Numerous approximations of Riemann-Stieltjes double integrals

Abstract

The concept of Riemann-Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$; where $f$ is called the integrand, $u$ is called the integrator, plays an important role in Mathematics. The approximation problem of the Riemann-Stieltjes integral $\int_a^b {f\left( t \right)du\left( t \right)}$ in terms of the Riemann-Stieltjes sums have been considered recently by many authors. However, a small attention and a few works have been considered for mappings of two variables; i.e., The approximation problem of the Riemann-Stieltjes double integral $\int_a^b {\int_c^d {f\left( {t,s} \right)d_s d_t u\left( {t,s} \right)} }$ in terms of the Riemann-Stieltjes double sums. This study is devoted to obtain several bounds for $\int_a^b {\int_c^d {f\left( {t,s} \right)d_s d_t u\left( {t,s} \right)} }$ under various assumptions on the integrand $f$ and the integrator $u$. Mainly, the concepts of bounded variation and bi-variation are used at large in the thesis. Several proposed cubature formula are introduced to approximate such double integrals. For mappings of two variables several inequalities of Trapezoid, Gr\"{u}ss and Ostrowski type for mappings of bounded variation, bounded bi-variation, Lipschitzian and monotonic are introduced and discussed. Namely, Trapezoid-type rules for $\mathcal{RS}$-Double integrals are proved, and therefore the classical Hermite-Hadamard inequality for mappings of two variables is established. A Korkine type identity is used to obtain several Gr\"{u}ss type inequalities for integrable functions. Finally, approximating real functions of two variables which possess $n$-th partial derivatives of bounded bi-variation, Lipschitzian and absolutely continuous are established and investigated.