{"paper":{"title":"Convergence of the solutions of the discounted equation: the discrete case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.OC","authors_text":"Albert Fathi, Andrea Davini, Maxime Zavidovique, Renato Iturriaga","submitted_at":"2016-07-28T02:09:38Z","abstract_excerpt":"We derive a discrete version of the results of our previous work. If $M$ is a compact metric space, $c : M\\times M \\to \\mathbb R$ a continuous cost function and $\\lambda \\in (0,1)$, the unique solution to the discrete $\\lambda$-discounted equation is the only function $u_\\lambda : M\\to \\mathbb R$ such that $$\\forall x\\in M, \\quad u_\\lambda(x) = \\min_{y\\in M} \\lambda u_\\lambda (y) + c(y,x).$$ We prove that there exists a unique constant $\\alpha\\in \\mathbb R$ such that the family of $u_\\lambda+\\alpha/(1-\\lambda)$ is bounded as $\\lambda \\to 1$ and that for this $\\alpha$, the family uniformly conv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.08295","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":""},"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"}