Fixed point index theory for decomposable multivalued maps and applications to discontinuous ϕ-Laplacian problems

Compression–Expansion Fixed Point Theorems for Decomposable Maps and Applications to Discontinuous $$\phi $$-Laplacian problems

“…The transversality condition (3.7) is the key ingredient to show that all solutions of (3.2) are Carathéodory solutions of (3.1). The proof of this result can be looked up in [27,Lemma 3.3].…”

“…In the recent paper [27], the authors define a fixed point index theory for the composition of two multivalued maps. We recall here its definition and its main properties.…”

“…In recent years, a lot of attention have been paid to the existence of solutions to boundary value problems with φ-Laplacian (see, for instance, [5,9,16,18,19,27] and the references therein). Here, we study in a unified way the classical homeomorphism φ : R → R, the singular homeomorphism φ : (−a, a) → R and the bounded one φ : R → (−b, b).…”

“…To overcome this difficulty we prove a new version of the compression-expansion fixed point theorem in cones for decomposable maps, that is, compositions of two usc multivalued maps which cover the fixed point problem above. Our approach is based on a suitable computation of the fixed point index for this class of maps, which was developed in [27]. Some previous results on fixed point theory for decomposable maps can be found in the pioneering paper [24] and in [11].…”

Existence of Solutions for Inclusion Problems in Musielak-Orlicz-Sobolev Space Setting

“…Precup and Rodríguez-López [4] proved the existence of solutions for an inclusion problem driven by a ϕ-Laplacian operator depending on the differential of the solution. Chen and Tang [5] discussed periodic solutions for a differential inclusion problem involving the pðtÞ-Laplacian.…”