{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T21:01:09Z","timestamp":1765486869075,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,22]],"date-time":"2023-01-22T00:00:00Z","timestamp":1674345600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In the present article, we explore the correlation between the sign of a Liouville\u2013Caputo-type difference operator and the monotone behavior of the function upon which the difference operator acts. Finally, an example is also provided to demonstrate the application and the validation of the results which we have proved herein.<\/jats:p>","DOI":"10.3390\/axioms12020114","type":"journal-article","created":{"date-parts":[[2023,1,23]],"date-time":"2023-01-23T01:36:26Z","timestamp":1674437786000},"page":"114","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Study of Monotonicity Analysis for the Delta and Nabla Discrete Fractional Operators of the Liouville\u2013Caputo Family"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2058-216X","authenticated-orcid":false,"given":"Christopher S.","family":"Goodrich","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari Mohan","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan"},{"name":"Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0223-4711","authenticated-orcid":false,"given":"Eman","family":"Al-Sarairah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Khalifa University, Abu Dhabi P.O. Box 127788, United Arab Emirates"},{"name":"Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 33011, Ma\u2019an 71111, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0365-0282","authenticated-orcid":false,"given":"Y. S.","family":"Hamed","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,22]]},"reference":[{"key":"ref_1","first-page":"165","article-title":"A transform method in discrete fractional calculus","volume":"2","author":"Atici","year":"2007","journal-title":"Int. J. Differ. Equ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2009.4.3","article-title":"Discrete fractional calculus with the nabla operator","volume":"2009","author":"Atici","year":"2009","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1090\/S0002-9939-08-09626-3","article-title":"Initial value problems in discrete fractional calculus","volume":"137","author":"Atici","year":"2009","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4149320","DOI":"10.1155\/2017\/4149320","article-title":"Arbitrary order fractional difference operators with discrete exponential kernels and applications","volume":"2017","author":"Abdeljawad","year":"2017","journal-title":"Discrete Dyn. Nat. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"406757","DOI":"10.1155\/2012\/406757","article-title":"On the definitions of nabla fractional operators","volume":"2012","author":"Abdeljawad","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.chaos.2017.04.006","article-title":"Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel","volume":"102","author":"Abdeljawad","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3355","DOI":"10.1140\/epjst\/e2018-00004-2","article-title":"Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 < \u03b1 < 5\/2","volume":"226","author":"Abdeljawad","year":"2017","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7461","DOI":"10.1002\/mma.5869","article-title":"Ulam-Hyers stability of Caputo fractional difference equations","volume":"42","author":"Chen","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"110","DOI":"10.2298\/AADM110131002F","article-title":"Fractional h-difference equations arising from the calculus of variations","volume":"5","author":"Ferreira","year":"2011","journal-title":"Appl. Anal. Discret. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3809","DOI":"10.1090\/proc\/12895","article-title":"The Poisson distribution, abstract fractional difference equations, and stability","volume":"145","author":"Lizama","year":"2017","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1697","DOI":"10.1007\/s11071-014-1250-3","article-title":"Discrete chaos in fractional delayed logistic maps","volume":"80","author":"Wu","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Goodrich, C.S., and Peterson, A.C. (2015). Discrete Fractional Calculus, Springer.","DOI":"10.1007\/978-3-319-25562-0"},{"key":"ref_13","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier. North-Holland Mathematics Studies."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"093143","DOI":"10.1063\/5.0098375","article-title":"Caputo-Hadamard fractional differential equation on time scales: Numerical scheme, asymptotic stability, and chaos","volume":"32","author":"Wu","year":"2022","journal-title":"Chaos"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2250145","DOI":"10.1142\/S0218348X22501456","article-title":"Hadamard fractional calculus on time scales","volume":"30","author":"Song","year":"2022","journal-title":"Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/s00013-014-0620-x","article-title":"A monotonicity result for discrete fractional difference operators","volume":"102","author":"Dahal","year":"2014","journal-title":"Arch. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"139","DOI":"10.2298\/AADM150218007A","article-title":"Analysis of discrete fractional operators","volume":"9","author":"Atici","year":"2015","journal-title":"Appl. Anal. Discrete Math."