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Theoretical And Numerical Computations Of Convexity Analysis For Fractional Differences Using Lower Boundedness

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  • PSHTIWAN OTHMAN MOHAMMED

    (Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq)

  • DUMITRU BALEANU

    (��Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey‡Institute of Space Sciences, R76900 Magurele-Bucharest, Romania§Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, Beirut 11022801, Lebanon)

  • EMAN AL-SARAIRAH

    (�Department of Mathematics, Khalifa University, P. O. Box 127788, Abu Dhabi, UAE∥Department of Mathematics, Al-Hussein Bin Talal University, P. O. Box 20, Ma’an 71111, Jordan)

  • THABET ABDELJAWAD

    (*Department of Mathematics and Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia††Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea‡‡Department of Medical Research, China Medical University, Taichung 40402, Taiwan§§Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa)

  • NEJMEDDINE CHORFI

    (�¶Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia)

Abstract

This study focuses on the analytical and numerical solutions of the convexity analysis for fractional differences with exponential and Mittag-Leffler kernels involving negative and nonnegative lower bounds. In the analytical part of the paper, we will give a new formula for ∇2 of the discrete fractional differences, which can be useful to obtain the convexity results. The correlation between the nonnegativity and negativity of both of the discrete fractional differences, (aCFR∇αf)(t)and(aABR∇αf)(t), with the convexity of the functions will be examined. In light of the main lemmas, we will define the two decreasing subsets of (2, 3), namely ℋk,𠜖 and ℳk,𠜖. The decrease of these sets enables us to obtain the relationship between the negative lower bound of (aCFR∇αf)(t) and the convexity of the function on a finite time set given by Na+1P := {a + 1,a + 2,…,P}, for some P ∈ Na+1 := {a + 1,a + 2,…}. Besides, the numerical part of the paper is dedicated to examine the validity of the sets ℋk,𠜖 and ℳk,𠜖 in certain regions of the solutions for different values of k and 𠜖. For this reason, we will illustrate the domain of the solutions by means of several figures in which the validity of the main theorems are explained.

Suggested Citation

  • Pshtiwan Othman Mohammed & Dumitru Baleanu & Eman Al-Sarairah & Thabet Abdeljawad & Nejmeddine Chorfi, 2023. "Theoretical And Numerical Computations Of Convexity Analysis For Fractional Differences Using Lower Boundedness," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(08), pages 1-12.
    • Handle: RePEc:wsi:fracta:v:31:y:2023:i:08:n:s0218348x23401837
    DOI: 10.1142/S0218348X23401837
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    Cited by:

    1. Almusawa, Musawa Yahya & Mohammed, Pshtiwan Othman, 2023. "Approximation of sequential fractional systems of Liouville–Caputo type by discrete delta difference operators," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).

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