http://scitecresearch.com/journals/index.php/jprm Journal of Progressive Research in Mathematics Scitech Research Organisation en-US Journal of Progressive Research in Mathematics 2395-0218 http://scitecresearch.com/journals/index.php/jprm/article/view/2225 <p>For discrete fractional operators with exponential kernels, positivity, monotonicity, and convexity findings are taken into consideration in this paper. Our findings cover both sequential and non-sequential scenarios and show how fractional differences with other kinds of kernels and the exponential kernel example are comparable and different. This demonstrates that the qualitative information gathered in the exponential kernel case does not match other situations perfectly.</p> Sherif Amirov Ihsan Hasan Saadullah ##submission.copyrightStatement## http://creativecommons.org/licenses/by-nc/4.0 2024-06-17 2024-06-17 21 1 1 14 http://scitecresearch.com/journals/index.php/jprm/article/view/2226 <p>Nearly every theory in mathematics has a discrete equivalent that simplifies it theoretically and practically so that it may be used in modeling real-world issues. With discrete calculus, for instance, it is possible to find the "difference" of any function from the first order up to the n-th order. On the other hand, it is also feasible to expand this theory using discrete fractional calculus and make n any real number such that the 1⁄2-order difference is properly defined. This article is divided into five chapters, each of which develops the most straightforward discrete fractional variational theory while illustrating some fundamental concepts and features of discrete fractional calculus. It is also investigated how the idea may be applied to the development of tumors. The first section provides a succinct introduction to the discrete fractional calculus and several key mathematical concepts that are utilized often in the subject. We demonstrate in section 2 that if the Caputo-Fabrizio nabla fractional difference operator &nbsp;of order &nbsp;and commencing at is positive for &nbsp; then &nbsp;is -increasing. On the other hand, if &nbsp;is rising and , then . Additionally, a result of monotonicity for the Caputo-type fractional difference operator is established. We show a fractional difference version of the mean-value theorem as an application and contrast it to the traditional discrete fractional instance.</p> Sherif Amirov Waad Shaban Mahw Alzebari ##submission.copyrightStatement## http://creativecommons.org/licenses/by-nc/4.0 2024-07-04 2024-07-04 21 1 15 23