ABSTRACT
In the first part of this chapter, we give some variants of the Schauder and
Krasnosel’skii fixed point theorems in Dunford-Pettis spaces for weakly com-
pact operators. Precisely, if an operator A acting on a Banach space X having
the property of Dunford-Pettis, leaves a subset M of X invariant, then A has, at least, a fixed point in M. In addition, if B is a contraction map of M into X , Ax+By ∈ M for x, y in M and if (I −B)−1A is a weakly compact operator, then A + B has, at least, a fixed point in M. Both of these two theorems can be used to resolve some open problems (see Chapter 5).
In the second part of this chapter, we establish new variants of fixed point
theorems in general Banach spaces. Furthermore, nonlinear Leray-Schauder
alternatives for the sum of two weakly sequentially continuous mappings are
presented. This notion of weakly sequential continuity seems to be the most
convenient in use. Moreover, it is not always possible to show that a given
operator between Banach spaces is weakly continuous. Quite often, its weakly
sequential continuity presents no problem. Finally, we establish fixed point
theorems for multi-valued maps with weakly sequentially closed graphs.
