Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications. (Q1399381)

Let \(E\) be a Banach space with a partial ordering \(\leq\) induced by a cone \(P\) of \(E\). Fix \(u_0\in E\) and put \(K=\{x\in E:x\geq u_0\}\). The author establishes two fixed point theorems for multivalued increasing operators with nonempty weakly closure values \(A:E\rightarrow E\) satisfying the following conditions: (i) \(Ax\) is a totally ordered subset of \(E\) for any \(x\in K\); (ii) \(Au_0\subset K\); (iii) if \(C=\{x_n\}\subset K\) is countable, totally ordered and \(C\) is contained either in the weak closure or in the strong closure of the set \(\{x_1\}\cup A(C)\), then \(C\) is weakly relatively compact. As application of one these fixed point theorems, the author obtains an existence theorem for the solutions of Hammerstein type integral inclusions of the form \(u(t)\in\int_0^T k(t,s) g(s,u(s))\,ds\) almost everywhere on \([0,T]\).