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...

this paper studies a fractional boundary value problem of nonlineardifferential equations of arbitrary orders. new existence and uniquenessresults are established using banach contraction principle. other existenceresults are obtained using schaefer and krasnoselskii fixed point theorems.in order to clarify our results, some illustrative examples are alsopresented.

Lidskii's theorem on eigenvalue perturbation is proved via a non-smooth mean value theorem.

The purpose of this paper was to prove formally, using the Mizar language, Arithmetic Mean/Geometric Mean theorem known maybe better under the name of AM-GM inequality or Cauchy mean theorem. It states that the arithmetic mean of a list of a non-negative real numbers is greater than or equal to the geometric mean of the same list. The formalization was tempting for at least two reasons: one of ...

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...

1. Differentiability Versus Continuity 2 2. Differentiation Rules 3 3. Optimization 7 3.1. Intervals and interior points 7 3.2. Functions increasing or decreasing at a point 7 3.3. Extreme Values 8 3.4. Local Extrema and a Procedure for Optimization 10 3.5. Remarks on finding roots of f ′ 12 4. The Mean Value Theorem 12 4.1. Statement of the Mean Value Theorem 12 4.2. Proof of the Mean Value Th...