{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T07:25:56Z","timestamp":1778570756254,"version":"3.51.4"},"reference-count":20,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,7,23]],"date-time":"2024-07-23T00:00:00Z","timestamp":1721692800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"YiBin University","award":["412-2021QH027"],"award-info":[{"award-number":["412-2021QH027"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we focus on a fractional differential equation \u20090CD\u03b1u(t)+q(t)u(t)=0 with boundary value conditions u(0)=\u03b4u(1),u\u2032(0)=\u03b3u\u2032(1). The paper begins by pointing out the inadequacies of the study conducted by Ma and Yangin establishing Lyapunov-type inequalities. It then discusses the properties of its Green\u2019s function and investigates extremum problems related to several linear functions. Finally, thorough classification and analysis of various cases for parameters \u03b4 and \u03b3 are conducted. As a result, a comprehensive solution corresponding to the Lyapunov-type inequality is obtained.<\/jats:p>","DOI":"10.3390\/sym16080941","type":"journal-article","created":{"date-parts":[[2024,7,24]],"date-time":"2024-07-24T15:47:32Z","timestamp":1721836052000},"page":"941","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Refinement of a Lyapunov-Type Inequality for a Fractional Differential Equation"],"prefix":"10.3390","volume":"16","author":[{"given":"Hongying","family":"Xiao","sequence":"first","affiliation":[{"name":"Faculty of Science, Yibin University, Yibin 644000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhaofeng","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuanyuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, China Three Gorges University, Yichang 443002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyou","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computing Sciences, Hunan University of Science and Technology, Xiangtan 411201, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,23]]},"reference":[{"key":"ref_1","first-page":"27","article-title":"Probleme g\u00e9n\u00e9ral de la stabilit\u00e9 du mouvement, (French Translation of a Russian paper dated 1893)","volume":"2","author":"Liapunov","year":"1907","journal-title":"Ann. Fac. Sci. Univ. Toulouse"},{"key":"ref_2","first-page":"231","article-title":"Recent developments of Lyapunov-type inequalities","volume":"5","author":"Tiryaki","year":"2010","journal-title":"Adv. Dyn. Syst. Appl."},{"key":"ref_3","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1007\/978-3-030-27407-8_24","article-title":"Recent developments of Lyapunov-type inequalities for fractional differential equations","volume":"151","author":"Ntouyas","year":"2019","journal-title":"Differ. Integral Inequalities"},{"key":"ref_5","first-page":"43","article-title":"Lyapunov-type inequalities for fractional differential equations: A survey","volume":"16","author":"Ntouyas","year":"2021","journal-title":"Surv. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ntouyas, S.K., Ahmad, B., and Tariboon, J. (2022). A Survey on Recent Results on Lyapunov-Typ eInequalities for Fractional Differential Equations. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6050273"},{"key":"ref_7","first-page":"443","article-title":"Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions","volume":"18","author":"Jleli","year":"2015","journal-title":"Math. Inequal. Appl."},{"key":"ref_8","first-page":"468536","article-title":"Lyapunov-type inequality for a fractional differential equation under a Robin boundary conditions","volume":"2015","author":"Jleli","year":"2015","journal-title":"J. Funct. Spaces"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1186\/s13662-015-0430-x","article-title":"Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions","volume":"2015","author":"Rong","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_10","first-page":"139","article-title":"A Lyapunov-type inequalities for a fractional differential equation under Sturm\u2013Liouville boundary conditions","volume":"20","author":"Wang","year":"2017","journal-title":"Math. Inequal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1186\/s13660-015-0769-2","article-title":"Lyapunov-type inequalities for a class of fractional differential equations","volume":"2015","author":"Samet","year":"2015","journal-title":"J. Inequal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1002\/mma.3972","article-title":"Hartman\u2013Wintner-type inequalities for a class of nonlocal fractional boundary value problems","volume":"40","author":"Cabrera","year":"2017","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"7269","DOI":"10.3934\/mbe.2021359","article-title":"Analysis of some Katugampola fractional differential equations with fractional boundary conditions","volume":"18","author":"Lupinska","year":"2021","journal-title":"Math. Biosci. Eng."},{"key":"ref_14","first-page":"2335","article-title":"Lyapunov type inequality in the frame of generalized Caputo derivative","volume":"14","author":"Jarad","year":"2021","journal-title":"Discret. Contin. Dyn. Syst. Ser. S"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1074","DOI":"10.3934\/math.2022064","article-title":"New Lyapunov-type inequalities for fractional multi-point boundary value problems involving Hilfer\u2013Katugampola fractional derivative","volume":"7","author":"Zhang","year":"2022","journal-title":"AIMS Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.22199\/issn.0717-6279-4366","article-title":"Lyapunov-type inequality for higher order left and right fractional p-Laplacian problems","volume":"40","author":"Cabada","year":"2021","journal-title":"Proyecciones"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"7020","DOI":"10.3934\/mbe.2023303","article-title":"Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse","volume":"20","author":"Li","year":"2023","journal-title":"Math. Biosci. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Fan, H., Rao, Y., Shi, K., and Wen, H. (2023). Global synchronization of fractional-order multi-delay coupled neural networks with multi-link complicated structures via hybrid impulsive control. Mathematics, 11.","DOI":"10.3390\/math11143051"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Fan, H., Tang, J., Shi, K., and Zhao, Y. (2023). Hybrid impulsive feedback control for drive-response synchronization of fractional-order multi-link Memristive neural networks with multi-delays. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7070495"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1186\/s13660-020-02448-z","article-title":"Lyapunov-type inequality and solution for a fractional differential equation","volume":"2020","author":"Ma","year":"2020","journal-title":"J. Inequalities Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/8\/941\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:21:49Z","timestamp":1760109709000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/8\/941"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,23]]},"references-count":20,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["sym16080941"],"URL":"https:\/\/doi.org\/10.3390\/sym16080941","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,23]]}}}