| Fractional calculus is a theoretical study of differential and integral for arbitrary order,it extends and expands on the basis of integer order calculus.Compared with the traditional integral order calculus,fractional calculus has a clearer and more accurate description of the characterization of complex systems.And the mathematical model established by fractional calculus is more flexible.With the numerous applications of the fractional calculus in the field of biomedical,chaos theory,rheology,optics,economics,etc.,mathematical researchers have shown great interest in the qualitative research of fractional differential equations abstracted from practical problems,such as the existence,uniqueness,stability,oscillation of initial value problems and boundary value problems.These theoretical studies efficiently solve the complex problems in the real world,so the theoretical study of fractional differential equations is of great significance in solving practical problems in many fields.In recent years,in the study of fractional calculus theory,we find that integral inequalities(such as:Opial-type inequality,Gronwall-type inequality,Lyapunov-type inequality,Cauchy-Schwarz-type inequality,etc.)are indispensable in the study of the qualitative and quantitative properties of differential equations,especially in exploring the existence,asymptotic behavior and the estimation of solutions.Therefore,the studies of various types of integral inequalities have become a hot topic for many scholars to explore.In this paper,we study the oscillation criteria of differential equation involving Modi-fied Riemann-Liouville fractional derivative,and discuss the application of Opial inequal-ity in conformable fractional calculus and the generalization of its discrete form.The following is divided into four chapters to elaborate.In the first chapter,we summarize the general situation of the research on the frac-tional calculus theory and the Opial-type inequality at home and abroad and some related definitions and lemmas which will be used in this paper.In the second chapter,we establish the oscillation criteria of nonlinear neutral dif-ferential equation involving Modified Riemann-Liouville fractional derivative Dtα[a(t)Dtα(x(t)+ p(t)x(Υ(t))]+f(t,x(σ(t)))= 0,t ≥ t0>0,0<α<1.We will reduce the fractional differential equation to integer order differential equation by appropriate variable transform.Then,we establish some new oscillation criteria through the method of reduction of order,Riccati transformation and a series of derivation.In the third chapter,we study the application of Opial type inequality in the con-formable fractional calculus.By using the related lemma and algebraic inequality,several generalizations of Opial type inequality are obtained.In the fourth chapter,we discuss the discrete form of Opial type inequality.Using the Holder inequality and the related algebraic inequality,the discrete form of Opial type inequality is improved. |
