Asymptotically Regular Mappings

Abstract

We shall study in this chapter the existence of fixed points for a different class of mappings, called asymptotically regular mappings. The concept of asymptotically regular mappings is due to Browder and Petryshyn [BP]. Some fixed point theorems for this class of mappings can be found in [Gr1], [Gr2] and references therein. The fixed point theorems which we shall study are based upon results in [DX]. As we shall see, there is a strong connection between these results and those in Chapter VIII. In particular, in some of them the role of the Clarkson modulus of convexity will be played by the moduli of near uniform convexity. In Section 1 we define a new geometric coefficient in Banach spaces which plays the role of the Lifshitz characteristic for asymptotically regular mappings, and we prove the corresponding version for these mappings of Theorem VIII.1.4. In Section 2 we study some relationships between the new coefficient and either the modulus of NUC or the weakly convergent sequence coefficient. We also find a simpler expression for the new coefficient in Banach spaces with the uniform Opial property. Moreover we prove that, in contrast to the Lifshitz characteristic, the new coefficient is easy to compute in lp-spaces. We recall that the Lifshitz characteristic is only known in some renorming of Hilbert spaces.