Nonlinear complementarity problem of mathematical programming in Banach space (Q1093556)

Short proofs of some existence theorems for the nonlinear complementarity problem in mathematical programming in a reflexive real Banach space B are given for an arbitrary closed convex cone K under much weaker assumptions (in the absence of boundedness) of the operator T from K into the dual space \(B^*\). The result proved is: Theorem. Let \(T: K\to B^*\) be hemicontinuous and monotone. Then there exists an \(x_ 0\in K\) such that \(x_ 0\in K\), \(Tx_ 0\in K^*\), \((Tx_ 0,x_ 0)\) (where \(K^*\) denotes the dual or polar of K) under each (any one) of the following conditions: (i) T is coercive (in particular T is \(\alpha\)-monotone), (ii) \(T0\in K^*\) (in particular \(T0=0),\) (iii) for some \(r>0\) there exists an \(u\in D^ 0_ r=\{x\in K:\| x\| <r\}\) with (Tx,x-u)\(\geq 0\) for all \(x\in S_ r=\{x\in K:\| x\| =r\}.\) Meanwhile the author has also proved that the result is also true for (iv) there exists an \(x\in K\) with Tx\(\in int K^*.\) It should also be noted that the result is not true if there exists \(x\in K\) with \(Tx\in K^*\), as it was claimed by \textit{A. T. Dash} and \textit{S. Nanda} [J. Math. Anal. Appl. 98, 328-331 (1984; Zbl 0547.90099)].