{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T00:58:24Z","timestamp":1776473904302,"version":"3.51.2"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,8,16]],"date-time":"2021-08-16T00:00:00Z","timestamp":1629072000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,8,16]],"date-time":"2021-08-16T00:00:00Z","timestamp":1629072000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1007\/s40314-021-01595-3","type":"journal-article","created":{"date-parts":[[2021,8,16]],"date-time":"2021-08-16T13:02:57Z","timestamp":1629118977000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":84,"title":["Qualitative analysis of caputo fractional integro-differential equations with constant delays"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8310-0266","authenticated-orcid":false,"given":"Martin","family":"Bohner","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2965-4561","authenticated-orcid":false,"given":"Osman","family":"Tun\u00e7","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2909-8753","authenticated-orcid":false,"given":"Cemil","family":"Tun\u00e7","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,8,16]]},"reference":[{"issue":"4","key":"1595_CR1","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/s10474-005-0250-7","volume":"109","author":"R Agarwal","year":"2005","unstructured":"Agarwal R, Bohner M, Domoshnitsky A, Goltser Y (2005) Floquet theory and stability of nonlinear integro-differential equations. Acta Math Hungar 109(4):305\u2013330. https:\/\/doi.org\/10.1007\/s10474-005-0250-7","journal-title":"Acta Math Hungar"},{"key":"1595_CR2","doi-asserted-by":"publisher","unstructured":"Agarwal R, Hristova S, O\u2019Regan D (2018) Lyapunov functions and stability of Caputo fractional differential equations with delays. Differ Equ Dyn Syst. https:\/\/doi.org\/10.1007\/s12591-018-0434-6","DOI":"10.1007\/s12591-018-0434-6"},{"issue":"9","key":"1595_CR3","doi-asserted-by":"publisher","first-page":"2951","DOI":"10.1016\/j.cnsns.2014.01.022","volume":"19","author":"N Aguila-Camacho","year":"2014","unstructured":"Aguila-Camacho N, Duarte-Mermoud MA, Gallegos JA (2014) Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul 19(9):2951\u20132957. https:\/\/doi.org\/10.1016\/j.cnsns.2014.01.022","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"1","key":"1595_CR4","first-page":"19","volume":"13","author":"F Alahmadi","year":"2018","unstructured":"Alahmadi F, Raffoul YN, Alharbi S (2018) Boundedness and stability of solutions of nonlinear Volterra integro-differential equations. Adv Dyn Syst Appl 13(1):19\u201331","journal-title":"Adv Dyn Syst Appl"},{"key":"1595_CR5","doi-asserted-by":"publisher","unstructured":"Andreev AS, Peregudova OA (2018) On the stability and stabilization problems of Volterra integro-differential equations. Russ J Nonlinear Dyn 14(3):387\u2013407. https:\/\/doi.org\/10.20537\/nd180309","DOI":"10.20537\/nd180309"},{"issue":"104985","key":"1595_CR6","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.cnsns.2019.104985","volume":"82","author":"A Babaei","year":"2020","unstructured":"Babaei A, Moghaddam BP, Banihashemi S, Tenreiro Machado JA (2020) Numerical solution of variable-order fractional integro-partial differential equations via Sinc collocation method based on single and double exponential transformations. Commun Nonlinear Sci Numer Simul 82(104985):21. https:\/\/doi.org\/10.1016\/j.cnsns.2019.104985","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"2","key":"1595_CR7","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/S0893-9659(98)00146-3","volume":"12","author":"N Bellomo","year":"1999","unstructured":"Bellomo N, Firmani B, Guerri L (1999) Bifurcation analysis for a nonlinear system of integro-differential equations modelling tumor-immune cells competition. Appl Math Lett 12(2):39\u201344. https:\/\/doi.org\/10.1016\/S0893-9659(98)00146-3","journal-title":"Appl