| Based on interval analysis theory,inequality theory,and calculus theory,this paper discusses the continuous and discrete Opial-type inequalities,and extends some classic Opial-type inequalities.At the same time,the papers published by Costa et al in 2019 and Zhao et al in 2022 in Fuzzy Set and System were also generalized.The conclusions obtained provide a powerful tool for solving problems of interval differential equation and interval difference equation.Among them,this paper is divided into six chapters.The first chapter introduces the research background and current situation,the research content and the structure of the paper,the main results and significance.The second chapter reviews the basic concepts and theorems of interval analysis.In this paper,the basic operation of interval,the definition and basic properties of the interval-valued function of one variable and two variables are given.In Chapter 3,based on H?lder’s inequality,Young’s inequality,gH-derivative of interval numbers and the basic theory of interval analysis,some new interval sequence Opial-type inequalities are obtained,some classical Opial-type inequalities are generalized and some examples are given to verify the conclusions.The fourth chapter based on Cauchy-Schwarz inequality and quasi-norm,we obtain some new interval Opial-type inequality.This paper generalizes the classical Opial inequality,improves the coefficients of the right end of the interval Opial inequality by Costa and Zhao.Several examples are given.In Chapter 5,based on the integral theory of interval-valued functions in two variables,some results published by Pachpatte in the Journal of Mathematical Analysis and Applications on Opial inequality in two variables are generalized,and some examples are given to verify the results obtained.The sixth chapter is the summary of this paper and the prospect of future research. |
