International Journal of Nonlinear Analysis and Applications

Mixed nonlocal boundary value problem for implicit fractional differential equation involving both retarded and advanced arguments

Document Type : Review articles

Author

Department of Mathematics, Higher Normal School of Technological Education- Skikda, Algeria

Abstract

In this paper, we investigate the existence and uniqueness of solutions for nonlinear implicit Hilfer-Hadamard fractional differential equations involving both retarded and advanced arguments and nonlocal mixed boundary conditions. We also use the Banach contraction mapping principle and Schaefer’s fixed point theorem to show the existence and uniqueness of solutions. The results obtained here extend the work of Benchohra et al. [10] and Haoues et al. [18]. An example is also
given to illustrate the results.

Keywords

[1] S. Abbas, M. Benchohra and J. Henderson, Existence and attractivity results for Hilfer fractional differential equations, J. Math. Sci. 243 (2019), 347–357.
[2] S. Abbas, M. Benchohra and J.R. Graef, Weak solutions to implicit differential equations involving the Hilfer fractional derivative, Nonlinear Dyn. Syst. Theory 18 (2018), no. 1, 1-–11.
[3] S. Abbas, M. Benchohra, J.-E. Lagreg, A. Alsaedi and Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Difference. Equ. 180 (2017), 1–14.
[4] S. Abbas, M. Benchohra, J.E. Lagreg and Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations, Analysis and stability, Chaos Solitons Fractals 102 (2017), 47–71.
[5] B. Ahmadn and S.K. Ntouyas, Initial value problem of fractional order Hadamard-type functional differential equations, Electron. J. Differ. Equ. 77 (2015), 1–9.
[6] B. Ahmad and S.K. Ntouyas, Hilfer–Hadamard fractional boundary value problems with nonlocal mixed boundary conditions, Fractal Fractional 5 (2021), 195.
[7] R.P. Agarwal, S.K. Ntouyas, B. Ahmad and A.K. Alzahrani, Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments, Adv. Difference. Equ. 2016 (2016), 92.
[8] Y. Arioua and N. Benhamidouche, Boundary value problem for Caputo-Hadamard fractional differential equations, Surv. Math. Appl. 12 (2017), 103-–115.
[9] S.P. Bhairat, Existence and continuation of solution of Hilfer fractional differential equations, J. Math. Model. 7 (2019), no. 1, 1–20.
[10] M. Benchohra S. Bouriah and J. Henderson, Nonlinear implicit Hadamard’s fractional differential equations with retarded and advanced arguments, Azer. J. Math. 8 (2018), no. 2.
[11] M. Benchohra, S. Bouriah and J.R. Graef, Boundary value problems for nonlinear implicit Caputo–HadamardType fractional differential equations with impulses, Mediterr. J. Math. 14 (2017), no. 206, 1–21.
[12] M. Benchohra, J. R. Graef, N. Guerraiche and S. Hamani, Nonlinear boundary value problems for fractional differential inclusions with Caputo-Hadamard derivatives on the half line, AIMS Math. 6 (2021), no. 6, 6278—6292.
[13] M. Benchohra, J. Lazreg and G. N’G´uer´ekata, Nonlinear implicit Hadamard’s fractional differential equations on Banach space retarded and advanced arguments, Int. J. Evol. Equ. 10 (2015), no. 3-4, 283—295.
[14] P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2019), 1—27.
[15] A. Boutiara, K. Guerbati and M. Benbachir, Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Math. 5 (2019), no. 1, 259—272.
[16] Y.Y. Gambo, F. Jarad, D. Baleanu and T. Abdeljawad, On Caputo modification of the Hadamard fractional derivatives, Adv. Differ. Equ. 2014 (2014), no. 1, 1–12.
[17] A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York 2003.
[18] M. Haoues, A. Ardjouni and A. Djoudi, Existence and uniqueness of solutions for the nonlinear retarded and advanced implicit Hadamard fractional differential equations with nonlocal conditions, nonlinear stud. 27 (2020), no. 2, 433–445.
[19] R. Hilfer, Threefold introduction to fractional derivatives, Anomalous Transport: Foundations and Applications, 2008, pp. 17—73.
[20] R. Hilfer, Applications of fractional calculus in physics, Singapore: World Sc. Publ. Co., 2000.
[21] F. Jarad, T. Abdeljawad and D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ. 2012 (2012), no. 1, Article 142, 8 pages.
[22] M.D. Kassim and N.E. Tatar, Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal. 2013 (2013), 1–12.
[23] A.A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191–1204.
[24] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of the Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier: Amsterdam, The Netherlands, 2006.
[25] K. Oldham and J. Spanier, The Fractional Calculus, Acad. Press, New York, 1974.
[26] D.S. Oliveira and E.C. Oliveira, Hilfer-Katugampola fractional derivative, eprint arXiv:1705.07733v1 [math.CA].2017.
[27] I. Podlubny, Fractional Differential Equations, in: Mathematics in Science and Engineering, 198, Acad. Press, New York, 1999.
[28] M.D. Qassim, K.M. Furati and N.E. Tatar, On a differential equation involving Hilfer-Hadamard fractional derivative, Abst. Appl. Anal. 2012 (2012).
[29] D. Qian, Z. Gong and C. Li, A generalized Gr¨onwall inequality and its application to fractional differential equations with Hadamard derivatives, Proc. the 3rd. Conf. on Nonlinear Sci. Complexity, Cankaya University, 2010.
[30] A.Y.A. Salamooni and D.D. Pawar, Existence and uniqueness of nonlocal boundary conditions for Hilfer–Hadamard-type fractional differential, Adv. Differ. Equ. 2021 (2021).
[31] A.Y.A. Salamooni and D.D. Pawar Existence and stability results for Hilfer–Katugampola-type fractional implicit differential equations with nonlocal conditions, J. Nonlinear Sci. Appl. 14 (2021), no. 3, 124-–138.
[32] A.Y.A. Salamooni and D.D. Pawar, Existence and uniqueness of boundary value problems for Hilfer-Hadamardtype fractional differential equations, Ganita 70 (2020), no. 2, 1-–16.
[33] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon Breach, Tokyo-Paris-Berlin, 1993.
[34] D. Vivek, K. Shah and K. Kanagarajan, Dynamical analysis of Hilfer–Hadamard type fractional pantograph equations via successive approximation, J. Taibah Univ. Sci. 13 (2019), no. 1, 225—230.
[35] D. Viveka, K. Kanagarajana and E.M. Elsayedb, Nonlocal initial value problems for implicit differential equations with Hilfer–Hadamard fractional derivative, Nonlinear Anal. Model. Control. 23 (2018), no. 3, 341—360.
[36] D. Vivek, K. Kanagrajan and E.M. Elsayed, Some existence and stability results for Hilfer fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math. 15 (2018).
[37] J.R. Wang, Ch. Zhu and M. Feckan, Solvability of fully nonlinear functional equations involving Erd´elyi-Kober fractional integrals on the unbounded interval, Optim. 63 (2014), 1235–1248.
[38] J.R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput. 266 (2015), 850–859.
[39] Y. Zhou. Basic theory of fractional differential equations. World Scientific, Singapore, 2014.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 2697-2708
  • Receive Date: 09 April 2022
  • Revise Date: 26 June 2022
  • Accept Date: 15 July 2022