The method of upper and lower solutions and impulsive fractional differential inclusions

Existence Results for Impulsive Fractional Differential Inclusions with Two Different Caputo Fractional Derivatives

“…If = 1, = 0, the boundary value condition becomes (0) − ( ) = , and our system (1) reduces to [20]. Thus, our problem (1) gives generalizations of [16,19,20].…”

“…Remark 15. If = 0, = 0, the boundary value condition becomes (0) = 0 , and our system (1) reduces to [16,19]. If = 1, = 0, the boundary value condition becomes (0) − ( ) = , and our system (1) reduces to [20].…”

On the solutions for impulsive fractional functional differential equations

“…However, the theory of the IVP for fractional functional differential equations and impulsive fractional differential equations is still in the initial stage. For impulsive fractional differential equations involving the Caputo fractional derivative, the existence and uniqueness of solutions for the IVP for nonlinear impulsive fractional differential equations are discussed in [13], the existence of solutions for a class of the IVP for impulsive fractional differential equations are investigated in [14] by using the method of upper and lower solutions. For impulsive fractional differential equations involving the Caputo fractional derivative, the existence and uniqueness of solutions for the IVP for nonlinear impulsive fractional differential equations are discussed in [13], the existence of solutions for a class of the IVP for impulsive fractional differential equations are investigated in [14] by using the method of upper and lower solutions.…”

Generalized monotone iterative method for integral boundary value problems with causal operators

“…Recently, the existence of solutions for such problems have received a great deal of attentions; for details see for example [1,2,10,11,12,19,20,21]. It is well known that monotone iterative technique is quite useful, see [3,13,14,15,16,17] and references therein. In [5,6,7,9,22], this method, combining upper and lower solutions, has been successfully applied to obtain the existence of extremal solutions for boundary value problems with integral boundary conditions.…”