In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{On fixed point theorems for monotone increasing vector valued mappings via scalarizing}, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

in this paper, we develop two piecewise polynomial methods for the numerical evaluation of cauchy principal value integrals of oscillatory kind. the two piecewisepolynomial quadratures are compact, easy to implement, and are numerically stable. two numerical examples are presented to illustrate the two rules developed, the convergence of the two schemes is proved and some error bounds obtained,