Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional Calculus
In interval analysis, the fuzzy inclusion relation and the fuzzy order relation are two different concepts. Under the inclusion connection, convexity and non-convexity form a substantial link with various types of inequalities. Moreover, convex fuzzy-interval-valued functions are well known in conve...
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MDPI AG
2022-02-01
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| Series: | Mathematics |
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| author | Muhammad Bilal Khan Gustavo Santos-García Hatim Ghazi Zaini Savin Treanță Mohamed S. Soliman |
| author_facet | Muhammad Bilal Khan Gustavo Santos-García Hatim Ghazi Zaini Savin Treanță Mohamed S. Soliman |
| author_sort | Muhammad Bilal Khan |
| collection | DOAJ |
| description | In interval analysis, the fuzzy inclusion relation and the fuzzy order relation are two different concepts. Under the inclusion connection, convexity and non-convexity form a substantial link with various types of inequalities. Moreover, convex fuzzy-interval-valued functions are well known in convex theory because they allow us to infer more exact inequalities than convex functions. Most likely, integral operators play significant roles to define different types of inequalities. In this paper, we have successfully introduced the Riemann–Liouville fractional integrals on coordinates via fuzzy-interval-valued functions (FIVFs). Then, with the help of these integrals, some fuzzy fractional Hermite–Hadamard-type integral inequalities are also derived for the introduced coordinated convex FIVFs via a fuzzy order relation (FOR). This FOR is defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula>-cuts or level-wise by using the Kulish–Miranker order relation. Moreover, some related fuzzy fractional Hermite–Hadamard-type integral inequalities are also obtained for the product of two coordinated convex fuzzy-interval-valued functions. The main results of this paper are the generalization of several known results. |
| first_indexed | 2024-03-09T21:31:46Z |
| format | Article |
| id | doaj.art-b8c963c62e8d4de4a4da9b9f42326a04 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:31:46Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b8c963c62e8d4de4a4da9b9f42326a042023-11-23T20:56:13ZengMDPI AGMathematics2227-73902022-02-0110453410.3390/math10040534Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional CalculusMuhammad Bilal Khan0Gustavo Santos-García1Hatim Ghazi Zaini2Savin Treanță3Mohamed S. Soliman4Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, PakistanFacultad de Economía y Empresa and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, 37007 Salamanca, SpainDepartment of Computer Science, College of Computers and Information Technology, Taif University, P.O. Box 11099, Taif 21944, Saudi ArabiaDepartment of Applied Mathematics, Politehnica University of Bucharest, 060042 Bucharest, RomaniaDepartment of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi ArabiaIn interval analysis, the fuzzy inclusion relation and the fuzzy order relation are two different concepts. Under the inclusion connection, convexity and non-convexity form a substantial link with various types of inequalities. Moreover, convex fuzzy-interval-valued functions are well known in convex theory because they allow us to infer more exact inequalities than convex functions. Most likely, integral operators play significant roles to define different types of inequalities. In this paper, we have successfully introduced the Riemann–Liouville fractional integrals on coordinates via fuzzy-interval-valued functions (FIVFs). Then, with the help of these integrals, some fuzzy fractional Hermite–Hadamard-type integral inequalities are also derived for the introduced coordinated convex FIVFs via a fuzzy order relation (FOR). This FOR is defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula>-cuts or level-wise by using the Kulish–Miranker order relation. Moreover, some related fuzzy fractional Hermite–Hadamard-type integral inequalities are also obtained for the product of two coordinated convex fuzzy-interval-valued functions. The main results of this paper are the generalization of several known results.https://www.mdpi.com/2227-7390/10/4/534fuzzy Riemann–Liouville fractional integralsHermite–Hadamard-type inequalityfuzzy-interval-valued functions |
| spellingShingle | Muhammad Bilal Khan Gustavo Santos-García Hatim Ghazi Zaini Savin Treanță Mohamed S. Soliman Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional Calculus Mathematics fuzzy Riemann–Liouville fractional integrals Hermite–Hadamard-type inequality fuzzy-interval-valued functions |
| title | Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional Calculus |
| title_full | Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional Calculus |
| title_fullStr | Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional Calculus |
| title_full_unstemmed | Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional Calculus |
| title_short | Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional Calculus |
| title_sort | some new concepts related to integral operators and inequalities on coordinates in fuzzy fractional calculus |
| topic | fuzzy Riemann–Liouville fractional integrals Hermite–Hadamard-type inequality fuzzy-interval-valued functions |
| url | https://www.mdpi.com/2227-7390/10/4/534 |
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