The existence of solutions to a homogeneous Dirichlet problem for a p-Laplacian differential inclusion is studied via a fixed-point type theorem concerning operator inclusions in Banach spaces. Some meaningful special cases are then worked out.

We study the existence of solutions of the semilinear equations (1) in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy are arbitrarily nonnegative constants, . The proofs are based upon degree theoretic arguments.

The local Hopf Bifurcation theorem is extended to implicit differential equations in Rn, of the form ẋ = f(x,ẋ, α), which are not solvable for the variable ẋ. The proof uses the S1 -degree of convex-valued mappings. An example of an implicit differential equation in R3 to which the presented theorem applies is provided.

We give mild sufficient conditions on a nonlinear functional to have eigenvalues. These results are intended for the study of boundary value problems for semilinear elliptic equations.

Let X be a Banach space, D an open subset of X and Y a complete metric space. Assume that Y is metrically convex. For closed, locally m-expansive and mapping open subsets of D onto open subsets of Y, is is shown that y ∊ T(D) if and only if there exists x0 ∊ D such that d(Tx0, y) ≤ d(Tx, y) for all x ≤ ∂D.

In this paper we continue our study of the solvability of nonlinear equations involving uniform limits of A-proper and pseudo A-proper maps under a new growth condition (1) that we began in [14,15]. Applications of our results to quasimonotone, ball-condensing pertubations of c -accretive maps and maps of semibounded variation and of type (M) are also given.