Solvability of Anti-periodic BVPs for Impulsive Fractional Differential Systems Involving Caputo and Riemann–Liouville Fractional Derivatives

Abstract

Sufficient conditions are given for the existence of solutions of anti-periodic value problems for impulsive fractional differential systems involving both Caputo and Riemann–Liouville fractional derivatives. We allow the nonlinearities $p(t)f(t,x,y,z,w)$ and $q(t)g(t,x,y,z,w)$ in fractional differential equations to be singular at $t=0$ and $t=1$. Both $f$ and $g$ may be super-linear and sub-linear. The analysis relies on some well known fixed point theorems. The initial value problem discussed may be seen as a generalization of some ecological models. An example is given to illustrate the efficiency of the main theorems. Many unsuitable lemmas in recent published papers are pointed out in order not to mislead readers. A conclusion section is given at the end of the paper.

Basic Concepts of Riemann–Liouville Fractional Differential Equations with Non-Instantaneous Impulses

“…As is shown in [12] the integral representation of the solution with impulsive condition (49) is not correct (see Remark 4.3 [12]). In [8] the initial value problems of nonlinear impulsive RL fractional differential equations are studied with instantaneous impulsive conditions of the type…”

“…Caputo fractional derivatives have some properties similar to ordinary derivatives (such as the derivative of a constant) which lead to similar initial value problems as well as similar impulsive conditions (instantaneous and non-instantaneous). In the literature many types of initial value problems and boundary value problems for Caputo fractional differential equations with instantaneous and non-instantaneous impulses are studied (see, for example, [7][8][9]). For Riemann-Liouville (RL) fractional differential equations with instantaneous impulses several results are obtained in [8,[10][11][12].…”

“…In the literature many types of initial value problems and boundary value problems for Caputo fractional differential equations with instantaneous and non-instantaneous impulses are studied (see, for example, [7][8][9]). For Riemann-Liouville (RL) fractional differential equations with instantaneous impulses several results are obtained in [8,[10][11][12]. However for the RL fractional derivative, the physical interpretation of the initial condition (see [1]) requires different impulsive conditions, so as a result the statement of the problem considered is crucial.…”

“…The integral representation of the solution of instantaneous impulsive RL fractional derivative of order q ∈ (0, 1) is given in [8] in the case when the impulsive conditions are…”

Heat Transfer in a Fractional Nanofluid Flow through a Permeable Medium

“…For example, valuable impact can be seen in fluid mechanics. Moreover, the influence of noninteger modeling is seen in viscoelasticity, control theory, and electrochemistry [15][16][17]. Advancements in fractional modeling are carried out with the passage of time and the inclusion of new fractional derivatives in the literature, for example, Yang's fractional derivative, Atangana-Baleanu derivatives, and Caputo derivatives [18,19].…”