An analytical study of bifurcation problems for equations involving Fredholm mappings

Let us consider equations in the formwhere Λ is an open subset of a normed space. For any fixed λ ∊ Λ, T, L(λ,.) and M(λ,.) are mappings from the closure D0 of a neighbourhood D0 of the origin in a Banach space X into another Banach space Y with T(0) = L(λ, 0) = M(λ, 0) = 0. Let λ be a characteristic value of the pair (T, L) such that T − L(λ,.) is a Fredholm mapping with nullity p and index s, p> s≧ 0. Under sufficient hypotheses on T, L and M, (λ, 0) is a bifurcation point of the above equations. Some well-known results obtained by Crandall and Rabinowitz [2], McLeod and Sattinger [5] and others will be generalised. The results in this paper are extensions of the results obtained by the author in [7].

On the existence of sulutions of the equation Lx ∞ Nx and a coincidence degree theory

The coincidence degree for the pair (L, N) developed by Mawhin (1972) provides a method for proving the existence of solutions of the equation Lx = Nx where L: dom L ⊂ X → Z is a linear Fredholm mapping of index zero and is a (possiblv nonlinear) mapping and Ω is a bounded open subset of X, X and Z being normed linear spaces over the reals. In this paper we have extended the coincidence degree for the pair (L, N) to solve the equation , where L: dom L ⊂ X → Z is a linear Fredholm mapping of index zero, and X, Z and Ω are as above, CK(Z) being the set of compact convex subsets of Z.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 47 H 15, 47 A 50; secondary 47 H 10, 47 A 55.