Properties of interval-valued function space under the gH-difference and their application to semi-linear interval differential equations

Advances in Difference Equations

Zhuhong Zhang¹  Juan Tao² 

[1] Department of Big Data Science and Engineering, College of Big Data and Information Engineering, Guizhou University, Guiyang, P.R. China;Department of Mathematics, College of Science, Guizhou University, Guiyang, P.R. China

关键词: interval-valued function space;    Hausdorff-Pompeiu metric;    gH-derivative;    semi-linear interval differential equation;    strong solution;    65G40;    46C99;    34A12;   

DOI  :  10.1186/s13662-016-0759-9

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

The conventional subtraction arithmetic on interval numbers makes studies on interval systems difficult because of irreversibility on addition, whereas the gH-difference as a popular concept can ensure interval analysis to be a valuable research branch like real analysis. However, many properties of interval numbers still remain open. This work focuses on developing a complete normed quasi-linear space composed of continuous interval-valued functions, in which some fundamental properties of continuity, differentiability, and integrability are discussed based on the gH-difference, the gH-derivative, and the Hausdorff-Pompeiu metric. Such properties are adopted to investigate semi-linear interval differential equations. While the existence and uniqueness of the (i)- or (ii)-solution are studied, a necessary condition that the (i)- and the (ii)-solutions to be strong solutions is obtained. For such a kind of equation it is demonstrated that there exists at least a strong solution under certain assumptions.

【 授权许可】

CC BY