},{"key":"ref_18","first-page":"1","article-title":"Some relations between the Caputo fractional difference operators and integer-order differences","volume":"2015","author":"Baoguo","year":"2015","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1007\/s13398-021-01015-5","article-title":"Second and third order forward difference operator: What is in between?","volume":"115","author":"Bravo","year":"2021","journal-title":"Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"6346","DOI":"10.1002\/mma.8174","article-title":"Qualitative properties of nonlocal discrete operators","volume":"45","author":"Bravo","year":"2022","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1080\/10236198.2018.1561883","article-title":"Mixed order monotonicity results for sequential fractional nabla differences","volume":"25","author":"Dahal","year":"2019","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1224","DOI":"10.1080\/10236198.2016.1188089","article-title":"Monotonicity and convexity for nabla fractional (q, h)-differences","volume":"22","author":"Du","year":"2016","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1515\/fca-2020-0028","article-title":"Asymptotic stability of fractional difference equations with bounded time delays","volume":"23","author":"Wang","year":"2020","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_24","first-page":"125079","article-title":"Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities","volume":"375","author":"Du","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1986","DOI":"10.1080\/10236198.2017.1380635","article-title":"A sharp convexity result for sequential fractional delta differences","volume":"23","author":"Goodrich","year":"2017","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1080\/10236198.2015.1011630","article-title":"Convexity for nabla and delta fractional differences","volume":"21","author":"Baoguo","year":"2015","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4961","DOI":"10.3934\/dcds.2020207","article-title":"Positivity, monotonicity, and convexity for convolution operators","volume":"40","author":"Goodrich","year":"2020","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_28","first-page":"47","article-title":"Monotonicity and convexity for nabla fractional q-differences","volume":"25","author":"Baoguo","year":"2016","journal-title":"Dynam. Syst. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3671","DOI":"10.2298\/FIL1712671A","article-title":"Monotonicity results for delta and nabla Caputo and Riemann fractional differences via dual identities","volume":"31","author":"Abdeljawad","year":"2017","journal-title":"Filomat"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1307","DOI":"10.1515\/fca-2019-0069","article-title":"Methods of upper and lower solutions for nonlinear Caputo fractional difference equations and its applications","volume":"22","author":"Chen","year":"2019","journal-title":"Frac. Calc. Appl. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Mohammed, P.O., Abdeljawad, T., and Hamasalh, F.K. (2021). On discrete delta Caputo-Fabrizio fractional operators and monotonicity analysis. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5030116"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3058","DOI":"10.3934\/era.2022155","article-title":"Positivity analysis for the discrete delta fractional differences of the Riemann-Liouville and Liouville-Caputo types","volume":"30","author":"Mohammed","year":"2022","journal-title":"Electron. Res. Arch."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1207","DOI":"10.1002\/mma.6823","article-title":"Monotonicity results for nabla fractional h-difference operators","volume":"44","author":"Liu","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Mohammed, P.O., Almutairi, O., Agarwal, R.P., and Hamed, Y.S. (2022). On convexity, monotonicity and positivity analysis for discrete fractional operators defined using exponential kernels. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6020055"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Mohammed, P.O., Srivastava, H.M., Baleanu, D., Jan, R., and Abualnaja, K.M. (2022). Monotonicity results for nabla Riemann-Liouville fractional differences. Mathematics, 10.","DOI":"10.3390\/math10142433"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2010.02.009","article-title":"Modeling with discrete fractional equations","volume":"369","author":"Atici","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1515\/cmb-2019-0002","article-title":"A study on discrete and discrete fractional pharmacokinetics-pharmacodynamics models for tumor growth and anti-cancer effects","volume":"7","author":"Atici","year":"2019","journal-title":"Comput. Math. Biophys."},{"key":"ref_38","first-page":"313","article-title":"A new approach for modeling with discrete fractional equations","volume":"151","author":"Atici","year":"2017","journal-title":"Fund. Inform."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.chaos.2018.09.022","article-title":"Different type kernel h\u2013fractional differences and their fractional h-sums","volume":"116","author":"Abdeljawad","year":"2018","journal-title":"Chaos Solit. Fract."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1155\/2013\/406910","article-title":"On delta and nabla Liouville-Caputo fractional differences and dual identities","volume":"2013","author":"Abdeljawad","year":"2013","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1090\/S0025-5718-1988-0929549-2","article-title":"On a new definition of the fractional difference","volume":"50","author":"Gray","year":"1988","journal-title":"Math. Comp."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1186\/s13662-015-0496-5","article-title":"Caputo type fractional difference operator and its application on discrete time scales","volume":"2015","author":"Rahmat","year":"2015","journal-title":"Adv. Diff. 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