A new conformable nabla derivative and its application on arbitrary time scales

Abstract

In this article, we introduce a new type of conformable derivative and integral which involve the time scale power function $\widehat{\mathcal{G}}_{\eta }(t, a)$ G ˆ η ( t , a ) for $t,a\in \mathbb{T}$ t , a ∈ T . The time scale power function takes the form $(t-a)^{\eta }$ ( t − a ) η for $\mathbb{T}=\mathbb{R}$ T = R which reduces to the definition of conformable fractional derivative defined by Khalil et al. (2014). For the discrete time scales, it is completely novel, where the power function takes the form $(t-a)^{(\eta )}$ ( t − a ) ( η ) which is an increasing factorial function suitable for discrete time scales analysis. We introduce a new conformable exponential function and study its properties. Finally, we consider the conformable dynamic equation of the form $\bigtriangledown _{a}^{\gamma }y(t)=y(t, f(t))$ ▽ a γ y ( t ) = y ( t , f ( t ) ) , and study the existence and uniqueness of the solution. As an application, we show that the conformable exponential function is the unique solution to the given dynamic equation. We also examine the analogue of Gronwall’s inequality and its application on time scales.

Novel Fractional Dynamic Hardy–Hilbert-Type Inequalities on Time Scales with Applications

“…Rahmat et al [11] presented a new type of conformable nabla derivative and integral which involve the time-scale power function G n (t, s) for s, t ∈ T. Definition 9. Suppose [s, t] ⊂ T and s < t. The generalized time-scale power function G n : T × T −→ R + for n ∈ N 0 is defined by…”

Gamma-Nabla Hardy–Hilbert-Type Inequalities on Time Scales

“…A time scale T is an arbitrary nonempty closed subset of the set of real numbers R. In the manuscript, we use the notation ∇ (γ,a) for the nabla conformable fractional derivative on time scales instead of ∇ γ a for simplification. For more details on nabla conformable fractionals, please see [25].…”

On nabla conformable fractional Hardy-type inequalities on arbitrary time scales

“…where f : R + → R. Benkhettou et al [38] introduced a conformable calculus on an arbitrary time scale, which is a natural extension of the conformable calculus.…”

“…In [12,13], Bohner and Peterson introduced the most basic concepts and definitions related to the theory of time scales. Next, some basic definitions and concepts about the fractional analysis, which are used in this manuscript, were given and adapted from [12,13,34,38]. Any nonempty arbitrary closed subset of the real numbers is called a time scale T. We assume that T has the standard topology on the real numbers R. Now, let σ : T → T be the forward jump operator defined by…”