Forced oscillation of certain fractional differential equations

Abstract

The paper deals with the forced oscillation of the fractional differential equationm-q a x is the Riemann-Liouville fractional integral of order m -q of x, and b k (k = 1, 2, . . . , m) are/is constants/constant. We obtain some oscillation theorems for the equation by reducing the fractional differential equation to the equivalent Volterra fractional integral equation and by applying Young's inequality. We also establish some new oscillation criteria for the equation when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. The results obtained here improve and extend some existing results. An example is given to illustrate our theoretical results. MSC: 34A08; 34C10

Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives

“…The objective of this paper is to study the oscillation of conformable fractional differential equations of the form (1). This will generalize the results obtained in [13,14] when we take a = 0. This paper is organized as follows.…”

“…3. As ρ → 1 in these theorems, we get the results obtained in [13] and [14] when a = 0. The main approach is based on applying Young's inequality which will help us in obtaining sharper conditions.…”

“…Then the corresponding integral operators are evaluated by the help of Laplace transforms for functions whose convolu-tion with the nonsingular kernel vanishes at the starting point a. The oscillation theory for fractional differential and difference equations was studied by some authors (see [13][14][15][16][17][18][19][20][21]), thus several definitions of fractional derivatives and fractional integral operators exist in the literature. In this article, we study the oscillation of fractional operators defined by the first approach mentioned above.…”

Oscillation criteria for a certain class of fractional order integro-diferential equations

“…Following the work developed in [15], Chen et al in the [6] studied the fractional differential equation…”

Oscillation of Solutions to Fractional Partial Differential Equations with Several Delays

“…In recent years, oscillatory behavior of solutions of fractional ordinary differential equations have been studied by authors [3][4][5][6][7][8][9][10][11]. However, there is a scarcity in the study of oscillation theory of fractional partial differential equations up to now, we refer to [12][13][14][15][16].…”