| Summary: | Monotonicity analysis of delta fractional sums and differences of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>υ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> on the time scale <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mspace width="0.166667em"></mspace><mi mathvariant="normal">Z</mi></mrow></semantics></math></inline-formula> are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional <i>h</i>-difference and delta Caputo fractional <i>h</i>-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula>-increasing on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="normal">M</mi><mrow><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi><mo>,</mo><mi>h</mi></mrow></msub></semantics></math></inline-formula>, where the delta Riemann–Liouville fractional <i>h</i>-difference of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula> of a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> starting at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi></mrow></semantics></math></inline-formula> is greater or equal to zero, and then, we can show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula>-increasing on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="normal">M</mi><mrow><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi><mo>,</mo><mi>h</mi></mrow></msub></semantics></math></inline-formula>, where the delta Caputo fractional <i>h</i>-difference of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula> of a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> starting at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi></mrow></semantics></math></inline-formula> is greater or equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mo>Γ</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>υ</mi><mo>)</mo></mrow></mfrac><msubsup><mrow><mo>(</mo><mi>z</mi><mo>−</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mi>h</mi></mrow><mrow><mo>(</mo><mo>−</mo><mi>υ</mi><mo>)</mo></mrow></msubsup><mi>y</mi><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for each <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>∈</mo><msub><mi mathvariant="normal">M</mi><mrow><mi>a</mi><mo>+</mo><mi>h</mi><mo>,</mo><mi>h</mi></mrow></msub></mrow></semantics></math></inline-formula>. Conversely, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>υ</mi><mspace width="0.166667em"></mspace><mi>h</mi><mo>)</mo></mrow></semantics></math></inline-formula> is greater or equal to zero and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> is increasing on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="normal">M</mi><mrow><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi><mo>,</mo><mi>h</mi></mrow></msub></semantics></math></inline-formula>, we show that the delta Riemann–Liouville fractional <i>h</i>-difference of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula> of a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> starting at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi></mrow></semantics></math></inline-formula> is greater or equal to zero, and consequently, we can show that the delta Caputo fractional <i>h</i>-difference of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula> of a function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> starting at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi></mrow></semantics></math></inline-formula> is greater or equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mo>Γ</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>υ</mi><mo>)</mo></mrow></mfrac><msubsup><mrow><mo>(</mo><mi>z</mi><mo>−</mo><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mi>h</mi></mrow><mrow><mo>(</mo><mo>−</mo><mi>υ</mi><mo>)</mo></mrow></msubsup><mi>y</mi><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>υ</mi><mi>h</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="normal">M</mi><mrow><mi>a</mi><mo>,</mo><mi>h</mi></mrow></msub></semantics></math></inline-formula>. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>h</mi><mspace width="0.166667em"></mspace><mi mathvariant="normal">Z</mi></mrow></semantics></math></inline-formula> utilizing the monotonicity results.
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