New Poisson\u2013Sch type inequalities and their applications in quantum calculus

Tao Liu,Chen Xin-Juan ,Yifan Xing

doi:10.1186/s13660-018-1735-6

Abstract

The Poisson type inequalities, which were improved by Shu, Chen, and Vargas-De-Teón (J. Inequal. Appl. 2017:114, 2017), are generalized by using Poisson identities involving modified Poisson kernel functions with respect to a cone. New generalizations of improved Poisson–Sch type inequalities are obtained by using the generalized Montgomery identity associated with the Schrödinger operator. As applications in quantum calculus, we estimate the size of weighted Schrödingerean harmonic Bergman functions in the upper half space.

Highlights

The Poisson–Sch inequality problem has many applications, e.g., second-order irreversible reactions, obstacle problems, the diffusion problem involving Michaelis–Menten, and reservoir simulation, see, for example, [11, 16,17,18] and the references therein for details
Various extensions and generalizations of the classical variational inequality models and complementarity problems have emerged in mechanics, nonlinear programming, physics, optimization and control, economics, transportation, finance, structural, elasticity, and applied sciences; see [7, 12, 17, 18] and the references therein for more details
There are a number of numerical methods, such as descent and decomposition, neutral differential equations, for the solution of Poisson–Sch inequality models and complementarity problems [16, 17]

Summary

Introduction

The Poisson–Sch inequality problem has many applications, e.g., second-order irreversible reactions, obstacle problems, the diffusion problem involving Michaelis–Menten, and reservoir simulation, see, for example, [11, 16,17,18] and the references therein for details. There are a number of numerical methods, such as descent and decomposition, neutral differential equations, for the solution of Poisson–Sch inequality models and complementarity problems [16, 17]. Based on the assumption of the convex set, the sequence generated by the proposed method converges to the unique solution of the Poisson–Sch inequality problem. The auxiliary principle method has been used to the Poisson–Sch inequality problem, the origin of which can be traced back to the reference by Lions and Stampacchia [16].

Results

Conclusion