Approximate Fixed Point Theory for Countably Condensing Maps and Multimaps and Applications

Abstract

In this paper, we establish an approximate fixed point theorem for Γ-τ-sequentially continuous countably ${\Phi }_{\tau }$-condensing mappings. As application, we obtain an approximate fixed point result for demicontinuous mappings and use this to prove new results in asymptotic fixed point theory. Moreover, we obtain new fixed point results for some countably ${\Phi }_{\tau }$-condensing mappings and multivalued mappings and fixed point results of Krasnoselskii-Daher type for the sum of two τ-sequentially continuous mappings in non-separable Hausdorff topological vector spaces. We also present a multivalued version of an approximation result of Ky Fan for τ-Γ-s.l.sc multivalued mappings. Apart from that, we show the applicability of our results to the theory of Volterra integral equations in Banach spaces and we prove the existence of limiting-weak solutions for differential equations in Banach spaces not necessarily reflexive. Our results extend the results of Banaś and Ben Amar, Barroso and Seda.

Keywords:

• Fixed point
• approximate point
• τ-measure of noncompactness
• τ-sequentially continuous
• countably ${\Phi }_{\tau }$-condensing
• τ-weakly closed graph
• τ-closed graph

MATHEMATICS SUBJECT CLASSIFICATION:

• 47H09
• 47H10
• 47H30