Several Classes Of Boundary Value Problems Of Fractional Differential Equations And The Monotone Iterative Technique

Mathematical modeling of the real world in biology,chemistry,physics,control theory,mechanics,dynamical systems,and so on,often involves fractional calculus.The characteristic of fractional calculus is that there are many different fractional derivatives,like Caputo,Hadamard,Riemann-Liouville(R-L),Caputo-Hadamard types,so the scholars have the opportunity to choose the most appropriate operators to describe complex problems in the real world.As a powerful tool for obtaining extremal solutions to nonlinear problems,the monotone iterative technique is widely used to study the integer-order and fractional differential equations.In this paper,several classes of fractional differential equations with R-L and Caputo conformable derivatives are studied,where R-L and Caputo conformable derivatives are new R-L type and Caputo type fractional differential operators generated by conformable derivative.The main contents of this paper are as follows:The Caputo conformable differential equations with p-Laplacian operator and integral boundary condition are studied.The uniqueness of solution for the nonlinear and linear Cauchy problems is proved by using the Banach fixed point theorem.Based on two comparison principles,the extremal solutions for this problem are obtained by using the monotone iterative technique.A class of nonlinear R-L conformable fractional differential equations is studied.The existence of two solutions for this problem is proved by using Guo-Krasnosel’skii’s(G-K)fixed point theorem.Based on a comparison principle,the extremal solutions for the problem are obtained by using the monotone iterative technique.The coupled system of R-L conformable fractional differential equations is studied.The existence and uniqueness of the solutions for the system are obtained by using G-K and Perov’s fixed point theorem,and the extremal solutions of the system are obtained by using the monotone iterative technique.