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

Fractional calculus is still being developed continuously and its operators are used to model complex systems where the kernel of the fractional operators reflects the non-locality [2, 3]. The singularity of the kernel of the fractional operators has recently motivated researchers to present new types of fractional operators with nonsingular kernels and their discrete versions [4, 5, 6, 7, 8, 9, 10, 11, 12]. This new trend added another approach in defining fractional derivatives and integrals. In classical fractional operators, the fractionalizing process depends on iterating the weighted usual integrals or some local type derivatives in the way to get the factorial function and then replace it by the gamma function. The theory of fractional calculus with operators having nonsingular kernels depends on a limiting approach via dirac delta functions. Indeed, the fractional derivative with the nonsingular kernel is first defined so that in the limiting case \(\alpha\rightarrow 0\) we get the function itself and when \(\alpha\rightarrow1\) we get the usual derivative of the function. Then the corresponding integral operators are evaluated by the help of Laplace transforms for functions whose convolution 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. Namely, we investigate the oscillation of a class of generalized fractional derivatives defined in [1] by iterating the local conformable derivative developed in [22]. We shall name this derivative the conformable fractional derivative (CFD). Although both the CFD with its Caputo setting and the Katugampola-type derivative studied in [23, 24, 25] coincide when \(a=0\), they are very different from each other. In fact, the kernel of CFDs depends on the end points a and b which causes many differences from the Katugampola-type one. We shall study the oscillation of conformable fractional differential equation of the form where \(m=\lceil\alpha\rceil\), \({}_{a}\mathfrak{D}^{\alpha,\rho}\) is the left-fractional conformable derivative of order \(\alpha\in\mathbb{C}\), \(\operatorname{Re}(\alpha)\geq0\) in the Riemann–Liouville setting and \({}_{a}\mathfrak {I}^{\alpha,\rho}\) is the left-fractional conformable integral operator.

The left conformable derivative starting from a of a function \(f:[a,\infty)\rightarrow\mathbb{R}\) of order \(0<\rho\leq1\) is defined by If \((T^{\rho}f)(t)\) exists on \((a,b)\), then \(({}_{a}T^{\rho}f)(a)=\lim_{t \to a^{+}}{({}_{a}T^{\rho}f)(t)}\).

If f is differentiable, then The corresponding left conformable integral is defined as For the extension to the higher order \(\rho>1\), see [22].

The following identity (see [1]) is essential to solving linear conformable fractional differential equations:

Using Lemma 3.1, we can write the solution representation of (1) as where \(F(t,x)=r(t)+f_{2}(t,x)-f_{1}(t,x)\) and \(\rho>0\).

We prove our results under the following assumptions: where \(p_{1}, p_{2} \in C([0,\infty),(0,\infty))\) and β, γ are positive constants.

Define and

In this section, we study the oscillation of conformable fractional differential equations in the Caputo setting of the form where \(m=\lceil\alpha\rceil\) and \({}_{a}^{C}\mathfrak{D}^{\alpha,\rho}\) is defined by (4).

Using Lemma 4.1, the solution representation of (29) can be written as where \(F(t,x)=r(t)+f_{2}(t,x)-f_{1}(t,x)\) and \(\rho>0\).

Define