This paper is devoted to the nabla unification of discrete and continuous Hardy-Copson type inequalities. Some obtained inequalities are counterparts their delta versions while others new even for discrete, continuous, cases. Moreover, these dynamic not only generalize unify related ones in literature but also improve them special

The stability of the nabla discrete distributed-order nonlinear dynamic systems is investigated in this paper. Firstly, a sufficient condition for asymptotic proposed based on Lyapunov direct method. In addition, some properties operators are derived. Based these properties, simpler criterion provided to determine such systems. Finally, two examples given illustrate validity results.

Given a monogenic function on the quaternionic algebra \({{\mathbb {H}}}\), Clifford {R}}}_n\) or octonionic {O}}}\) we prove that \(|\nabla ^m f|^\alpha \) is subharmonic for some \(\alpha >0\) where \(\nabla f\) mth order gradient of f. We find also optimal value \). This generalization result Calderon and Zygmund.

and Applied Analysis 3 and define recursively a∇−nf t ∫ t a a∇−n 1f τ ∇τ 2.4 for n 2, 3, . . .. Then we have the following. Proposition 2.1 Nabla Cauchy formula . Let n ∈ Z , a, b ∈ T and let f : T → R be ∇-integrable on a, b ∩ T. If t ∈ T, a ≤ t ≤ b, then a∇−nf t ∫ t a ̂ hn−1 ( t, ρ τ ) f τ ∇τ . 2.5 Proof. This assertion can be proved by induction. If n 1, then 2.5 obviously holds. Let n ≥ 2 an...

In this paper, we deal with the following double phase problem $$\begin{aligned} \left\{ \begin{array}{lll} -\text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) =&{} \gamma \left( \displaystyle \frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle \frac{|u|^{q-2}u}{|x|^q}\right) \\ &{}+f(x,u) &{} \text{ in } \Omega ,\\ u=0&{} \partial , \end{array} \right. \end{aligned}$$ where...

We generalize an idea in the works of Landauer and Bennett on computations, Hill's chemical kinetics, to emphasize importance kinetic cycles mesoscopic nonequilibrium thermodynamics (NET). For continuous stochastic systems, a NET phase space is formulated terms cycle affinity $\nabla\wedge\big(\mathbf{D}^{-1}\mathbf{b}\big)$ vorticity potential $\mathbf{A}(\mathbf{x})$ stationary flux $\mathbf{...