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October  2021, 14(10): 3703-3718. doi: 10.3934/dcdss.2021020

Some new bounds analogous to generalized proportional fractional integral operator with respect to another function

1. 

Government College University, Faisalabad, Pakistan

2. 

Department of Mathematics, Faculty of Arts and Sciences, Cankaya University Ankara, Turkey

3. 

Department of Medical Research, China Medical University Hospital, Taichung 40402, Taiwan, École Normale Supérieure, Moulay Ismail University, Meknes, Morocco

* Corresponding author: Zakia Hammouch.

Received  November 2019 Revised  January 2020 Published  October 2021 Early access  March 2021

The present article deals with the new estimates in the view of generalized proportional fractional integral with respect to another function. In the present investigation, we focus on driving certain new classes of integral inequalities utilizing a family of positive functions $ n(n\in\mathbb{N}) $ for this newly defined operator. From the computed outcomes, we concluded some new variants for classical generalized proportional fractional and other integrals as remarks. These variants are connected with some existing results in the literature. Certain interesting consequent results of the main theorems are also pointed out.

Citation: Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3703-3718. doi: 10.3934/dcdss.2021020
References:
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T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016.  Google Scholar

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T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 2017 (2017), 78-87.  doi: 10.1186/s13662-017-1126-1.  Google Scholar

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T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.  Google Scholar

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M. Adil KhanY.-M. ChuT. U. Khan and J. Khan, Some new inequalities of Hermite-Hadamard type for $s$-convex functions with applications, Open Math., 15 (2017), 1414-1430.   Google Scholar

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M. Adil Khan, Y.-M. Chu, A. Kashuri, R. Liko and G. Ali, Conformable fractional integrals versions of Hermite-Hadamard inequalities and their generalizations, J. Funct. Spaces, 2018 (2018), 6928130, 9 pp. doi: 10.1155/2018/6928130.  Google Scholar

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M. Adil KhanA. IqbalM. Suleman and Y.-M. Chu, Hermite-Hadamard type inequalities for fractionalintegrals via Green's function, J. Inequal. Appl., 2018 (2018), 161-176.  doi: 10.1186/s13660-018-1751-6.  Google Scholar

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R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simulat, 44 (2017), 460-481.  doi: 10.1016/j.cnsns.2016.09.006.  Google Scholar

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J. AlzabutT. AbdeljawadF. Jarad and W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequal. Appl., 2019 (2019), 101-113.  doi: 10.1186/s13660-019-2052-4.  Google Scholar

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D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137.   Google Scholar

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A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), 3-24.  doi: 10.1051/mmnp/2018010.  Google Scholar

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S. Bhatter, A. Mathur, D. Kumar and J. Singh, A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memory, Physica A, 537 (2020), 122578, 13 pp. doi: 10.1016/j.physa.2019.122578.  Google Scholar

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Y.-M. ChuM. Adil KhanT. Ali and S. S. Dragomir, Inequalities for $GA$-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 93-105.  doi: 10.1186/s13660-017-1371-6.  Google Scholar

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Z. Dahmani, New classes of integral inequalities of fractional order, Matematiche, 69 (2011), 237-247.  doi: 10.4418/2014.69.1.18.  Google Scholar

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F. JaradT. Abdeljawad and J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Spec. Top., 226 (2017), 3457-3471.  doi: 10.1140/epjst/e2018-00021-7.  Google Scholar

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F. JaradM. A. Alqudah and T. Abdeljawad, On more general forms of proportional fractional operators, Open Math., 18 (2020), 167-176.   Google Scholar

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F. JaradU. UgurluT. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ, 2017 (2017), 247-263.  doi: 10.1186/s13662-017-1306-z.  Google Scholar

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H. Kalsoom, S. Rashid, M. Idrees, Y.-M. Chu and D. Baleanu, Two variable quantum integral inequalities of Simpson-type based on higher order generalized strongly preinvex and quasi preinvex functions, Symmetry, 12 (2020), 51. doi: 10.3390/sym12010051.  Google Scholar

