Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type

Saleh S. Redhwan; Sadikali L. Shaikh; Mohammed S. Abdo; Saleh S. Redhwan; Sadikali L. Shaikh; Mohammed S. Abdo

doi:10.3934/math.2020240

AIMS Mathematics

2020, Volume 5, Issue 4: 3714-3730. doi: 10.3934/math.2020240

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Research article

Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type

1.
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University Aurangabad, (M.S), 431001, India
2.
Department of Mathematics, Maulana Azad College of arts, Science and Commerce, RozaBagh, Aurangabad 431004 (M.S.), India
3.
Department of Mathematics, Hodeidah University, Al-Hodeidah, Yemen

Received: 30 December 2019 Accepted: 25 March 2020 Published: 20 April 2020
MSC : 34B15, 34B18, 26A33, 34A12

This paper deals with a nonlinear implicit fractional differential equation with the anti-periodic boundary condition involving the Caputo-Katugampola type. The existence and uniqueness results are established by applying the fixed point theorems of Krasnoselskii and Banach. Further, by using generalized Gronwall inequality the Ulam-Hyers stability results are proved. To demonstrate the effectiveness of the main results, appropriate examples are granted.
- fractional differential equations,
- Katugampola fractional operator,
- Ulam-Hyers stability,
- fixed point theorems,
- fractional Gronwall inequality
Citation: Saleh S. Redhwan, Sadikali L. Shaikh, Mohammed S. Abdo. Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type[J]. AIMS Mathematics, 2020, 5(4): 3714-3730. doi: 10.3934/math.2020240

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Abstract

This paper deals with a nonlinear implicit fractional differential equation with the anti-periodic boundary condition involving the Caputo-Katugampola type. The existence and uniqueness results are established by applying the fixed point theorems of Krasnoselskii and Banach. Further, by using generalized Gronwall inequality the Ulam-Hyers stability results are proved. To demonstrate the effectiveness of the main results, appropriate examples are granted.

