The purpose of this paper is to present some new contraction mappings via control functions. In addition, fixed point results for Θ,α,θ,Ψ contraction, rational and almost are obtained. Moreover, under types (I), (II), (III) Θ,θ,Ψυ0, several circle solutions provided in the setting a G-Metric space. Our extend, unify, generalize many previously published papers direction. examples show reliabili...

Let (X,m, H) be a hereditary m-space. A subset of X is said to θ-H-compact relative if for every cover U by m(θ)-open sets X, there exists finite U0 such that \ ∪ ∈ H. We obtain several properties these sets. Also, we define and investigate two kinds strong forms “θ-H-compact X”.

This supplementary appendix contains additional materials for the paper “Robustness and Separation in Multidimensional Screening.” Section B contains proofs of auxiliary results not included in the main paper. Section C details how the generalized virtual values coincide with traditional (ironed) virtual values in the single-good monopoly problem. Theorems, equations, and sections in the main p...