The results about \(L\)-semi-inner and \(T\)-semi-inner product spaces [see G. Lumer, Trans. Am. Math. Soc. 100, 29-43 (1961; Zbl 0102.327); R. A. Tapia, Proc. Am. Math. Soc. 41, 569-574 (1973; Zbl 0286.46025)] are used for obtaining some theorems of surjectivity for a class of nonlinear operators mapping a Banach space \(X\) into its dual.