Bifurcation and asymptotic bifurcation for equations involving A-proper mappings with applications to differential equations

Abstract

Let X, Y be real Banach spaces, T: X → Y A-proper, and C: X → Y compact. Section 1 of this paper is devoted to the study of bifurcation and asymptotic bifurcation problems for Eq. (∗): Tx − λCx = 0. In Theorem 1 it is shown that if T(0) = C(0) = 0 and T and C have F-derivatives T 0 and C 0 at 0 with T 0 A-proper and injective, then each eigenvalue of T 0 x − λC 0 x = 0 of odd multiplicity is a bifurcation point for Eq. (∗). Theorem 2 shows that if T and C have asymptotic derivatives T ∞ and C ∞, then each eigenvalue of T ∞ x − λC ∞ x = 0 of odd multiplicity is an asymptotic bifurcation point for Eq. (∗). Special cases are treated when Y = X and T = I − F with F k-ball-contractive or when Y ≠ X and T is either of type ( S) or of strongly accretive type. Section 2 is devoted to applications of Theorems 1 and 2 to bifurcation problems involving elliptic operators. The usefulness of Theorems 1 and 2 stems from the fact that they are directly applicable to differential eigenvalue problems without the preliminary reduction of Eq. (∗) to equivalent problems involving compact operators. Moreover, in some cases they are applicable in situations to which the known bifurcation results are not applicable.