","timestamp":1760112530000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/682"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,1]]},"references-count":39,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["axioms13100682"],"URL":"https:\/\/doi.org\/10.3390\/axioms13100682","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,10,1]]}}}