Fractional calculus in abstract space and its application in fractional Dirichlet type problems

With the development of nonlinear science, it is found that the fractional differential equation can more accurately describe the variation of natural phenomena.Therefore, the study of fractional differential equations and their boundary value problems is of great significance for solving nonlinear problems.The Dirichlet function, as an abstract mathematical model, has many unique properties in calculus.It points out special circumstances when describing many mathematical concepts, It can also be used to construct counterexamples in calculus and to clarify many fuzzy concepts to deepen the understanding of mathematical concepts.Therefore, this paper mainly studies the necessary and sufficient conditions for the controllability of fractional linear differential systems in abstract space.The finite difference decomposition method of fractional calculus in the abstract space for the Dirichlet function equation is also studied.The existence of solutions for boundary value problems of fractional differential equations with Dirichlet boundary value condition in the abstract space is discussed by using the critical point theory. (C) 2019 Elsevier Ltd. All rights reserved.