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# Fixed Point Theory under Weak Topology

DOI link for Fixed Point Theory under Weak Topology

Fixed Point Theory under Weak Topology book

# Fixed Point Theory under Weak Topology

DOI link for Fixed Point Theory under Weak Topology

Fixed Point Theory under Weak Topology book

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## 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.