Let $K$ be a closed convex subset of a Hilbert space $H$ and $T:K\u22b8K$ a nonexpansive multivalued map with a unique fixed point $z$ such that $\left\{z\right\}=T\left(z\right)$. It is shown that we can construct a sequence of approximating fixed points sets converging in the sense of Mosco to $z$.

We establish new existence results for nontrivial solutions of some integral inclusions of Hammerstein type, that are perturbed with an affine functional. In order to use a theory of fixed point index for multivalued mappings, we work in a cone of continuous functions that are positive on a suitable subinterval of $[0,1]$. We also discuss the optimality of some constants that occur in our theory. We improve, complement and extend previous results in the literature.

Utilizing the theory of fixed point index for compact maps, we establish new results on the existence of positive solutions for a certain third order boundary value problem. The boundary conditions that we study are of nonlocal type, involve Stieltjes integrals and are allowed to be nonlinear.

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