An analysis of exponential kernel fractional difference operator for delta positivity

Positivity analysis for a fractional difference operator including an exponential formula in its kernel has been examined. A composition of two fractional difference operators of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ν</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> (ν ,μ ) in the sense of Liouville–Caputo type operators has been analysed in cases when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ν</m:mi> <m:mo>≠</m:mo> <m:mi>μ</m:mi> </m:math> ν ≠ μ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ν</m:mi> <m:mo>=</m:mo> <m:mi>μ</m:mi> </m:math> ν =μ . Due to the kernel of the fractional difference operator being convergent, there has been a restriction in the domain of the solution. Incidentally, a negative lower bounded condition has been carried out through analysing the positivity results. For a better understanding, an increasing function has been considered as a test for the main results.