Math Lett"},{"key":"1595_CR8","doi-asserted-by":"publisher","first-page":"899","DOI":"10.1016\/j.amc.2014.06.060","volume":"243","author":"L Berezansky","year":"2014","unstructured":"Berezansky L, Braverman E, Idels L (2014) New global exponential stability criteria for nonlinear delay differential systems with applications to BAM neural networks. Appl Math Comput 243:899\u2013910. https:\/\/doi.org\/10.1016\/j.amc.2014.06.060","journal-title":"Appl Math Comput"},{"key":"1595_CR9","unstructured":"Berezansky L, Domoshnitsky A (2019) On stability of a second order integro-differential equation. Nonlinear Dyn Syst Theory 19(1-SI):117\u2013123"},{"key":"1595_CR10","doi-asserted-by":"publisher","unstructured":"Bohner M, Tun\u00e7 O (2021) Qualitative analysis of integro-differential equations with variable retardation. Discrete Contin Dyn Syst Ser B. https:\/\/doi.org\/10.3934\/dcdsb.2021059","DOI":"10.3934\/dcdsb.2021059"},{"key":"1595_CR11","unstructured":"Burton TA (2005) Volterra integral and differential equations, Mathematics in Science and Engineering, vol 202, 2nd edn. Elsevier B. V, Amsterdam"},{"issue":"3","key":"1595_CR12","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1216\/JIE-2018-30-3-347","volume":"30","author":"YK Chang","year":"2018","unstructured":"Chang YK, Ponce R (2018) Uniform exponential stability and applications to bounded solutions of integro-differential equations in Banach spaces. J Integral Equ Appl 30(3):347\u2013369. https:\/\/doi.org\/10.1216\/JIE-2018-30-3-347","journal-title":"J Integral Equ Appl"},{"key":"1595_CR13","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/j.amc.2014.12.010","volume":"254","author":"B Chen","year":"2015","unstructured":"Chen B, Chen J (2015) Razumikhin-type stability theorems for functional fractional-order differential systems and applications. Appl Math Comput 254:63\u201369. https:\/\/doi.org\/10.1016\/j.amc.2014.12.010","journal-title":"Appl Math Comput"},{"issue":"4","key":"1595_CR14","doi-asserted-by":"publisher","first-page":"1516","DOI":"10.1016\/j.jksus.2018.11.012","volume":"31","author":"M El Hajji","year":"2019","unstructured":"El Hajji M (2019) Boundedness and asymptotic stability of nonlinear Volterra integro-differential equations using Lyapunov functional. J King Saud Univ Sci 31(4):1516\u20131521","journal-title":"J King Saud Univ Sci"},{"key":"1595_CR15","doi-asserted-by":"publisher","unstructured":"Hale JK, Verduyn Lunel SM (1993) Introduction to functional-differential equations, Applied Mathematical Sciences, vol 99. Springer, New York. https:\/\/doi.org\/10.1007\/978-1-4612-4342-7","DOI":"10.1007\/978-1-4612-4342-7"},{"key":"1595_CR16","doi-asserted-by":"publisher","first-page":"393","DOI":"10.2528\/PIER08012502","volume":"81","author":"S Hatamzadeh-Varmazyar","year":"2008","unstructured":"Hatamzadeh-Varmazyar S, Naser-Moghadasi N, Babolian E, Masouri Z (2008) Numerical approach to survey the problem of electromagnetic scattering from resistive strips based on using a set of orthogonal basis functions. Prog Electromagn Res 81:393\u2013412","journal-title":"Prog Electromagn Res"},{"key":"1595_CR17","unstructured":"Haykin S (2009) Neural networks and learning machines, 3rd edn. Pearson Prentice Hall"},{"key":"1595_CR18","unstructured":"Hristova S, Tun\u00e7 C (2019) Stability of nonlinear Volterra integro-differential equations with Caputo fractional derivative and bounded delays. Electron J Differ Equ pp Paper No. 30, 11"},{"key":"1595_CR19","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/j.cam.2016.11.011","volume":"317","author":"S Kheybari","year":"2017","unstructured":"Kheybari S, Darvishi MT, Wazwaz AM (2017) A semi-analytical approach to solve integro-differential equations. J Comput Appl Math 317:17\u201330. https:\/\/doi.org\/10.1016\/j.cam.2016.11.011","journal-title":"J Comput Appl