Study On Existence, Uniqueness And Globally Attractivity Of Solutions Of A Volterra Integral Equation Of Fractional Order

In this paper, by using fxed point theory and a new method, we study thefollowing Volterra integral equation of fractional order:x（t）=p（t）+f （t, x（t））/Γ（α） t0u（t, s, x（s））/（t s）^1-αds, t∈R₊:=[0,∞)where1> α>0, f∈C（R+×R, R）, u∈C（R₊×R₊×R, R）, and p∈C（R₊）:=C（R₊, R） is a bounded function, and we obtain two results whichrefer to the existence, uniqueness and globally attractivity of solutions of theabove Volterra integral equation of fractional order.To prove that our obtained results are interesting, we recall the related knownresults and introduct the new method, this method is applied to this equationfrstly（see chapter1）.This paper is divided into6chapters. In chapter1, we introduce the researchbackground, the most recent results for properties of solutions of the Volterra inte-gral equation of fractional order, the new method, our main results, the provingmethod, the structure of this paper; In chapter2, we introduce some relatedmarks which are used in the paper, some defnitions and some lemmas; In chap-ter3, we study the properties of solutions of the above Volterra integral equationof fractional order and obtain an existence and globally attractivity result forthe above Volterra integral equation of fractional order; In chapter4, we studythe properties of solutions of the above Volterra integral equation of fractionalorder and obtain an existence and uniqueness result for the above Volterra inte-gral equation of fractional order; In chapter5, we apply our conclusions by twoexamples; In chapter6, we summarize the paper.