Study Of Properties And Applications Of Fractional Integral Operator

Fractional Calculus originated in the era of Leibniz, and it has been greatly de-veloped in recent decades. Fractional models provide an excellent instrument for thedescription of memory and hereditary properities of various materials. In the fieldsof viscoelastic, controlled systems, electrochemistry, etc, scientists have proposed andstudied a large number of fractional order models.In the first chapter, we introduce some of the history and current conditions inthe Fractional Calculus and Fractional Di?erential Equations. At the first , we givea detailed description of the fractional calculus applied in various engineer fields andother subjects, Then we also introduce some past work about existence and uniquenessof solution of fractional di?erential equation.In the second chapter, according to the related properties of the Riesz potential,we point out that the fractional integral operator Iλis bounded and compact fromLp(0, T ) to Lq(0, T ). Then, we used the basic tools of functional analysis to studythe fractional integral operator's boundness and compactness from weighted L_p spaceto weighted Lq space. And, we partly discuss the limiting cases of the indexs p, q,β.When p, q,βsatisfy certain conditions, the relevant spaces could even be embeddedinto H o¨lder spaces.In the third chapter, we apply these results to a class of nonlinear fractionaldi?erential equation. Through using Schauder fixed point theorem, we get the existenceof solutions in L_p(0, T ) and L_β~p(0, T ), then the regularity of solutions is discussed.