(\(GDDVIP;\lambda\)) and the variable step iterative method for \(T\)-\(\eta-\)invex function of order \(\lambda\) in Hilbert spaces (Q2860991)

The authors study a variational inequality problem: Find \(x_0 \in K\) such thatNEWLINENEWLINE(VIP) \(\langle T(x_0), x-x_0 \rangle \geq 0\) \(\forall x \in K\),NEWLINENEWLINEwhere \(X\) is a reflexive real Banach space with its dual \(X^*\), \(K\) is a nonempty subset of \(X\), \(T:K \rightarrow X^*\) is a nonlinear mapping. The pair \(\langle f,x\rangle\) denotes the value of \(f \in X^*\) at \(x \in K\).NEWLINENEWLINEThe concept of invexity of a function brought a new edge to generalize the variational inequality problem which is a general case of an optimization problem, a complementarity problem and a fixed point problem. For the generalization of the differentiable invex function, the authors introduce the \(T-\eta\)-invex as an operator invex function in ordered topological vector spaces, \(H\)-differentiable manifolds, \(n\)-manifolds and \(S^n\).NEWLINENEWLINEThe iterative method for \(T-\eta\)-invex function of order \(\lambda\) is proposed. For general purpose, the authors have obtained the numerical solution for the \(T\)-\(\eta\)-invex function of order \(\lambda\) using the variable step iterative method with some additional conditions. The authors prove an existence theorem of \(T\)-\(\eta\)-invex function of order \(\lambda\) followed by a concrete example.