{"paper":{"title":"The Euler-Lagrange and Legendre Necessary Conditions for Fractional Calculus of Variations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Elimhan N. Mahmudov, Shakir Sh. Yusubov, Shikhi Sh. Yusubov","submitted_at":"2025-06-07T09:44:37Z","abstract_excerpt":"In this paper, we study the problems of minimizing a functional depending on the Caputo fractional derivative of order $0< \\alpha \\leq 1$ and the Riemann- Liouville fractional integral of order $\\beta >0$ under certain constraints. A fractional analogue of the Du Bois-Reymond lemma is proved. Using this lemma for various weak local minimum problems, the Euler-Lagrange equation is derived in integral form. Some serious works in the literature claim that the standard proof of the Legendre condition in the classical case $\\alpha=1$ cannot be adapted to the fractional case $0<\\alpha <1$ with final"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.06736","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.06736/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}