{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:05:56Z","timestamp":1760148356693,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,23]],"date-time":"2023-04-23T00:00:00Z","timestamp":1682208000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Lyapunov functions\/functionals have found their footing in Volterra integro-differential equations. This is not the case for integral equations, and it is therefore further explored in this paper. In this manuscript, we utilize Lyapunov functionals combined with Laplace transform to qualitatively analyze the solutions of the integral equation In addition, we extend our method to nonlinear integral equations, integral equations with infinite delay, and integral equations with several kernels. We mention that Laplace transform has been used to solve integral equations of convolution types but has never been applied directly to integral equations that are not of the convolution type. In addition, our method allows us to find the upper estimates, and our necessary conditions are easy to verify.<\/jats:p>","DOI":"10.3390\/axioms12050410","type":"journal-article","created":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T02:31:05Z","timestamp":1682303465000},"page":"410","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Lyapunov Functionals in Integral Equations"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2154-9049","authenticated-orcid":false,"given":"Youssef N.","family":"Raffoul","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joseph","family":"Raffoul","sequence":"additional","affiliation":[{"name":"Electrical and Computer Engineering Department, University of Dayton, Dayton, OH 45469-2316, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,23]]},"reference":[{"key":"ref_1","unstructured":"Burton, T.A. (2008). Liapunov Functionals for Integral Equations, Trafford Publishing."},{"key":"ref_2","unstructured":"Brauer, F., and Nohel, J.A. (1969). Qualitative Theory of Ordinary Differential Equations, Dover."},{"key":"ref_3","unstructured":"Coddington, E.A., and Levinson, N. (1955). Theory of Ordinary Differential Equations, McGraw-Hill Book Company."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Cushing, J.M. (1977). Integro-Differential Equations and Delay Models in Population Dynamics, Springer. Lecture Notes in Biomathematics.","DOI":"10.1007\/978-3-642-93073-7"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Driver, R.D. (1978). Introduction to Ordinary Differential Equations, Harper & Row, Publishers.","DOI":"10.1007\/978-1-4684-9467-9_5"},{"key":"ref_6","first-page":"137","article-title":"Qualitative Analysis of Dynamic Equations on Time Scales Using Lyapunov Functions","volume":"14","author":"Messina","year":"2022","journal-title":"Differ. Equ. Appl."},{"key":"ref_7","unstructured":"Miller, R.K. (1971). Nonlinear Volterra Integral Equations, Benjamin."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Raffoul, Y. (2023). Advanced Differential Equations, Academic Press.","DOI":"10.1016\/B978-0-32-399280-0.00014-0"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Wang, M., Saleem, N., Bashir, S., and Zhou, M. (2022). Fixed point of modified F-Contraction with an application. Axioms, 11.","DOI":"10.3390\/axioms11080413"},{"key":"ref_10","first-page":"19","article-title":"Boundedness and Stability of Solutions of Nonlinear Volterra Integro-Differential Equations","volume":"13","author":"Alhamadi","year":"2018","journal-title":"Adv. Dyn. Syst. Appl."},{"key":"ref_11","unstructured":"Burton, T.A. (1985). Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press."},{"key":"ref_12","unstructured":"Burton, T.A. (2006). Stability by Fixed Point Theory for Functional Differential Equations, Dover."},{"key":"ref_13","first-page":"2","article-title":"Bounded Solutions of Almost Linear Volterra Equation","volume":"7","author":"Islam","year":"2012","journal-title":"Adv. Dyn. Syst. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.joems.2015.08.001","article-title":"New stability and boundedness results to Volterra integro-differential equations with delay","volume":"24","author":"Tunc","year":"2016","journal-title":"J. Egypt. Math. Soc."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/410\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:21:33Z","timestamp":1760124093000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/410"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,23]]},"references-count":14,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["axioms12050410"],"URL":"https:\/\/doi.org\/10.3390\/axioms12050410","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,4,23]]}}}