An extension of Browder fixed point theorem and applications (Q2747010)

In [Math. Ann. 177, 283-301 (1968; Zbl 0176.45204)] \textit{F. E. Browder} proved the following theorem: If \(K\) is nonempty compact convex subset of a normed linear space \(X\) and \(T: K\to 2^K\) is a multivalued map such that:NEWLINENEWLINENEWLINE(i) for each \(x\in K\), \(T(x)\) is non-empty and convex,NEWLINENEWLINENEWLINE(ii) for each \(y\in K\), \(T^{-1}(y)\) is relatively openNEWLINENEWLINENEWLINEthen there exists an element \(x\in K\) such that \(x\in T(x)\).NEWLINENEWLINENEWLINEIn the present paper the authors proved the following extension of the results:NEWLINENEWLINENEWLINETheorem 3.1. Let \(K\) be a non-empty compact convex subset of a normed linear space \(X\). Let \(T_1: K\to 2^X\) and \(T_2: X\to 2^K\) be such thatNEWLINENEWLINENEWLINE(a) for each \(x\in K\), \(T_1(x)\) is non-empty and convex,NEWLINENEWLINENEWLINE(b) for each \(y\in X\), \(T^{-1}_1(y)\) is relatively open in \(K\),NEWLINENEWLINENEWLINE(c) \(T_2(y)\) is non-empty closed and convex for each \(y\in T_1(x)\) and \(x\in K\),NEWLINENEWLINENEWLINE(d) \(T_2\) is upper semicontinuous.NEWLINENEWLINENEWLINEThen there exists an \(x\in K\) such that \(x\in (T_2\circ T_1)(x)\).NEWLINENEWLINENEWLINEAs an application to the above theorem a variational inequality of Fan type is established and further a fuzzy coincidence theorem due to S. Chang is derived.