Math"},{"issue":"1","key":"1595_CR20","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1109\/21.87054","volume":"18","author":"B Kosko","year":"1988","unstructured":"Kosko B (1988) Bidirectional associative memories. IEEE Trans Systems Man Cybernet 18(1):49\u201360. https:\/\/doi.org\/10.1109\/21.87054","journal-title":"IEEE Trans Systems Man Cybernet"},{"issue":"5","key":"1595_CR21","doi-asserted-by":"publisher","first-page":"1810","DOI":"10.1016\/j.camwa.2009.08.019","volume":"59","author":"Y Li","year":"2010","unstructured":"Li Y, Chen Y, Podlubny I (2010) Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput Math Appl 59(5):1810\u20131821. https:\/\/doi.org\/10.1016\/j.camwa.2009.08.019","journal-title":"Comput Math Appl"},{"key":"1595_CR22","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1016\/j.aml.2015.06.018","volume":"51","author":"S Liu","year":"2016","unstructured":"Liu S, Jiang W, Li X, Zhou XF (2016) Lyapunov stability analysis of fractional nonlinear systems. Appl Math Lett 51:13\u201319. https:\/\/doi.org\/10.1016\/j.aml.2015.06.018","journal-title":"Appl Math Lett"},{"issue":"3","key":"1595_CR23","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1007\/s11075-019-00712-y","volume":"83","author":"P Mokhtary","year":"2020","unstructured":"Mokhtary P, Moghaddam BP, Lopes AM, Tenreiro Machado JA (2020) A computational approach for the non-smooth solution of non-linear weakly singular Volterra integral equation with proportional delay. Numer Algorithms 83(3):987\u20131006. https:\/\/doi.org\/10.1007\/s11075-019-00712-y","journal-title":"Numer Algorithms"},{"issue":"5","key":"1595_CR24","doi-asserted-by":"publisher","first-page":"6433","DOI":"10.1007\/s40314-018-0698-z","volume":"37","author":"ZS Mostaghim","year":"2018","unstructured":"Mostaghim ZS, Moghaddam BP, Haghgozar HS (2018) Numerical simulation of fractional-order dynamical systems in noisy environments. Comput Appl Math 37(5):6433\u20136447. https:\/\/doi.org\/10.1007\/s40314-018-0698-z","journal-title":"Comput Appl Math"},{"issue":"3","key":"1595_CR25","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1007\/s40306-017-0243-y","volume":"43","author":"PHA Ngoc","year":"2018","unstructured":"Ngoc PHA, Anh TT (2018) New stability criteria for nonlinear Volterra integro-differential equations. Acta Math Vietnam 43(3):485\u2013501. https:\/\/doi.org\/10.1007\/s40306-017-0243-y","journal-title":"Acta Math Vietnam"},{"key":"1595_CR26","doi-asserted-by":"publisher","unstructured":"Nikan O, Avazzadeh Z (2021) An improved localized radial basis-pseudospectral method for solving fractional reaction-subdiffusion problem. Results Phys 23:104048. https:\/\/doi.org\/10.1016\/j.rinp.2021.104048","DOI":"10.1016\/j.rinp.2021.104048"},{"key":"1595_CR27","doi-asserted-by":"publisher","unstructured":"Nikan O, Avazzadeh Z, Tenreiro Machado JA (2021a) An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model. J King Saud Univ Sci 33(1):101243. https:\/\/doi.org\/10.1016\/j.jksus.2020.101243","DOI":"10.1016\/j.jksus.2020.101243"},{"key":"1595_CR28","doi-asserted-by":"publisher","unstructured":"Nikan O, Avazzadeh Z, Tenreiro Machado JA (2021b) A local stabilized approach for approximating the modified time-fractional diffusion problem arising in heat and mass transfer. J Adv Res. https:\/\/doi.org\/10.1016\/j.jare.2021.03.002","DOI":"10.1016\/j.jare.2021.03.002"},{"issue":"105755","key":"1595_CR29","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1016\/j.cnsns.2021.105755","volume":"99","author":"O Nikan","year":"2021","unstructured":"Nikan O, Avazzadeh Z, Tenreiro Machado JA (2021c) Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport. Commun Nonlinear Sci Numer Simul 99(105755):22. https:\/\/doi.org\/10.1016\/j.cnsns.2021.105755","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"1595_CR30","unstructured":"Podlubny I (1999) Fractional differential equations, Mathematics