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A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 2006.  Google Scholar

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R. KhalilM. Al HoraniA. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.  Google Scholar

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D. KumarJ. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods. Appl. Scis., 43 (2020), 443-457.   Google Scholar

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D. KumarJ. SinghK. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, Inter. J. Heat. Mass. Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094.  Google Scholar

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M. A. Latif, S. Rashid, S. S. Dragomir and Y.-M. Chu, Hermite-Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications, J. Inequal. Appl, 2019 (2019), 317. doi: 10.1186/s13660-019-2272-7.  Google Scholar

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W. J. LiuQ. A. Ngo and V. N. Huy, Several interesting integral inequalities, J. Math. Inequal., 3 (2009), 201-212.  doi: 10.7153/jmi-03-20.  Google Scholar

[25]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 87-92.   Google Scholar

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K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.  Google Scholar

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D. Nie, S. Rashid, A. O. Akdemir, D. Baleanu and J. -B. Liu, On some new weighted inequalities for differentiable exponentially convex and exponentially quasi-convex functions with applications, Mathematics, 7 (2019), 727. doi: 10.3390/math7080727.  Google Scholar

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M. A. Noor, K. I. Noor and S. Rashid, Some new classes of preinvex functions and inequalities, Mathematics, 7 (2019), 29. doi: 10.3390/math7010029.  Google Scholar

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D. Oregan and B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl., 2015 (2015), 247-257.  doi: 10.1186/s13660-015-0769-2.  Google Scholar

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K. M. Owolabi, Analysis and numerical simulation of multicomponent system with Atangana-Baleanu fractional derivative, Chaos, Solitons & Fractals, 115 (2018), 127-134.  doi: 10.1016/j.chaos.2018.08.022.  Google Scholar

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K. M. Owolabi, Numerical patterns in system of integer and non-integer order derivatives, Chaos, Solitons & Fractals, 115 (2018), 143-153.  doi: 10.1016/j.chaos.2018.08.010.  Google Scholar

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K. M. Owolabi, Mathematical modelling and analysis of love dynamics: A fractional approach, Physica A: Stat. Mech. Appl., 525 (2019), 849-865.  doi: 10.1016/j.physa.2019.04.024.  Google Scholar

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K. M. Owolabi and A. Atangana, Computational study of multi-species fractional reaction-diffusion system with ABC operator, Chaos. Solitons & Fractals., 128 (2019), 280-289.  doi: 10.1016/j.chaos.2019.07.050.  Google Scholar

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K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Physica A: Stat. Mech. Appl., 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.  Google Scholar

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G. RahmanT. AbdeljawadF. JaradA. Khan and K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Diff. Eqs, 2019 (2019), 454-464.  doi: 10.1186/s13662-019-2381-0.  Google Scholar

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show all references

References:
[1]

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66.  doi: 10.1016/j.cam.2014.10.016.  Google Scholar

[2]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 2017 (2017), 78-87.  doi: 10.1186/s13662-017-1126-1.  Google Scholar

[3]

T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.  Google Scholar

[4]

M. Adil KhanY.-M. ChuT. U. Khan and J. Khan, Some new inequalities of Hermite-Hadamard type for $s$-convex functions with applications, Open Math., 15 (2017), 1414-1430.   Google Scholar

[5]

M. Adil Khan, Y.-M. Chu, A. Kashuri, R. Liko and G. Ali, Conformable fractional integrals versions of Hermite-Hadamard inequalities and their generalizations, J. Funct. Spaces, 2018 (2018), 6928130, 9 pp. doi: 10.1155/2018/6928130.  Google Scholar

[6]

M. Adil KhanA. IqbalM. Suleman and Y.-M. Chu, Hermite-Hadamard type inequalities for fractionalintegrals via Green's function, J. Inequal. Appl., 2018 (2018), 161-176.  doi: 10.1186/s13660-018-1751-6.  Google Scholar

