%0 Journal Article
%T Moore-Penrose Inverse and Semilinear Equations
%A Hugo Leiva
%A Ra¨²l Manzanilla
%J Advances in Linear Algebra & Matrix Theory
%P 11-17
%@ 2165-3348
%D 2018
%I Scientific Research Publishing
%R 10.4236/alamt.2018.81002
%X In this paper, we study the existence of solutions for the semilinear equation <img src=\"Edit_fcfafc28-ce72-4080-ab47-990cb017223a.bmp\" alt=\"\" />, where <em>A</em> is a <img src=\"Edit_5da83526-1cbc-4961-83b8-d87a7acadcfc.bmp\" alt=\"\" />, <img src=\"Edit_f34bf6ec-96a3-4fed-945f-5278bd5ba87a.bmp\" alt=\"\" />, <img src=\"Edit_320b2e23-d602-4a19-97a2-3c647407cee7.bmp\" alt=\"\" /> and <img src=\"Edit_87ff7696-ea2b-4b1c-ad6a-082e191ceb5f.bmp\" alt=\"\" /> is a nonlinear continuous function. Assuming that the Moore-Penrose inverse <em>A</em><sup>T</sup>(<em>AA</em><sup>T</sup>)<sup>-1&nbsp;</sup>exists (<em>A </em>denotes the transposed matrix of <em>A</em>) which is true whenever the determinant of the <img src=\"Edit_77c9b8e7-5dbc-4650-8fdb-fabfcd0b8f24.bmp\" alt=\"\" /> matrix <em>AA</em><sup>T </sup>is different than zero, and the following condition on the nonlinear term <img src=\"Edit_b41d91f7-cc7a-454b-b959-69c6bd65a492.bmp\" alt=\"\" /> satisfied <img src=\"Edit_b53b8e0e-d717-4305-82cb-5ece1746d037.bmp\" alt=\"\" />. We prove that the semilinear equation has solutions for all<img src=\"Edit_87cbc1bc-ac4d-4fce-bd7a-60277a51c7a7.bmp\" alt=\"\" />. Moreover, these solutions can be found from the following fixed point relation <img src=\"Edit_cc5ee4eb-2146-43a2-b996-a2d83109329d.bmp\" alt=\"\" />.
%K Semilinear Equations
%K Moore-Penrose Inverse
%K Rothe¡¯s Fixed Point Theorem
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=81794