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In this paper, a vector optimization problem over cones is considered, where the functions involved are η-semidifferentiable. Necessary and sufficient optimality conditions are obtained. A dual is formulated and duality results are proved using the concepts of cone ρ-semilocally preinvex, cone ρ-semilocally quasi-preinvex and cone ρ-semilocally pseudo-preinvex functions.

In this paper, we introduce two types of proper quasimonotone maps over cones for a vector-valued bifunction and discuss their relations with generalized monotone maps, namely cone pseudomonotone and cone quasimonotone maps. Strong vector variational like inequality problems of the Stampacchia and the Minty type have been defined. These problems are a generalization of the classical Stampacchia and Minty problems and encompass many problems studied in the literature. A generalization of celebrated Minty lemma, relating the solutions of the two problems, has been proved. Existence results for strong Stampacchia and Minty type vector variational like inequality problems have been established using the notions of proper quasimonotone maps over cones. Gap functions have also been proposed for both problems.

In order to estimate the solution of the zero for the nonlinear systems, we conduct the local convergence investigation in this paper. In contrast to the Lipschitz condition used in the preceding study, we have used the Hölder continuity requirement. Additionally, we use a derivative approximation to take the derivative free iterative technique with the same order. A computed radius of convergence balls based on the Hölder constant is also provided. No Taylor's series approximation on a higher order Fréchet derivate is used in this investigation. To broaden the relevance of our work, a comparison of convergence ball radii is also provided. This highlights the uniqueness of this paper.