Monotonicity results for CFC nabla fractional differences with negative lower bound
Christopher S. Goodrich, Jagan Mohan Jonnalagadda
Abstract
Abstract We consider the sequential CFC-type nabla fractional difference <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mo stretchy="false">(</m:mo> <m:mi>CFC</m:mi> </m:msup> <m:msubsup> <m:mo>∇</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>ν</m:mi> </m:msubsup> <m:mmultiscripts> <m:mo>∇</m:mo> <m:mi>a</m:mi> <m:mi>μ</m:mi> <m:mprescripts/> <m:none/> <m:mi>CFC</m:mi> </m:mmultiscripts> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {(^{\mathrm{CFC}}\nabla^{\nu}_{a+1}{}^{\mathrm{CFC}}\nabla^{\mu}_{a}u)(t)} and show that one can derive monotonicity-type results even in the case where this difference satisfies a strictly negative lower bound. This illustrates some dissimilarities between the integer-order and fractional-order cases.