A coincidence degree for bifurcation problems
This article deals with coincidence degree for pairs \((L, A)\) of operators in a Hilbert space \(H\); the first operator is assumed to be Leray-Schauder type, the second one compact and monotone. This degree is defined by the equation \[ d_\lambda\bigl( (L,A),D\bigr)= d_{LS}\bigl(I-(I+ \lambda A)^{-1}(I-L), D,0\bigr) \] and has the properties of the classical topological degree; it can be used in order to study bifurcation phenomena, for equations of the type \[ Lu+\lambda Au+F(\lambda,u)+ G(\lambda,u)=0 \] with \(F( \lambda,u)\) and \(G(\lambda,u)\) satisfying the conditions \(\|F(\lambda,u) \|\leq k\|u\|\) for some \(k>0\) and \(\|G(\lambda,u)\|= o(\|u \|)\) for \(u\to 0\) uniformly in \(\lambda\); in particular, analogues to Krasnosel'skiĭ and Rabinowitz theorems about bifurcation are given. At the end of the article, using the Browder-Ton theorem about imbedding reflexive separable Banach spaces into Hilbert spaces, the main results are enlarged to vector fields of monotone type defined from a Banach space into its dual.
scientific article; zbMATH DE number 1903500