Properties of the preinvex and quasi preinvex multifunctions (Q2762843)

Let \(Y\) be a real vector space and \(\eta\colon X\times Y\rightarrow Y\) be a function. A set \(E\subset Y\) is \(\eta\)-invex, cf. \textit{D. Bhatia and A. Mehra} [J. Math. Anal. Appl. 214, No. 2, 599-612 (1997; Zbl 0894.90142)], if and only if for \(x_1,x_2\in E\), \(\lambda\in[0,1]\), we have \(x_2+\lambda \eta(x_1,x_2)\in E\).NEWLINENEWLINENEWLINELet \(X\) be a nonempty \(\eta\)-invex subset of a real vector space \(Y\), \(Z\) a Hausdorff topological real vector space with a partial order \(\geq\) generated by a convex cone \(C\subset Z\) and \(F\colon X\rightarrow 2^Z\) a multifunction. Then \(F\) is \(C\)-\(\eta\)-preinvex if and only if for any \(x_1,x_2\in X\), \(\lambda\in[0,1]\), \(y_1\in Fx_1\), \(y_2\in Fx_2\), there exists \(y_3\in F(x_2+\lambda\eta(x_1,x_2))\) such that \(y_3\leq \lambda y_1+(1-\lambda)y_2\). \(F\) is quasi \(C\)-\(\eta\)-preinvex if and only if for any \(a\in Z\), the set \(\{x\in X\mid\) there exists \(y\in F(x)\) with \(y\leq a\}\) is \(\eta\)-invex.NEWLINENEWLINENEWLINEIn this paper the author proves that a \(C\)-preinvex multifunction on is also quasi \(C\)-preinvex and the composition of an increasing convex function with a \(C\)-preinvex multifunction is also \(C\)-preinvex.