Ostrowskiʹs inequality for vector-valued functions and applications

In this paper, we obtain some Ostrowski type inequalities for vector-valued functions which both generalise and improve the scalar case and show that the midpoint inequality is the best possible inequality in the class. A quadrature formula of the Riemann type for the Bochner integral and the error bounds are considered. Applications to operator inequalities and numerical approximation for the mild solution of inhomogeneous vector-valued differential equations are also pointed out. In the last section, a numerical example is analysed.