On the oscillation of Hadamard fractional differential equations

Abstract

Hadamard fractional derivatives are nonlocal fractional derivatives with singular logarithmic kernel with memory, and hence they are suitable to describe complex systems. In this paper, sufficient conditions are established for the oscillation of solutions fractional differential equations in the frame of left Hadamard fractional derivatives of order α ∈ C, Re(α) ≥ 0. The results are also obtained for fractional Hadamard derivatives in the Caputo setting. Examples are provided to illustrate the applicability of the main results.

Oscillation for a Class of Right Fractional Differential Equations on the Right Half Line with Damping

“…This is in contradiction with (19). Hence, (23) holds. Define the function as generalized Riccati substitution…”

“…Suppose that ( ) is a nonoscillation solution of (12); without loss of generality, we may assume that ( ) is an eventually positive solution of (12). Proceeding the same as in the proof of Theorem 1, we get (23). Define the function ( ) as follows…”

“…In 2018, Bahaaeldin Abdalla and Thabet Abdeljawad [23] studied the oscillation of Hadamard fractional differential equation of the form…”

Oscillatory Properties of Odd-Order Delay Differential Equations with Distribution Deviating Arguments

“…The memory effect may also appear non-locally when certain fractional derivatives with power, exponential and Mittag-Leffler laws are possibly applied. So, it would be interesting to extend the results of this paper to the fractional delay differential equations, see [28][29][30][31][32].…”

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, .…”