for each x [member of] [a, b], provided f is of bounded variation on [a, b], while u: [a, b] [right arrow] R is r-H-Holder continuous, i.

for each x [member of] [a, b], provided f is of bounded variation on [a, b] while u: [a, b] [right arrow] R is r-H-Holder continuous.

Let TV[0,1], respectively AC[0,1], denote the class of all functions of

bounded variation, respectively the absolutely continuous functions on [0,1].

Among the topics are variation on the p-Laplacian, extremal functions in Poincare-Sobolev inequalities for functions of

bounded variation, homocline type solutions for a class of differential equations with periodic coefficients, the cooperative case of quasilinear and singular systems, weighted asymmetric problems for an indefinite elliptic operator, multiple non-trivial solutions of the Dirichlet problem for the prescribed mean curvature equation, and existence of nodal solutions for some nonlinear elliptic problems.

As our main results will demonstrate, the perimeter which can be computed from the covariogram is the one from the theory of functions of

bounded variation (Ambrosio et al.

The results are stronger than that in paper [5], meanwhile the results are essential generalization of continuous dependence of

bounded variation solutions on parameters for Kurzweil equations in paper [6].

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (where the sum is taken over {b'} which form a finite Borel partition of b, for all Borel sets bin B; then [mu] is said to be of

bounded variation.

By definition for [PHI] (u) = u we have the ordinary

bounded variation V (f, [DELTA]) of the function f on a segment [DELTA].

4), we briefly review functions of

bounded variation.

Let TV [0,1] denote the class of all functions of

bounded variation on [0,1], i.

b] [PSI] (t) df (t), which exists, since f is of

bounded variation and [PSI] is differentiable on (a, b).

In this paper, we establish some weighted generalizations of open Newton-Cotes type inequalities for mappings of

bounded variation, and give several applications for r-moment, expectation of a continuous random variable.