Partial Gateaux and Frechet Derivatives and Applications to Variational Analysis

In this paper, we define partial Gateaux and Frechet derivatives for multivariable and single-valued mappings between Banach spaces.

We will prove some properties of partial Gateaux and Frechet derivatives.

By these definitions, we find the explicit formulas for some polynomial type operators with two variables from lp by lp to lp with respect to each variable.

Then, we will introduce the concepts of generalized partially critical points and partial ordered extrema in partially ordered Banach spaces.

By these concepts, we will investigate the connection between generalized partially critical points and partial ordered-extrema of two-variable and single-valued mappings in partially ordered Banach spaces.

These results extend the connection between critical points and extrema of real valued functions in calculus.

We will give some applications of partial Gateaux and Frechet derivatives to ordered optimizations and variational inequality problems between partially ordered Banach spaces.

Finally, we study the connection between partial Gateaux and Frechet derivatives and ordered monotone of two-variable and single-valued mappings in partially ordered Banach spaces.