The approximate fixed point property in Hausdorff topological vector spaces and applications

Abstract

Let l be a compact convex subset of a Hausdorfftopological vector space $(\mathcal{E},\tau)$ and $\sigma$ anotherHausdorff vector topology in $\mathcal{E}$. We establish an approximatefixed point result for sequentially continuous maps f:(l,$\sigma$)$\to$ (l,$\tau$). As application, weobtain the weak-approximate fixed point property for demicontinuousself-mapping weakly compact convex sets in general Banach spaces and use this toprove new results in asymptotic fixed point theory. These resultsare also applied to study the existence of limiting-weak solutions for differentialequations in reflexive Banach spaces.