[7]

R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simulat, 44 (2017), 460-481.  doi: 10.1016/j.cnsns.2016.09.006.  Google Scholar

[8]

J. AlzabutT. AbdeljawadF. Jarad and W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequal. Appl., 2019 (2019), 101-113.  doi: 10.1186/s13660-019-2052-4.  Google Scholar

[9]

D. R. Anderson and D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109-137.   Google Scholar

[10]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 13 (2018), 3-24.  doi: 10.1051/mmnp/2018010.  Google Scholar

[11]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus, Models and Numerical Methods, World Scientific: Singapore, 2012. doi: 10.1142/9789814355216.  Google Scholar

[12]

S. Bhatter, A. Mathur, D. Kumar and J. Singh, A new analysis of fractional Drinfeld-Sokolov-Wilson model with exponential memory, Physica A, 537 (2020), 122578, 13 pp. doi: 10.1016/j.physa.2019.122578.  Google Scholar

[13]

Y.-M. ChuM. Adil KhanT. Ali and S. S. Dragomir, Inequalities for $GA$-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 93-105.  doi: 10.1186/s13660-017-1371-6.  Google Scholar

[14]

Z. Dahmani, New classes of integral inequalities of fractional order, Matematiche, 69 (2011), 237-247.  doi: 10.4418/2014.69.1.18.  Google Scholar

[15]

F. JaradT. Abdeljawad and J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives, Eur. Phys. J. Spec. Top., 226 (2017), 3457-3471.  doi: 10.1140/epjst/e2018-00021-7.  Google Scholar

[16]

F. JaradM. A. Alqudah and T. Abdeljawad, On more general forms of proportional fractional operators, Open Math., 18 (2020), 167-176.   Google Scholar

[17]

F. JaradU. UgurluT. Abdeljawad and D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ, 2017 (2017), 247-263.  doi: 10.1186/s13662-017-1306-z.  Google Scholar

[18]

H. Kalsoom, S. Rashid, M. Idrees, Y.-M. Chu and D. Baleanu, Two variable quantum integral inequalities of Simpson-type based on higher order generalized strongly preinvex and quasi preinvex functions, Symmetry, 12 (2020), 51. doi: 10.3390/sym12010051.  Google Scholar

[19]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, 2006.  Google Scholar

[20]

R. KhalilM. Al HoraniA. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.  Google Scholar

[21]

D. KumarJ. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Math. Methods. Appl. Scis., 43 (2020), 443-457.   Google Scholar

[22]

D. KumarJ. SinghK. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, Inter. J. Heat. Mass. Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094.  Google Scholar

[23]

M. A. Latif, S. Rashid, S. S. Dragomir and Y.-M. Chu, Hermite-Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications, J. Inequal. Appl, 2019 (2019), 317. doi: 10.1186/s13660-019-2272-7.  Google Scholar

[24]

W. J. LiuQ. A. Ngo and V. N. Huy, Several interesting integral inequalities, J. Math. Inequal., 3 (2009), 201-212.  doi: 10.7153/jmi-03-20.  Google Scholar

[25]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 87-92.   Google Scholar

[26]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.  Google Scholar

[27]

D. Nie, S. Rashid, A. O. Akdemir, D. Baleanu and J. -B. Liu, On some new weighted inequalities for differentiable exponentially convex and exponentially quasi-convex functions with applications, Mathematics, 7 (2019), 727. doi: 10.3390/math7080727.  Google Scholar

[28]

M. A. Noor, K. I. Noor and S. Rashid, Some new classes of preinvex functions and inequalities, Mathematics, 7 (2019), 29. doi: 10.3390/math7010029.  Google Scholar

[29]

D. Oregan and B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl., 2015 (2015), 247-257.  doi: 10.1186/s13660-015-0769-2.  Google Scholar

[30]

