Strong Surjections and Nearness

Abstract

Summary:We show that the properties of being strongly surjective or stable solvable (in the sense of Furi-Martelli-Vignoli) carry over to maps which are near in the sense of Campanato. Keywords: Near mappings, strong surjection, stable solvable map, implicit differential equations. Classification: 47H99, 47H15, 34A09, 34B15. Introduction. M. Furi, M. Martelli and A. Vignoli in [3]introduced the notions of strong surjection and stable solvable map between two normed spaces E and F in order to define the spectrum for a nonlinear operator. Also, these concepts are related to that of zero-epi map, which is due to the same authors [4] and is very important in the study of solvability of nonlinear equations. Near operators have been introduced by S. Campanato and also studied by A. Tarsia and S. Leonardi in [2,7,10,13,14] and have applications in nonlinear differential equations, too. We prove that the property of being a strong surjection or stable solvable is preserved by nearness and notice that this can be used in order to prove existence results for differential equations in implicit form.