{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,22]],"date-time":"2026-02-22T10:49:34Z","timestamp":1771757374929,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,12,5]],"date-time":"2023-12-05T00:00:00Z","timestamp":1701734400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071218"],"award-info":[{"award-number":["12071218"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This article is a study on the (k,s)-Riemann\u2013Liouville fractional integral, a generalization of the Riemann\u2013Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension of the graph of continuous functions after the extended integral. It is shown that the upper Box dimension of the (k,s)-Riemann\u2013Liouville fractional integral for any continuous functions is no more than the upper Box dimension of the functions on the unit interval I=[0,1], which indicates that the upper Box dimension of the integrand f(x) will not be increased by the \u03c3-order (k,s)-Riemann\u2013Liouville fractional integral ksD\u2212\u03c3f(x) where \u03c3&gt;0 on I. Additionally, we prove that the fractal dimension of ksD\u2212\u03c3f(x) of one-dimensional continuous functions f(x) is still one.<\/jats:p>","DOI":"10.3390\/sym15122158","type":"journal-article","created":{"date-parts":[[2023,12,5]],"date-time":"2023-12-05T02:55:32Z","timestamp":1701744932000},"page":"2158","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann\u2013Liouville Fractional Integral"],"prefix":"10.3390","volume":"15","author":[{"given":"Bingqian","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China"}]},{"given":"Wei","family":"Xiao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,5]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"New inequalities of ostrowski type for co-ordinated convex functions via fractional integrals","volume":"2","author":"Latif","year":"2012","journal-title":"J. Frac. Calc. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2403","DOI":"10.1016\/j.mcm.2011.12.048","article-title":"Hermite\u2013Hadamard\u2019s inequalities for fractional integrals and related fractional inequalities","volume":"57","author":"Sarikaya","year":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"065102","DOI":"10.1088\/1361-6471\/acbe58","article-title":"Fractional calculus within the optical model used in nuclear and particle physics","volume":"50","author":"Herrmann","year":"2023","journal-title":"J. Phys. G Nucl. Part. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Alsaedi, A., Alghanmi, M., Ahmad, B., and Ntouyas, S.K. (2018). Generalized Liouville-Caputo fractional differential equations and inclusions with nonlocal generalized fractional integral and multipoint boundary conditions. Symmetry, 10.","DOI":"10.3390\/sym10120667"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Jain, S., Cattani, C., and Agarwal, P. (2022). Fractional hypergeometric functions. Symmetry, 14.","DOI":"10.3390\/sym14040714"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Izadi, M., and Cattani, C. (2020). Generalized Bessel polynomial for multi-order fractional differential equations. Symmetry, 12.","DOI":"10.3390\/sym12081260"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Abdalla, M., Akel, M., and Choi, J. (2021). Certain matrix Riemann-Liouville fractional integrals associated with functions involving generalized Bessel matrix polynomials. Symmetry, 13.","DOI":"10.3390\/sym13040622"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Duan, J.S., and Chen, L. (2018). Solution of fractional differential equation systems and computation of matrix Mittag-Leffler functions. Symmetry, 10.","DOI":"10.3390\/sym10100503"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/j.jat.2008.08.012","article-title":"Box dimension and fractional integral of linear fractal interpolation functions","volume":"161","author":"Ruan","year":"2009","journal-title":"J. Approx."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1016\/j.chaos.2006.01.124","article-title":"The relationship between the fractal dimensions of a type of fractal functions and the order of their fractional calculus","volume":"34","author":"Liang","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","first-page":"297","article-title":"The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus","volume":"200","author":"Liang","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2741","DOI":"10.1016\/j.chaos.2009.03.180","article-title":"On the fractional calculus of Besicovitch function","volume":"42","author":"Liang","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1016\/j.chaos.2005.01.041","article-title":"What is the exact condition for fractional integrals and derivatives of Besicovitch functions to have exact Box dimension","volume":"26","author":"He","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","first-page":"227","article-title":"Von Koch curve and its fractional calculus","volume":"54","author":"Liang","year":"2011","journal-title":"Acta Math. Sin. Chin. Ser."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1142\/S0218348X95000175","article-title":"The relationship between fractional calculus and fractals","volume":"3","author":"Tatom","year":"1995","journal-title":"Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1651","DOI":"10.1515\/fca-2018-0087","article-title":"Fractal dimension of Riemann-Liouville fractional integral of 1-dimensional continuous functions","volume":"21","author":"Liang","year":"2018","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_17","first-page":"711","article-title":"On the fractional calculus functions of a type of Weierstrass function","volume":"25","author":"Yao","year":"2004","journal-title":"Chin. Ann. Math. Ser. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.chaos.2007.04.017","article-title":"The fractal dimensions of graphs of the Weyl-Marchaud fractional derivative of the Weierstrass-type function","volume":"35","author":"Yao","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1494","DOI":"10.1007\/s10114-016-6069-z","article-title":"Fractal dimensions of fractional integral of continuous functions","volume":"32","author":"Liang","year":"2016","journal-title":"Acta Math. Sin. Engl. Ser."