On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations

Abstract

We investigate the existence of solutions for two high-order fractional differential equations including the Caputo-Fabrizio derivative. In this way, we introduce some new tools for obtaining solutions for the high-order equations. Also, we discuss two illustrative examples to confirm the reported results. In this way one gets the possibility of utilizing some continuous or discontinuous mappings as coefficients in the fractional differential equations of higher order.

On high order fractional integro-differential equations including the Caputo–Fabrizio derivative

“…Some researchers tried to use it for solving different equations (see, for example, [2,9] and [14]). Recently, approximate solutions of some fractional differential equations have been reviewed (see, for example, [3,4,6,12,13] and [7]). Also, one is finding some new applications for fractional derivations (see, for example, [3]).…”

Optimality conditions for fractional differential inclusions with nonsingular Mittag–Leffler kernel

“…Furthermore, as far as the optimal control of problem (1.1)-(1.3) is concerned, one can refer to the methods of the Lagrange multiplier technique for the classical Caputo and Riemann-Liouville fractional time derivative presented by different authors (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]26] and the references therein).…”

“…The optimal control problems and calculus of variation for variational equality with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel are studied in many papers (see, for example, [1,2,10,11,21] and the references therein).…”

Existence of solutions for fractional multi-point boundary value problems at resonance with three-dimensional kernels

“…And the usual way to investigate the fractional BVPs and IVPs is nonlinear analysis such as variation method (see [12][13][14]), fixed-point theorems (see [15][16][17][18]), upper and lower solutions method (see [19,20]), coincidence degree theory (see [21][22][23][24][25]). Besides, for the recent advances in other techniques for solving nonlinear problems, see [26][27][28][29][30]. For example, De La Sen et al [29] .…”

“…In recent years, boundary value problems (BVPs) and initial value problems (IVPs) of fractional differential equations have been discussed widely, and numerous valuable results have been obtained (see [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]). And the usual way to investigate the fractional BVPs and IVPs is nonlinear analysis such as variation method (see [12][13][14]), fixed-point theorems (see [15][16][17][18]), upper and lower solutions method (see [19,20]), coincidence degree theory (see [21][22][23][24][25]).…”