Fractal Dimensions of Fractional Integral of Continuous Functions

作     者：Yong Shun LIANG Wei Yi SU 

作者机构：Institute of ScienceNanjing University of Science and Technology Department of MathematicsNanjing University 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2016年第32卷第12期

页      面：1494-1508页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Supported by National Natural Science Foundation of China(Grant Nos.11201230 and 11271182) 

主　　题：Holder condition fractional calculus fractal dimension bound variation 

摘      要：In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann-Liouville integral of a continuous function f（x） of order v（v 〉 0） which is written as D-Vf（x） has been proved to still be continuous and bounded. Furthermore, upper box dimension of D-v f（x） is no more than 2 and lower box dimension of D-v f（x） is no less than 1. If f（x） is a Lipshciz function, D-v f（x） also is a Lipshciz function. While f（x） is differentiable on [0, 1], D-v f（x） is differentiable on [0, 1] too. With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann-Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f（x） satisfying HSlder condition, upper box dimension of Riemann-Liouville fractional integral of f（x） seems no more than upper box dimension of f（x）. Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann-Liouville integral of a continuous function satisfying HSlder condition of order v（v 〉 0） is strictly less than 2 - v. Riemann-Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl-Marchaud fractional derivative of certain continuous functions have been estimated.