S-asymptotically ω-periodic Solutions And Bifurcation Of Two Class Of Fractional Derivative-Integral Equations

In the first part of this paper, we will study the existence and uniqueness of S-asymptotically ω-periodic solutions of the following integral-differential equation of Riemann-Liouville type: Where,1<α<2, A:D(A)∈X→X is a linear densely operator of sector type on Banach space X.f is a continuous function satisfying a suitable Lipshcitz condition.We will use the contraction mapping theory, divide the integral interval, and substitute the integral variable, to prove the function has a unique S-asymptotically ω-periodic solutions, when the function f satisfies Lipshcitz condition; On the other hand, we will take advantage of Gronwall-Bellman inequality to estimate the unique S-asymptotically ω-periodic solutions. When we take advantage of Gronwall-Bellman inequality, we can zoom inequality until it satisfies the conditions of Gronwall-Bellman inequality.When the L in the Lipshcitz condition is not a constant but a continuous function, we take advantage of lemma3.1and three mappings constructed to meet the conditions of the lemma3.1, such that one mapping has a unique fixed point.The second part of this paper will concerns the bifurcation of the following fractional differential equation of Caputo: where n-1<p≤n,m-1<μ≤m, n and m are positive integers, by linearization techniques, for the above problem at the neighborhood of equilibrium point of its, the characteristic equation and the necessary and sufficient condition for the solution of characteristic equation are founded, Moreover, all possible parameters influencing on the solutions are discussed, we define the1:2resonant double Hopf point and find the members of the set.This paper is divided into three chapters:The first chapter introduces the development history and the research significance of fractional calculus. The second chapter introduces the definition of S-asymptotically ω-periodic function, we proof the function has a unique S-asymptotically ω-periodic solutions and estimate the unique S-asymptotically ω-periodic solutions.The third chapter introduces the definition of R-L and Caputo fractional calculus, we discuss all parameters influence on the solutions, we define the1:2resonant double Hopf point and find the members of the set.