References

[1]	M. Hamid, M. Usman, Z. H. Khan, et al. Dual solutions and stability analysis of flow and heat transfer of Casson fluid over a stretching sheet, Phys. Lett. A, 383 (2019), 2400-2408. doi: 10.1016/j.physleta.2019.04.050
[2]	M. Hamid, M. Usman, Z. H. Khan, et al. Numerical study of unsteady MHD flow of Williamson nanofluid in a permeable channel with heat source/sink and thermal radiation, Europ. Phys. J. Plus, 133 (2018), 527. doi: 10.1140/epjp/i2018-12322-5
[3]	M. Hamid, M. Usman, T. Zubair, et al. Shape effects of MoS2 nanoparticles on rotating flow of nanofluid along a stretching surface with variable thermal conductivity: A Galerkin approach, Int. J. Heat Mass Transfer, 124 (2018), 706-714. doi: 10.1016/j.ijheatmasstransfer.2018.03.108
[4]	S. T. Mohyud-Din, M. Usman, K. Afaq, et al. Examination of carbon-water nanofluid flow with thermal radiation under the effect of Marangoni convection, Eng. Comput., 34 (2017), 2330-2343. doi: 10.1108/EC-04-2017-0135
[5]	M. Usman, M. M. Din, T. Zubair, et al. Fluid flow and heat transfer investigation of blood with nanoparticles through porous vessels in the presence of magnetic field, J. Algorithms Comput. Technol., 13 (2018), 1748301818788661.
[6]	M. Usman, M. Hamid, R. U. Haq, et al. Heat and fluid flow of water and ethylene-glycol based Cu-nanoparticles between two parallel squeezing porous disks: LSGM approach, Int. J. Heat Mass Transfer, 123 (2018), 888-895. doi: 10.1016/j.ijheatmasstransfer.2018.03.030
[7]	A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, 204, 2006.
[8]	A. B. Malinowska, T. Odzijewicz, D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations, Springer, Berlin, 2015.
[9]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[10]	R. P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095-1100. doi: 10.1016/j.camwa.2009.05.010
[11]	D. Baleanu, K. Diethelm, E. Scalas, et al. Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, 3, 2012.
[12]	D. Baleanu, O. G. Mustafa, R. P. Agarwal, On L^p -solutions for a class of sequential fractional differential equations, Appl. Math. Comput., 218 (2011), 2074-2081.
[13]	U. N. Katugampola, New approach to a generalized fractional integral. Appl. Math. Comput., 218 (2011), 860-865.
[14]	A. G. Butkovskii, S. S. Postnov, E. A. Postnova, Fractional integro-differential calculus and its control-theoretical applications i-mathematical fundamentals and the problem of interpretation, Autom. Remote Control, 74 (2013), 543-574. doi: 10.1134/S0005117913040012
[15]	S. Gaboury, R. Tremblay, B. J. Fugere, Some relations involving a generalized fractional derivative operator, J. Inequalities Appl., 167, 2013.
[16]	R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scienti c, River Edge, New Jerzey, 2 Eds., 2014.
[17]	G. Jumarie, On the solution of the stochastic differential equation of exponential growth driven by fractional brownian motion, Appl. Math. Lett., 18 (2005), 817-826. doi: 10.1016/j.aml.2004.09.012
[18]	U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.
[19]	U. N. Katugampola, Mellin transforms of the generalized fractional integrals and derivatives, Appl. Math. Comput., 257 (2015), 566-580.
[20]	U. N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, Preprint, arXiv:1411.5229, 2014.
[21]	R. Almeida, A Gronwall inequality for a general Caputo fractional operator, arXiv preprint arXiv:1705.10079, 2017.
[22]	D. S. Oliveira, E. Capelas de Oliveira, Hilfer-Katugampola fractional derivatives, Comput. Appl. Math., 37 (2018), 3672-3690. doi: 10.1007/s40314-017-0536-8
[23]	S. Abbas, M. Benchohra, J. R. Graef, et al. Implicit Fractional Differential and Integral Equations: Existence and Stability, Walter de Gruyter: London, UK, 2018.
[24]	M. Benchohra, S. Bouriah, M. A. Darwish, Nonlinear boundary value problem for implicit differential equations of fractional order in Banach spaces, Fixed Point Theory, 18 (2017), 457-470. doi: 10.24193/fpt-ro.2017.2.36
[25]	M. Benchohra, S. Bouriah, J. R. Graef, Nonlinear implicit differential equations of fractional order at resonance, Electron. J. Differential Equations, 324 (2016), 1-10.
[26]	M. Benchohra, J. E. Lazreg, Nonlinear fractional implicit differential equations, Commun. Appl. Anal., 17 (2013), 471-482.
[27]	E. Alvarez, C. Lizama, R. Ponce, Weighted pseudo anti-periodic solutions for fractional integrodifferential equations in Banach spaces, Appl. Math. Comput., 259 (2015), 164-172.
[28]	F. Chen, A. Chen, X. Wu, Anti-periodic boundary value problems with Riesz-Caputo derivative, Advances Difference Equations, 1 (2019), 119.
[29]	D. Yang, C. Bai, Existence of solutions for Anti-Periodic fractional differential inclusions involving Riesz-Caputo fractional derivative, Mathematics, 7 (2019), 630.
[30]	B. Ahmad, J. J. Nieto, Anti-periodic fractional boundary value problems, Comput. Math. Appl., 62 (2011), 1150-1156. doi: 10.1016/j.camwa.2011.02.034
[31]	M. Benchohra, J. E. Lazreg, Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative, Studia Universitatis Babes-Bolyai, Mathematica, 62 (2017), 27-38. doi: 10.24193/subbmath.2017.0003
[32]	L. Palve, M. S. Abdo, S. K. Panchal, Some existence and stability results of HilferHadamard fractional implicit differential fractional equation in a weighted space, preprint: arXiv:1910.08369v1 math. GM, (2019), 20.
[33]	I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 26 (2010), 103-107.
[34]	J. Wang, Y. Zhou, M. Medved, Existence and stability of fractional differential equations with Hadamard derivative, Topol. Methods Nonlinear Anal., 41 (2013), 113-133.
[35]	T. A. Burton, C. KirkÙ, A fixed point theorem of Krasnoselskii Schaefer type, Mathematische Nachrichten, 189 (1998), 23-31. doi: 10.1002/mana.19981890103
[36]	J. V. C. Sousa, E. C. Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, arXiv preprint arXiv:1709.03634, 2017.
[37]	M. Benchohra, S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroccan J. Pure Appl. Anal., 1 (2015), 22-37. doi: 10.7603/s40956-015-0002-9
[38]	A. Boutiara, K. Guerbati, M. Benbachir, Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Math., 5 (2019), 259-272.
[39]	S. S. Redhwan, S. L. Shaikh, M. S. Abdo, Theory of Nonlinear Caputo-Katugampola Fractional Differential Equations, arXiv preprint arXiv:1911.08884, 2019.

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