K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Math. Model. Nat. Phenom., 13 (2018), 7-34.  doi: 10.1051/mmnp/2018006.  Google Scholar

[31]

K. M. Owolabi, Analysis and numerical simulation of multicomponent system with Atangana-Baleanu fractional derivative, Chaos, Solitons & Fractals, 115 (2018), 127-134.  doi: 10.1016/j.chaos.2018.08.022.  Google Scholar

[32]

K. M. Owolabi, Numerical patterns in system of integer and non-integer order derivatives, Chaos, Solitons & Fractals, 115 (2018), 143-153.  doi: 10.1016/j.chaos.2018.08.010.  Google Scholar

[33]

K. M. Owolabi, Mathematical modelling and analysis of love dynamics: A fractional approach, Physica A: Stat. Mech. Appl., 525 (2019), 849-865.  doi: 10.1016/j.physa.2019.04.024.  Google Scholar

[34]

K. M. Owolabi and A. Atangana, Computational study of multi-species fractional reaction-diffusion system with ABC operator, Chaos. Solitons & Fractals., 128 (2019), 280-289.  doi: 10.1016/j.chaos.2019.07.050.  Google Scholar

[35]

K. M. Owolabi and A. Atangana, Numerical Methods for Fractional Differentiation, Springer Series in Computational Mathematics book series (SSCM), 2019. doi: 10.1007/978-981-15-0098-5.  Google Scholar

[36]

K. M. Owolabi and A. Atangana, On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems, Chaos, 29 (2019), 023111, 12 pp. doi: 10.1063/1.5085490.  Google Scholar

[37]

K. M. Owolabi and Z. Hammouch, Mathematical modeling and analysis of two-variable system with noninteger-order derivative, Chaos, 29 (2019), 013145, 15 pp. doi: 10.1063/1.5086909.  Google Scholar

[38]

K. M. Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order derivative, Physica A: Stat. Mech. Appl., 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.  Google Scholar

[39] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego CA, 1999.   Google Scholar
[40]

G. RahmanT. AbdeljawadA. Khan and K. S. Nisar, Some fractional proportional integral inequalities, J. Inequal. Appl., 2019 (2019), 244-257.  doi: 10.1186/s13660-019-2199-z.  Google Scholar

[41]

G. RahmanT. AbdeljawadF. JaradA. Khan and K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Diff. Eqs, 2019 (2019), 454-464.  doi: 10.1186/s13662-019-2381-0.  Google Scholar

[42]

S. RashidM. A. NoorK. I. Noor and F. Safdar, Integral inequalities for generalized preinvex functions, Punjab. Univ. J. Math., 51 (2019), 77-91.   Google Scholar

[43]

S. Rashid, M. A. Noor, K. I. Noor, F. Safdar and Y.-M. Chu, Hermite-Hadamard inequalities for the class of convex functions on time scale, Mathematics., 956 (2019). Google Scholar

[44]

S. Rashid, T. Abdeljawad, F. Jarad and M. A. Noor, Some estimates for generalized Riemann-Liouville fractional integrals of exponentially convex functions and their applications, Mathematics, 7 (2019), 807. doi: 10.3390/math7090807.  Google Scholar

[45]

S. RashidA. O. AkdemirF. JaradM. A. Noor and K. I. Noor, Simpson's type integral inequalities for $k$-fractional integrals and their applications, AIMS. Math., 4 (2019), 1087-1100.  doi: 10.3934/math.2019.4.1087.  Google Scholar

[46]

S. Rashid, A. O. Akdemir, M. A. Noor and K. I. Noor, Generalization of inequalities analogous to preinvex functions via extended generalized Mittag-Leffler functions, in Proceedings of the International Conference on Applied and Engineering Mathematics?Second International Conference, ICAEM 2019, Hitec Taxila, Pakistan, (2019), 256–263. doi: 10.1109/ICAEM.2019.8853807.  Google Scholar

[47]

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