{"paper":{"title":"Solutions and stability of a variant of Van Vleck's and d'Alembert's functional equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Elqorachi Elhoucien, Redouani Ahmed, Th. M. Rassais","submitted_at":"2016-08-09T21:49:30Z","abstract_excerpt":"In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$\\int_{S}f(\\sigma(y)xt)d\\mu(t)-\\int_{S}f(xyt)d\\mu(t) = 2f(x)f(y), \\;x,y\\in S,$$ where $S$ is a semigroup, $\\sigma$ is an involutive morphism of $S$, and $\\mu$ is a complex measure that is linear combinations of Dirac measures $(\\delta_{z_{i}})_{i\\in I}$, such that for all $i\\in I$, $z_{i}$ is contained in the center of $S$. (2) We determine the complex-valued continuous solutions of the following variant of d'Alembert's functional equation $$\\int_{S}f(xty)d\\upsilon(t)+\\int_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.03906","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}