Measures of Weak Noncompactness and Fixed Points

Abstract

The interaction between measures of weak noncompactness and fixed point theory is really strong and fruitful. In particular, measures of weak noncompactness play a significant role in topological fixed point problems. The purpose of this chapter is to exhibit the importance of the use of measures of weak noncompactness in topological fixed point theory and to demonstrate how the theory of measures of weak noncompactness will be applied in integral and partial differential equations. The theory of measures of weak noncompactness was initiated by De Blasi in the paper [28], where he introduced the first measure of weak noncompactness. De Blasi’s measure can be regarded as a counterpart of the classical Hausdorff measure of noncompactness. Unfortunately, it is not easy to construct formulas which allow to express the measure of weak noncompactness in a convenient form. For this reason, measures of weak noncompactness have been axiomatized [12] allowing thus several authors to construct measures of weak noncompactness in several Banach spaces [7, 11, 47, 48]. Measures of weak noncompactness have been successfully applied in operator theory, differential equations and integral equations. In particular, they enabled several authors to dispense with the lack of weak compactness in many practical situations. The material is far from exhausting the subject and basically we do not go into profound applications.