Systems of Inclusions Involving Fredholm Operators of Nonnegative Index and Nonconvex-Valued Maps

We study the existence of solutions to a system of two inclusions involving Fredholm operators of nonnegative (Fredholm) index and the so-called c-admissible maps: upper semicontinuous maps with values being continuous images of cell-like sets. The presented approach resembles the substitution technique, i.e. applies the so-called solution map. The appearance of the ‘dimension defect’ reflecting the nonnegativity of Fredholm indices requires the use of algebraic tools based on the cohomotopy theory, different from the usual homological ones present in the fixed point theory. An application to boundary value problems is provided as well as the brief exposition of basic facts from the cohomotopy theory. Mathematics Subject Classifications (2000) 47H04, 47A53, 55M20, 55Q55, 34A60.