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Báo cáo hóa học: " Research Article Bifurcation Results for a Class of Perturbed Fredholm Maps"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Bifurcation Results for a Class of Perturbed Fredholm Maps | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 752657 19 pages doi 10.1155 2008 752657 Research Article Bifurcation Results for a Class of Perturbed Fredholm Maps Pierluigi Benevieri1 and Alessandro Calamai2 1 Dipartimento di Matematica Applicata Universita degli Studi di Firenze Via S. Marta 3 50139 Firenze Italy 2 Dipartimento di Scienze Matematiche Universita Politecnica delle Marche Via Brecce Bianche 60131 Ancona Italy Correspondence should be addressed to Pierluigi Benevieri pierluigi.benevieri@unifi.it Received 3 March 2008 Revised 18 July 2008 Accepted 27 July 2008 Recommended by Fabio Zanolin We prove a global bifurcation result for an equation of the type Lx X h x k x 0 where L E F is a linear Fredholm operator of index zero between Banach spaces and given an open subset Q of E h k Q X 0 x F are C1 and continuous respectively. Under suitable conditions we prove the existence of an unbounded connected set of nontrivial solutions of the above equation that is solutions x A with A 0 whose closure contains a trivial solution x 0 . The proof is based on a degree theory for a special class of noncompact perturbations of Fredholm maps of index zero called a-Fredholm maps which has been recently developed by the authors in collaboration with M. Furi. Copyright 2008 P. Benevieri and A. Calamai. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction We study a bifurcation problem for the semilinear operator equation Lx X h x k xf 0 1.1 in Q X 0 to where Q is an open subset of a Banach space E L E F is a linear Fredholm operator of index zero between real Banach spaces and the maps h Q F and k Q F are of class C1 and continuous respectively. In addition we assume that for any nonnegative real A the map x Lx Xh x is a nonlinear Fredholm map of index zero.