Oscillation Of Several Types Of Fractional Differential Equations

The development of the oscillatory theory of differential equations benefits from the comparison theorem and separation theorem of the zero point distri-bution of solutions of homogeneous second order linear differential equations established by G.sturm.Because the oscillatory theory of differential equations have a profound physical background and mathematical model,making it an important branch of the qualitative theory of differential equations.With the continuous development of all kinds of new technology and pro-duction mode change,in many practical application about the oscillation and non-oscillation of solutions.Because of the theoretical basis of fractional or-der calculation and the good development prospect,many scholars have made in-depth research on the oscillation of fractional order differential equations in recent decades,and many excellent results and useful conclusions have been obtained.Moreover,the future of the oscillations of fractional order differential equations is still very broad,so this paper will continue to study this part.This paper refers to a large number of scholars' literature,discusses the oscillation of several kinds of fractional-order differential equations,mainly use the generalized Raccati transformation,the integral average technology,and cleverly use the variable substitution,obtains some new vibration criteria,and gives some applications.This paper includes the following four parts:Chapter 1 Preference.In this chapter,the main equations and some related definitions and properties are introduced.Chapter 2 We mainly discuss in the following conditions:?t0?f-1(exp(-?t0sp(v)dv))ds=?,and?t0?f-1(exp(-?t0sp(v)dv))ds<?.the oscillatory of the following equation:D-?+1y(t)·D-?y(t)-p(t)f(D-?y(t))+q(t)h(?t?(s-t)-?y(s)ds)=0,where 0<?<1 is a real number.D?y is the Liouville right-sided fractional derivative of y.By using the generalized Raccati transformation and the integral average technique,we study the oscillation of the above equations,and the results in many literatures are generalized.Chapter 3 We mainly study the oscillation behavior of fractional differ-ential equations with nonlinear terms in the form of:Dt?[?(t)k1(x(t),Dt?x(t))+k2(x(t),Dt?x(t))]-p(t)k3[x(t),Dt?x(t)]Dt?x(t)+F(t,x(t),x(?(t)))=e(t),for t?t0>0 and 0<?<1,where Dt?(·)denotes the Modified Riemann-Lioville fractional derivative with respect to the variable t.By using the properties of the Riemann-Liouville derivatives,variable substitution,and operator method,the oscillation of the solution is obtained.Chapter 4 We consider the oscillatory behavior about linear conformable fractional differential equations of Kamenev type of the following(p(t)y(?+1)(t))(?)+y(?+1)(t)+q(t)y(t)=0,t?t0,where p?C([t0,?),(0,?)),q?C([t0,?),R),0<??1.By using the special properties of conformable fractional derivatives,gen-eralized Raccati transformation and integral average technique,the oscillation of the solution is obtained.