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Wu, J.R. (2020). The effects of the Riemann-Liouville fractional integral on the Box dimension of fractal graphs of H\u00f6lder continuous functions. Fractals, 28.","DOI":"10.1142\/S0218348X20500528"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2350044","DOI":"10.1142\/S0218348X23500445","article-title":"Fractal dimension variation of continuous functions under certain operations","volume":"31","author":"Yu","year":"2023","journal-title":"Fractals"},{"key":"ref_22","first-page":"2440006","article-title":"Construction of monotonous approximation by fractal interpolation functions and fractal dimensions","volume":"31","author":"Yu","year":"2023","journal-title":"Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Yu, B.Y., and Liang, Y.S. (2022). On the lower and upper Box dimensions of the sum of two fractal functions. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6070398"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Yu, B.Y., and Liang, Y.S. (2022). Estimation of the fractal dimensions of the linear combination of continuous functions. Mathematics, 10.","DOI":"10.3390\/math10132154"},{"key":"ref_25","first-page":"2154","article-title":"Approximation with continuous functions preserving fractal dimensions of the Riemann-Liouville operators of fractional calculus","volume":"10","author":"Yu","year":"2023","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ross, B. (1975). Fractional Calculus and Its Applications, Springer.","DOI":"10.1007\/BFb0067095"},{"key":"ref_27","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience."},{"key":"ref_28","first-page":"860","article-title":"New approach to a generalized fractional integral","volume":"218","author":"Katugampola","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_29","first-page":"77","article-title":"(k,s)-Riemann-Liouville fractional integral and applications","volume":"45","author":"Sarikaya","year":"2016","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_30","first-page":"179","article-title":"On hypergeometric functions and Pochhammer k-symbol","volume":"15","author":"Pariguan","year":"2007","journal-title":"Divulg. Mat."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Samraiz, M., Umer, M., Kashuri, A., Abdeljawad, T., Iqbal, S., and Mlaiki, N. (2021). On weighted (k,s)-Riemann-Liouville fractional operators and solution of fractional kinetic equation. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5030118"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s40065-016-0158-9","article-title":"(k,s)-Riemann-Liouville fractional integral inequalities for continuous random variables","volume":"6","author":"Tomar","year":"2017","journal-title":"Arab. J. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1007\/s41478-021-00318-5","article-title":"Analytical properties of (k,s)-Riemann\u2013Liouville fractional integral and its fractal dimension","volume":"29","author":"Priya","year":"2021","journal-title":"J. Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/s41478-022-00451-9","article-title":"The relationship between the order of (k,s)-Riemann-Liouville fractional integral and the fractal dimensions of a fractal function","volume":"31","author":"Navish","year":"2023","journal-title":"J. Anal."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Falconer, K.J. (1990). Fractal Geometry: Mathematical Foundations and Applications, John Wiley Sons Inc.","DOI":"10.2307\/2532125"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2250094","DOI":"10.1142\/S0218348X22500943","article-title":"On Box dimension of Hadamard fractional integral (partly answer fractal calculus conjecture)","volume":"30","author":"Xiao","year":"2022","journal-title":"Fractals"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2150264","DOI":"10.1142\/S0218348X21502649","article-title":"Relationship of upper Box dimension between continuous fractal functions and their Riemann\u2013Liouville fractional integral","volume":"29","author":"Xiao","year":"2021","journal-title":"Fractals"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"4304","DOI":"10.1016\/j.na.2010.02.007","article-title":"Box dimensions of Riemann-Liouville farctional integrals of continuous functions of bounded variation","volume":"72","author":"Liang","year":"2010","journal-title":"Nonlinear Anal."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1007\/s10114-013-2044-0","article-title":"Some remarks on one-dimensional functions and their Riemann-Liouville fractional calculus","volume":"30","author":"Zhang","year":"2014","journal-title":"Acta Math. Sin. Engl. Ser."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1750047","DOI":"10.1142\/S0218348X17500475","article-title":"Fractal dimension of Riemann-Liouville fractional integral of certain unbounded variational continuous function","volume":"25","author":"Li","year":"2017","journal-title":"Fractal"},{"key":"ref_41","first-page":"220","article-title":"A note on Katugampola fractional calculus and fractal dimensions","volume":"339","author":"Verma","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"100106","DOI":"10.1016\/j.rineng.2020.100106","article-title":"An effective method to compute the box-counting dimension based on the mathematical definition and intervals","volume":"6","author":"Wu","year":"2020","journal-title":"Results Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2158\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:37:55Z","timestamp":1760132275000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2158"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,5]]},"references-count":42,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["sym15122158"],"URL":"https:\/\/doi.org\/10.3390\/sym15122158","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,5]]}}}