Some New Fractional Integral Inequalities And Their Applications

With the development of the theory of differential equation, integral inequality has been widely promoted. In recent decades, the theory of fractional differential calculus has gotten many important achievements in mathematics, signal processing systems, thermal and optical systems and other areas, which also have received great attention from scholars at domestic and aboard. Many practical problems can be abstracted into fractional equation model when it is combined with the classical theory of ordinary differential equation, and we have gotten a series of valuable results with the related researches. As an important tool, fractional integral inequality has gotten more attention of mathematicians in the study of the properties of solution about fractional differential equation. Various types of integral inequality and its promotion play an important role in the study of uniqueness, boundedness and continuous dependence on the initial values of solution about fractional differential equation.This paper is inspires by [2,3,11,17,30,31], we extend some known integral inequalities to fractional integral inequalities, then obtain some new results.The thesis is divided into four chapters according to the contents.Chapter 1 We introduce the background and the main contents of this paper.Chapter 2 Based on some known integral inequalities in [2], this chapter gives the existence of certain new results: restrictions on the right side of the equation by the different forms of the function f’ and the exponential function, we get the bound of the left side of the equation.Chapter 3 We study some new Gronwall-Bellman inequalities under the definition of the modified Riemann-Liouville fractional derivative, we extended them into the following integral inequalities: hen we apply those inequalities to the research on the boundedness, uniquencess and continuous dependence on the initial for solutions to fractional differential equations.Chapter 4 By using the properties of modified Riemann-Liouville fractional deriva-ive, some new delay integral inequalities have been studied: hen we offered explicit bounds for the unknown functions and applied the result to he research concerning the boundness, uniqueness and continuous dependence on the initial for solutions to certain fractional differential equations.