• Previous Article
    Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition
  • DCDS-S Home
  • This Issue
  • Next Article
    Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves
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 and Continuous Dynamical Systems - S, 2021, 14 (10) : 3703-3718. doi: 10.3934/dcdss.2021020
References:
[1]

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

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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. 

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[42]

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

[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).

[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.

[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.

[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.

[47]

S. Rashid, F. Jarad, M. A. Noor and H. Kalsoom, Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2020), 1225. doi: 10.3390/math7121225.

[48]

S. Rashid, M. A. Latif, Z. Hammouch and Y.-M. Chu, Fractional integral inequalities for strongly $h$-preinvex functions for a kth order differentiable functions, Symmetry, 11 (2019), 1448. doi: 10.3390/sym11121448.

[49]

S. Rashid, M. A. Noor and K. I. Noor, New Estimates for Exponentially Convex functions via conformable fractional operator, Fractal Fract., 3 (2019), 19. doi: 10.3390/fractalfract3020019.

[50]

S. RashidM. A. Noor and K. I. Noor, Some generalize Riemann-Liouville fractional estimates involving function having exponentially convexity property, Punjab. Univ. J. Math., 51 (2019), 1-15. 

[51]

S. RashidM. A. Noor and K. I. Noor, Fractional exponentially $m$-convex functions and inequalities, Int. J. Anal.Appl., 17 (2019), 464-478. 

[52]

S. Rashid, M. A. Noor and K. I. Noor, Inequalities pertaining fractional approach through exponentially convex functions, Fractal Fract., 37 (2019). doi: 10.3390/fractalfract3030037.

[53]

S. Rashid, M. A. Noor, K. I. Noor and A. O. Akdemir, Some new generalizations for exponentially $s$-convexfunctions and inequalities via fractional operators, Fractal Fract., 24 (2019). doi: 10.3390/fractalfract3020024.

[54]

S. RashidF. SafdarA. O. AkdemirM. A. Noor and K. I. Noor, Some new fractional integral inequalities for exponentially $m$-convex functions via extended generalized Mittag-Leffler function, J. Inequal. Appl., 2019 (2019), 299-316.  doi: 10.1186/s13660-019-2248-7.

[55]

G. Samko, A. A. Kilbas and I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, (1993)

[56]

J. SinghD. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Math. Modelling. Natural. Phenomena, 14 (2019), 303-326.  doi: 10.1051/mmnp/2018068.

[57]

Y. -Q. Song, M. Adil Khan, S. Zaheer Ullah and Y.-M. Chu, Integral inequalities involving strongly convex functions, J. Funct. Spaces, 2018 (2018), 6595921, 8 pp. doi: 10.1155/2018/6595921.

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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[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. 

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[42]

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

[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).

[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.

[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.

[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.

[47]

S. Rashid, F. Jarad, M. A. Noor and H. Kalsoom, Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2020), 1225. doi: 10.3390/math7121225.

[48]

S. Rashid, M. A. Latif, Z. Hammouch and Y.-M. Chu, Fractional integral inequalities for strongly $h$-preinvex functions for a kth order differentiable functions, Symmetry, 11 (2019), 1448. doi: 10.3390/sym11121448.

[49]

S. Rashid, M. A. Noor and K. I. Noor, New Estimates for Exponentially Convex functions via conformable fractional operator, Fractal Fract., 3 (2019), 19. doi: 10.3390/fractalfract3020019.

[50]

S. RashidM. A. Noor and K. I. Noor, Some generalize Riemann-Liouville fractional estimates involving function having exponentially convexity property, Punjab. Univ. J. Math., 51 (2019), 1-15. 

[51]

S. RashidM. A. Noor and K. I. Noor, Fractional exponentially $m$-convex functions and inequalities, Int. J. Anal.Appl., 17 (2019), 464-478. 

[52]

S. Rashid, M. A. Noor and K. I. Noor, Inequalities pertaining fractional approach through exponentially convex functions, Fractal Fract., 37 (2019). doi: 10.3390/fractalfract3030037.

[53]

S. Rashid, M. A. Noor, K. I. Noor and A. O. Akdemir, Some new generalizations for exponentially $s$-convexfunctions and inequalities via fractional operators, Fractal Fract., 24 (2019). doi: 10.3390/fractalfract3020024.

[54]

S. RashidF. SafdarA. O. AkdemirM. A. Noor and K. I. Noor, Some new fractional integral inequalities for exponentially $m$-convex functions via extended generalized Mittag-Leffler function, J. Inequal. Appl., 2019 (2019), 299-316.  doi: 10.1186/s13660-019-2248-7.

[55]

G. Samko, A. A. Kilbas and I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, (1993)

[56]

J. SinghD. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Math. Modelling. Natural. Phenomena, 14 (2019), 303-326.  doi: 10.1051/mmnp/2018068.

[57]

Y. -Q. Song, M. Adil Khan, S. Zaheer Ullah and Y.-M. Chu, Integral inequalities involving strongly convex functions, J. Funct. Spaces, 2018 (2018), 6595921, 8 pp. doi: 10.1155/2018/6595921.

[1]

Huaiyu Zhou, Jingbo Dou. Classifications of positive solutions to an integral system involving the multilinear fractional integral inequality. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022070

[2]

Yongliang Zhou, Yangkendi Deng, Di Wu, Dunyan Yan. Necessary and sufficient conditions on weighted multilinear fractional integral inequality. Communications on Pure and Applied Analysis, 2022, 21 (2) : 727-747. doi: 10.3934/cpaa.2021196

[3]

Jaydeep Swarnakar. Discrete-time realization of fractional-order proportional integral controller for a class of fractional-order system. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 309-320. doi: 10.3934/naco.2021007

[4]

Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom. More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2119-2135. doi: 10.3934/dcdss.2021063

[5]

Gennaro Infante. Positive and increasing solutions of perturbed Hammerstein integral equations with derivative dependence. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 691-699. doi: 10.3934/dcdsb.2019261

[6]

Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure and Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229

[7]

Jagannathan Gomatam, Isobel McFarlane. Generalisation of the Mandelbrot set to integral functions of quaternions. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 107-116. doi: 10.3934/dcds.1999.5.107

[8]

Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085

[9]

Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks and Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555

[10]

Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012

[11]

Gümrah Uysal. On a special class of modified integral operators preserving some exponential functions. Mathematical Foundations of Computing, 2022  doi: 10.3934/mfc.2021044

[12]

Kazuhiro Ishige, Tatsuki Kawakami, Kanako Kobayashi. Global solutions for a nonlinear integral equation with a generalized heat kernel. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 767-783. doi: 10.3934/dcdss.2014.7.767

[13]

William Thomson. For claims problems, another compromise between the proportional and constrained equal awards rules. Journal of Dynamics and Games, 2015, 2 (3&4) : 363-382. doi: 10.3934/jdg.2015011

[14]

Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032

[15]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248

[16]

Stanisław Migórski, Shengda Zeng. The Rothe method for multi-term time fractional integral diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 719-735. doi: 10.3934/dcdsb.2018204

[17]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3529-3539. doi: 10.3934/dcdss.2020432

[18]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations and Control Theory, 2022, 11 (1) : 225-238. doi: 10.3934/eect.2020109

[19]

Yixuan Wu, Yanzhi Zhang. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 851-876. doi: 10.3934/dcdss.2022016

[20]

Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1947-1955. doi: 10.3934/dcdss.2020152

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (351)
  • HTML views (463)
  • Cited by (0)

Other articles
by authors

[Back to Top]