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Exploring New Features of Structural Nabla Derivatives on Arbitrary Time Scales
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In this paper, we investigate the structural $\nabla$-derivative on
arbitrary time scales. We begin by recalling the classical $\nabla$-derivative and then extend it to the structural setting by incorporating a
structural function $p(\cdot)$ and a positive parameter $\alpha$. Several
illustrative examples are provided to emphasize the distinction between the
standard $\nabla$ derivative and its structural counterpart. Fundamental
properties are established, including monotonicity results and a chain rule
for function compositions. These results not only unify existing
approaches-such as the classical $\nabla$, weighted, and fractional-type
operators-but also open new perspectives for applications in dynamic
equations, stability analysis, and variational problems within the time
scale framework.
Bukhtishu Publishing Group
Title: Exploring New Features of Structural Nabla Derivatives on Arbitrary Time Scales
Description:
In this paper, we investigate the structural $\nabla$-derivative on
arbitrary time scales.
We begin by recalling the classical $\nabla$-derivative and then extend it to the structural setting by incorporating a
structural function $p(\cdot)$ and a positive parameter $\alpha$.
Several
illustrative examples are provided to emphasize the distinction between the
standard $\nabla$ derivative and its structural counterpart.
Fundamental
properties are established, including monotonicity results and a chain rule
for function compositions.
These results not only unify existing
approaches-such as the classical $\nabla$, weighted, and fractional-type
operators-but also open new perspectives for applications in dynamic
equations, stability analysis, and variational problems within the time
scale framework.
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