A Study of Positivity Analysis for Difference Operators in the Liouville–Caputo Setting

Abstract

The class of symmetric function interacts extensively with other types of functions. One of these is the class of positivity of functions, which is closely related to the theory of symmetry. Here, we propose a positive analysis technique to analyse a class of Liouville–Caputo difference equations of fractional-order with extremal conditions. Our monotonicity results use difference conditions ΔaLCμf(a+J0+1−μ)≥(1−μ)f(a+J0) and ΔaLCμf(a+J0+1−μ)≤(1−μ)f(a+J0) to derive the corresponding relative minimum and maximum, respectively. We find alternative conditions corresponding to the main conditions in the main monotonicity results, which are simpler and stronger than the existing ones. Two numerical examples are solved by achieving the main conditions to verify the obtained monotonicity results.