{"paper":{"title":"Spatially localized solutions of the Hammerstein equation with sigmoid type of nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Alexander V. Sobolev, Anna Oleynik, Arcady Ponosov, Vadim Kostrykin","submitted_at":"2015-11-19T17:37:46Z","abstract_excerpt":"We study the existence of fixed points to a parameterized Hammertstain operator $\\cH_\\beta,$ $\\beta\\in (0,\\infty],$ with sigmoid type of nonlinearity. The parameter $\\beta<\\infty$ indicates the steepness of the slope of a nonlinear smooth sigmoid function and the limit case $\\beta=\\infty$ corresponds to a discontinuous unit step function. We prove that spatially localized solutions to the fixed point problem for large $\\beta$ exist and can be approximated by the fixed points of $\\cH_\\infty.$ These results are of a high importance in biological applications where one often approximates the smoo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.06364","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"}