Existence And Stability Of Solutions For Caputo-Tempered Fractional Differential Equations

Fractional calculus is an important filed in applied mathematics and it is extension of the integer calculus and derivatives.It was first discussed in 1695 in a letter between L’Hosptial and Leibniz.Fractional and fractional differential equations have been applied widely to mathematics,physics,anomalous diffusion and material science in the past four decades.Up to now,more than one definition of fractional derivatives is proposed in the published studies,among,which the Liouville-Riemanns and Caputo’s fractional derivatives and integral are the most classical ones,and the investigation on the theory of fractional differential equations focus mainly on this type of the fractional derivatives and integral.At the same time,the concepts of the above-mentioned derivatives and integrals are also developed in applications,among which tempered fractional derivatives are recently introduced;and the studies of theory and applications of the differential equations involving the tempered fractional derivatives have become one of the important tropics.In this thesis,we mainly study the stability and existence of the solutions to the Caputo-type tempered fractional differential equations.It is divided into five chapters.The first chapter introduces the research background and current situation of Caputo-tempered fractional differential equations,and briefly summarizes the main work and results of this thesis.The second chapter introduces some notations,basic concepts and lemmas which are useful in this thesis.In the theoretical framework of Caputo-tempered fractional differential equations,Chapter three studies the stability of solutions in the space of absolutely continuous functions.The principle of comparison and the inequalities of the fractional derivative of composite functions are first established,on which the Lyapunov’s second method is generalized to the the case of the tempered fractional equations;and the sufficient conditions of the global existence of bounded solutions are derived and the criteria for the asymptotic stability of the solutions are obtained.In the fourth chapter,we discuss the asymptoticity of the solutions in the space of continuous functions.A principle of comparison is first established,by which the sufficient condition ensuring the separation of curves of solutions are derived.Finally,the criteria for the asymptoticity of the solutions are obtained by using of both principle of comparison and separation of curves of solutions and by means of the property of the Mittage-Leffler function and Gronwall inequalities,respectively.Based on Chapter three and four,we investigates in five Chapter the Ulam-Hyer stability of solutions to initial value problems of caputo-type tempered fractional differential equations.Using Gronwall inequality and the technique of inequalities,we obtain sufficient conditions to ensure the Ulam-Hyer stability of solutions defined on the bounded and unbounded intervals,respectively.The main results obtained in the thesis are new in the setting of the Caputo-tempered fractional differential equations.