Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions

The Green’s function of a class of two-term fractional differential equation boundary value problem and its applications

“…While there are a lot of works dealing with multi-term FDEs with initial conditions, the results dealing with boundary value problems of multi-term FDEs are relatively scarce. For some recent literature on Caputo type multi-term FBVPs, we mention the papers [8,9] and the references therein. In [20], we established some new positive properties of the Green's function for the Riemann-Liouville type FBVP, in which the linear operator contains two terms:…”

“…Compared with the existing works, this paper has the following features. Firstly, the fractional derivative discussed in this paper is the standard Riemann-Liouville derivative, which is different from [8,9], and the linear operator of the FBVP we are considered with contains two terms, which is different from [14,19,27,28,30]; in other words, we discuss different problem which has been seldom studied before. Secondly, some meaningful properties of the Green's function for the case that 2 < α < 3 are established; this is different from [20] since Ref.…”

On a nonlocal integral boundary value problem of nonlinear Langevin equation with different fractional orders

“…The interest in this topic results from the applicability of nonlocal and integral conditions in mathemati-cal modeling of many real world situations arising in applied and biological sciences. For details and examples, see [11][12][13][14][15][16][17][18][19][20][21][22][23] and the references cited therein.…”

Convergence analysis for iterative learning control of conformable fractional differential equations

“…As we all know Khalil et al introduced the concept of conformable (local version) fractional derivative, which coincides with the standard (nonlocal version) fractional derivatives on polynomials up to a constant multiple and also can be used to characterize fractional Newton mechanics and the model in mathematical biology . In particular, local version fractional derivative is well behaved and obeys the Leibniz rule and chain rule, which has been proved not well kept for nonlocal version fractional derivative like Riemann‐Liouville and Caputo derivatives . Meanwhile, it is a natural extension of the usual derivative and can be widely used to establish chain rule, exponential functions, Gronwall's inequality, integration by parts, Taylor power series expansions, Grünwald‐Letnikov approach, and calculus of variations for conformable version fractional calculus see, for example, Abdeljawad.…”

“…4 In particular, local version fractional derivative is well behaved and obeys the Leibniz rule and chain rule, which has been proved not well kept for nonlocal version fractional derivative like Riemann-Liouville and Caputo derivatives. [5][6][7] Meanwhile, it is a natural extension of the usual derivative and can be widely used to establish chain rule, exponential functions, Gronwall's inequality, integration by parts, Taylor power series expansions, Grünwald-Letnikov approach, and calculus of variations for conformable version fractional calculus (see, for example, Abdeljawad 8 ). In addition, Laplace transforms, 8 variation of constants methods, 9 and differential transform method 10 are used to find the representation and stability of solutions to linear conformable differential equations, and functional analysis method is used to deal with nonlinear conformable differential equations 11,12 respectively.…”