Solvability Of Some Weighted Fractional Nonlinear Differential Equations

In this paper,we mainly discuss the solvability of a class of singular nonlinear ψ-Riemann-Liouville fractional differential equation integral boundary value problems,a class of weighted fractional nonlinear differential equation periodic boundary value problems and a class of weighted fractional nonlinear integral and differential system periodic boundary value problems.Firstly,we construct the corresponding Green function for a class of singular nonlinear ψ-RiemannLiouville fractional boundary value problems,and obtain the existence and uniqueness of positive solutions by using the mixed monotone operator method,which generalizes the relevant conclusions about the existence and uniqueness of positive solutions for singular nonlinear Riemann-Liouville fractional differential equations.Secondly,we establish a suitable variational space and construct the corresponding variational functional for a class of periodic boundary value problems of weighted fractional order nonlinear differential equations in the variational space.The existence of the nontrivial weak solution of the problem is obtained by the mountain road lemma in critical point theory.Finally,we successfully find the workspace to study the periodic boundary value problems for a class of weighted fractional nonlinear integral and differential systems by using the critical point theory.The corresponding variational functions for the periodic boundary value problems of weighted fractional nonlinear integral and differential systems are constructed,and the sufficient conditions for the existence of at least three weak solutions for this class of boundary value problems are obtained by using the three critical point theorem.The conclusions of this paper perfect the theory of boundary value problems of fractional differential equations and expand the application of critical point theory in the study of the solvability of boundary value problems of weighted fractional differential equations.