Linear Multi-Dimensional Continuous Integral Inequalities

Yuming Qin
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Yuming Qin: Donghua University, Department of Applied Mathematics College of Science

Chapter Chapter 5 in Integral and Discrete Inequalities and Their Applications, 2016, pp 449-726 from Springer

Abstract: Abstract Gronwall’s one dimensional inequality (Theorem 1.1.1 ) [239], also known in a generalized form as Bellman’s lemma [61], has been extended to several independent variables by different authors. For example, in [140] Conlan and Diaz obtained a generalization of Gronwall’s inequality in n variables in order to prove uniqueness of solutions of a nonlinear partial differential equation. Walter [636] gave a more natural extension of Gronwall’s inequality in any number of variables by using the properties of monotone operator. By using the notion of a Riemann function, Snow [603] obtained corresponding inequalities in two independent variables for scalar and vector functions. It turns out to be that Snow’s technique in the scalar case can be employed to establish Gronwall’s inequality in n independent variables which coincides with the result given in [636] when a representation of the Riemann function is used.

Date: 2016
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DOI: 10.1007/978-3-319-33301-4_5

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