On the oscillation of higher order fractional difference equations with mixed nonlinearities

Abstract

Based on certain mathematical inequalities and Volterra sum equations, we establish oscillation criteria for higher order fractional difference equations with mixed nonlinearities. The problem is addressed for equations involving Riemann-Liouville and Caputo operators. Two examples are constructed to demonstrate the validity of the proposed assumptions. Our results improve those obtained in the previous works.

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

“…Case (2): Let α > 1. Then (18) and (19) are still true. Hence, from (28) and using (27), we find that…”

“…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.…”

On the Oscillation of Non-Linear Fractional Difference Equations with Damping

“…The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo's differential operator. Afterwards, several results have appeared and thus many types of fractional differential and difference equations have been investigated; the reader can consult the papers [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] where different approaches have been used to prove the main results. For the sake of completeness and comparison, we review some results in the sequel.…”

Forced oscillation of fractional differential equations via conformable derivatives with damping term

“…The oscillation theory for fractional differential and difference equations has been studied by some authors (see [19][20][21][22][23][24][25][26][27][28][29]). In [23] the authors studied the oscillation theory for fractional differential equations by considering fractional initial value problem of the form Recently, in [21] the authors studied the oscillation of a conformable initial value problem of the form ⎧ ⎨ ⎩ a D α,ρ x(t) + f 1 (t, x) = r(t) + f 2 (t, x), t > a, lim t→a + a J j-α,ρ x(t) = b j (j = 1, 2, .…”