in Science and Engineering, vol 198. Academic Press, Inc., San Diego, CA, an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications"},{"issue":"1","key":"1595_CR31","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1515\/msds-2016-0002","volume":"3","author":"Y Raffoul","year":"2016","unstructured":"Raffoul Y, Rai H (2016) Uniform stability in nonlinear infinite delay Volterra integro-differential equations using Lyapunov functionals. Nonauton Dyn Syst 3(1):14\u201323. https:\/\/doi.org\/10.1515\/msds-2016-0002","journal-title":"Nonauton Dyn Syst"},{"key":"1595_CR32","volume-title":"Integral equations and their applications","author":"M Rahman","year":"2007","unstructured":"Rahman M (2007) Integral equations and their applications. WIT Press, Southampton"},{"issue":"1","key":"1595_CR33","doi-asserted-by":"publisher","first-page":"55","DOI":"10.2307\/2044951","volume":"94","author":"M Rama Mohana Rao","year":"1985","unstructured":"Rama Mohana Rao M, Srinivas P (1985) Asymptotic behavior of solutions of Volterra integro-differential equations. Proc Am Math Soc 94(1):55\u201360. https:\/\/doi.org\/10.2307\/2044951","journal-title":"Proc Am Math Soc"},{"key":"1595_CR34","doi-asserted-by":"publisher","unstructured":"Sadati SJ, Baleanu D, Ranjbar A, Ghaderi R, Abdeljawad T (2010) Mittag-Leffler stability theorem for fractional nonlinear systems with delay. Abstr Appl Anal pp Art. ID 108651:7. https:\/\/doi.org\/10.1155\/2010\/108651","DOI":"10.1155\/2010\/108651"},{"issue":"2","key":"1595_CR35","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1016\/j.joems.2015.08.001","volume":"24","author":"C Tun\u00e7","year":"2016","unstructured":"Tun\u00e7 C (2016a) New stability and boundedness results to Volterra integro-differential equations with delay. J Egypt Math Soc 24(2):210\u2013213. https:\/\/doi.org\/10.1016\/j.joems.2015.08.001","journal-title":"J Egypt Math Soc"},{"issue":"2","key":"1595_CR36","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/j.joems.2014.12.010","volume":"24","author":"C Tun\u00e7","year":"2016","unstructured":"Tun\u00e7 C (2016b) A note on the qualitative behaviors of non-linear Volterra integro-differential equation. J Egypt Math Soc 24(2):187\u2013192. https:\/\/doi.org\/10.1016\/j.joems.2014.12.010","journal-title":"J Egypt Math Soc"},{"issue":"5","key":"1595_CR37","doi-asserted-by":"publisher","first-page":"1775","DOI":"10.18576\/amis\/100518","volume":"10","author":"C Tun\u00e7","year":"2016","unstructured":"Tun\u00e7 C (2016c) Properties of solutions of Volterra integro-differential equations with delay. Appl Math Inf Sci 10(5):1775\u20131780","journal-title":"Appl Math Inf Sci"},{"issue":"2","key":"1595_CR38","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/j.jtusci.2015.12.009","volume":"11","author":"C Tun\u00e7","year":"2017","unstructured":"Tun\u00e7 C (2017a) Qualitative properties in nonlinear Volterra integro-differential equations with delay. J Taibah Univ Sci 11(2):309\u2013314","journal-title":"J Taibah Univ Sci"},{"issue":"1","key":"1595_CR39","first-page":"121","volume":"26","author":"C Tun\u00e7","year":"2017","unstructured":"Tun\u00e7 C (2017b) Stability and boundedness in Volterra integro-differential equations with delay. Dyn Syst Appl 26(1):121\u2013130","journal-title":"Dyn Syst Appl"},{"issue":"4","key":"1595_CR40","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1016\/j.jksus.2017.05.003","volume":"30","author":"C Tun\u00e7","year":"2018","unstructured":"Tun\u00e7 C (2018) Asymptotic stability and boundedness criteria for nonlinear retarded Volterra integro-differential equations. J King Saud Univ Sci 30(4):531\u2013536","journal-title":"J King Saud Univ Sci"},{"issue":"3","key":"1595_CR41","doi-asserted-by":"publisher","first-page":"158","DOI":"10.1080\/25765299.2018.1509554","volume":"25","author":"C Tun\u00e7","year":"2018","unstructured":"Tun\u00e7 