On the existence of zeros of monotone operators in reflexive Banach spaces
Let \(E\) be a reflexive, strictly convex and smooth Banach space with dual \(E^*\), \(C\) be a nonempty bounded closed convex subset of \(E\) and \(T: E\to E^*\) be a hemicontinuous monotone mapping. The authors study the existence of zeros \(x\in C\) of the nonlinear equation \(T(x)= 0\). For this, they introduce a new sufficient condition of the following kind: \[ \text{inf}_{y\in C} V(y,(J- T)x)< V(x,(J- T)x)\quad \forall x\in C: J^{-1}(J- T)x\not\in C, \] where \(J: E\to E^*\) is the duality mapping of \(E\) and \(V:E\times E^*\to\mathbb{R}_+\) is the (generalized) quadratic distance function defined by \(V(x, x^*)= \| x\|^2- 2\langle x,x^*\rangle+\| x^*\|^2\). In the first part of the paper, this condition is characterized and compared with other known conditions. Moreover, examples of mappings which satisfy this new condition are provided. In the second part, the authors derive the existence theorem for zeros by means of this condition, using a known fixed point theorem of Takahashi. The result is used for the derivation of a surjectivity assertion for the mapping \(T\) and for a fixed point theorem for nonexpansive mappings in Hilbert spaces. In the last part, the authors present a sufficient and necessary condition for the existence of zeros of maximal monotone set-valued mappings.
scientific article; zbMATH DE number 5077912