Sentence examples for proved various from inspiring English sources

Niculescu [7] introduced the concept of relative convexity and proved various properties and generalizations of classical results for relative convexity.

In the past years, many researchers have proved various results on the theory of nonexpansive mappings (or contractions).

In recent past, several authors (e.g., [18 31]) proved various fixed point theorems employing relatively more general contractive conditions.

In [12], the authors defined expansive maps in (N -cone metric spaces aN -coneed various fixed point theoremetric

Wolfe [30] associated a dual problem with a primal nonlinear programming problem and proved various duality theorems under the assumptions of convexity.

In subsequent papers several authors proved various coupled and common coupled fixed point theorems (e.g., [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]).

In this section, we prove various gradient estimates for admissible solutions u of the oblique problem (1.1)–(1.1).

For the case of real invariants, the authors prove various monotonicity results for the means with respect to the invariants.

We also prove various improved Hardy inequalities on mean convex domains along the line of Brezis and Marcus (1997) [7].

This, together with standard techniques for the complex Monge Ampère operator, e.g. integrating by parts, is often used to prove various properties of the Green function.

In the first part of the paper we prove various results on regularity of Feynman Kac functionals of Hunt processes associated with time-dependent semi-Dirichlet forms.