C, Tun\u00e7 O (2018a) New qualitative criteria for solutions of Volterra integro-differential equations. Arab J Basic Appl Sci 25(3):158\u2013165","journal-title":"Arab J Basic Appl Sci"},{"issue":"2","key":"1595_CR42","first-page":"107","volume":"11","author":"C Tun\u00e7","year":"2018","unstructured":"Tun\u00e7 C, Tun\u00e7 O (2018b) New results on behaviors of functional Volterra integro-differential equations with multiple time-lags. Jordan J Math Stat 11(2):107\u2013124","journal-title":"Jordan J Math Stat"},{"issue":"1","key":"1595_CR43","first-page":"41","volume":"6","author":"C Tun\u00e7","year":"2018","unstructured":"Tun\u00e7 C, Tun\u00e7 O (2018c) New results on the stability, integrability and boundedness in Volterra integro-differential equations. Bull Comput Appl Math 6(1):41\u201358","journal-title":"Bull Comput Appl Math"},{"issue":"1","key":"1595_CR44","first-page":"253","volume":"6","author":"C Tun\u00e7","year":"2018","unstructured":"Tun\u00e7 C, Tun\u00e7 O (2018d) On the exponential study of solutions of Volterra integro-differential equations with time lag. Electron J Math Anal Appl 6(1):253\u2013265","journal-title":"Electron J Math Anal Appl"},{"key":"1595_CR45","doi-asserted-by":"publisher","unstructured":"Tun\u00e7 O (2020) On the qualitative analyses of integro-differential equations with constant time lag. Appl Math Inf Sci 14(1):57\u201363. https:\/\/doi.org\/10.18576\/amis\/140107","DOI":"10.18576\/amis\/140107"},{"key":"1595_CR46","doi-asserted-by":"publisher","unstructured":"Tun\u00e7 O, Atan O, Tun\u00e7 C, Yao JC (2021) Qualitative analyses of integro-fractional differential equations with Caputo derivatives and retardations via the Lyapunov\u2013Razumikhin method. Axioms 10(2). https:\/\/doi.org\/10.3390\/axioms10020058","DOI":"10.3390\/axioms10020058"},{"issue":"3","key":"1595_CR47","first-page":"325","volume":"9","author":"K Wang","year":"1993","unstructured":"Wang K (1993) Uniform asymptotic stability in functional-differential equations with infinite delay. Ann Differ Equ 9(3):325\u2013335","journal-title":"Ann Differ Equ"},{"issue":"1","key":"1595_CR48","first-page":"89","volume":"16","author":"Q Wang","year":"2000","unstructured":"Wang Q (2000) The stability of a class of functional differential equations with infinite delays. Ann Differ Equ 16(1):89\u201397","journal-title":"Ann Differ Equ"},{"issue":"1","key":"1595_CR49","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/S0096-3003(02)00527-1","volume":"146","author":"AM Wazwaz","year":"2003","unstructured":"Wazwaz AM (2003) The existence of noise terms for systems of inhomogeneous differential and integral equations. Appl Math Comput 146(1):81\u201392. https:\/\/doi.org\/10.1016\/S0096-3003(02)00527-1","journal-title":"Appl Math Comput"},{"key":"1595_CR50","doi-asserted-by":"publisher","unstructured":"Wazwaz AM (2011) Linear and nonlinear integral equations. Higher Education Press, Beijing. https:\/\/doi.org\/10.1007\/978-3-642-21449-3 methods and applications","DOI":"10.1007\/978-3-642-21449-3"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01595-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-021-01595-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-021-01595-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,24]],"date-time":"2021-09-24T17:32:15Z","timestamp":1632504735000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-021-01595-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,16]]},"references-count":50,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["1595"],"URL":"https:\/\/doi.org\/10.1007\/s40314-021-01595-3","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,16]]},"assertion":[{"value":"3 March 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 July 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 August 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"214"}}