{"paper":{"title":"On the stability of the existence of fixed points for the projection-iterative methods with relaxation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Adam Paszkiewicz, Andrzej Komisarski","submitted_at":"2014-05-20T18:40:26Z","abstract_excerpt":"We consider an $\\alpha$-relaxed projection $P_A^\\alpha:H\\to H$ given by $P_A^\\alpha(x)=\\alpha P_A(x)+(1-\\alpha)x$ where $\\alpha\\in[0,1]$ and $P_A$ is the projection onto a non-empty, convex and closed subset $A$ of the real Hilbert space $H$. We characterise all the sets $F\\subset[0,1]$ such that for some non-empty, convex and closed subsets $A_1,A_2,\\dots,A_k\\subset H$ the composition $P_{A_k}^\\alpha P_{A_{k-1}}^\\alpha\\dots P_{A_1}^\\alpha$ has a fixed point iff $\\alpha\\in F$. It proves, that if $\\dim H\\geq 3$ and $k\\geq3$ then the class of the derscribed above sets $F$ of coefficients $\\alpha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.5183","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"}