On the existence of mild solutions for totally nonlinear Caputo-Hadamard fractional differential equations
Year 2022,
, 161 - 168, 30.06.2022
Abdelouaheb Ardjouni
,
Abderrahim Guerfi
Abstract
The existence of mild solutions of a totally nonlinear Caputo-Hadamard fractional differential equation is
investigated using the Krasnoselskii-Burton fixed point theorem and some results are presented. Two example
are given to illustrate our obtained results.
References
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of Qualitative Theory of Di?erential Equations 2009(1) (2009), 1-20.
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functional differential equations, Electronic Journal of Differential Equations 2017(36) (2017), 1-11.
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Proyecciones 40(1) (2021), 139-152.
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(2021), 77-86.
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equations, Kragujevac Journal of Mathematics 45(6) (2021), 897-908.
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Journal of Matematik 7(2) (2019), 314-317.
- [8] A. Ardjouni, A. Djoudi, Approximating solutions of nonlinear hybrid Caputo fractional integro-differential equations via
Dhage iteration principle, Ural Mathematical Journal 5(1) 2019, 3-12.
- [9] A. Ardjouni, A. Djoudi, Existence and uniqueness of positive solutions for first-order nonlinear Liouville-Caputo fractional
differential equations, São Paulo J. Math. Sci. 14 (2020), 381-390.
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di?erential equations, Results in Nonlinear Analysis 2(3) (2019) 136-142.
- [11] A. Ardjouni, A. Lachouri, A. Djoudi, Existence and uniqueness results for nonlinear hybrid implicit Caputo-Hadamard
fractional differential equations, Open Journal of Mathematical Analysis 3(2) (2019), 106-111.
- [12] Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal.
Appl. 311 (2005) 495-505.
- [13] Z.B. Bai, T.T. Qiu, Existence of positive solution for singular fractional differential equation, Appl. Math. Comput. 215
(2009), 2761-2767.
- [14] H. Boulares, A. Ardjouni, Y. Laskri, Positive solutions for nonlinear fractional differential equations, Positivity 21 (2017),
1201?1212.
- [15] B. Bordj, A. Ardjouni, Periodic and asymptotically periodic solutions in nonlinear coupled Volterra integro-dynamic
systems with in nite delay on time scales, Advances in the Theory of Nonlinear Analysis and its Applications 5(2) (2021)
180-192.
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- [17] D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl.
204 (1996), 609-625.
- [18] C. Derbazi, Z. Baitiche, M. Benchohra, A. Cabada, Initial value problem for nonlinear fractional differential equations with
ψ-Caputo derivative via monotone iterative technique, Axioms 9(57) (2020), 55-67.
- [19] C. Derbazi, Z. Baitiche, M. Feckan, Some new uniqueness and Ulam stability results for a class of multiterms fractional
differential equations in the framework of generalized Caputo fractional derivative using the Φ-fractional Bielecki-type
norm, Turk. J. Math. 45 (2021), 2307-2322.
- [20] C. Derbazi, Z. Baitiche, A. Zada, Existence and uniqueness of positive solutions for fractional relaxation equation
in terms of ψ-Caputo fractional derivative, International Journal of Nonlinear Sciences and Numerical Simulation,
https://doi.org/10.1515/ijnsns-2020-0228.
- [21] E. Kaufmann, E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation,
Electron. J. Qual. Theory Differ. Equ. 3 (2008), 1-11.
- [22] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Ams-
terdam 2006.
- [23] C. Kou, H. Zhou, Y. Yan, Existence of solutions of initial value problems for nonlinear fractional differential equations on
the half-axis, Nonlinear Anal. 74 (2011), 5975-5986.
- [24] K.Q. Lan, W. Lin, Positive solutions of systems of Caputo fractional differential equations, Communications in Applied
Analysis 17(1) (2013), 61-86.
- [25] M. Matar, On existence of positive solution for initial value problem of nonlinear fractional differential equations of order
1 < α ≤ 2, Acta Math. Univ. Comenianae, LXXXIV(1) (2015), 51-57.
- [26] K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York,
1993.
- [27] S. Niyom, S.K. Ntouyas, S. Laoprasittichok, J. Tariboon, Boundary value problems with four orders of Riemann-Liouville
fractional derivatives, Adv. Difference Equ., 2016(165) (2016), 1-14.
- [28] S.K. Ntouyas, J. Tariboon, Fractional boundary value problems with multiple orders of fractional derivatives and integrals,
Electronic Journal of Differential Equations, 2017(100) (2017), 1-18.
- [29] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
- [30] C. Wang, R. Wang, S. Wang, C. Yang, Positive Solution of Singular Boundary Value Problem for a Nonlinear Fractional
Differential Equation, Bound. Value Probl. 2011 (2011), Art ID 297026.
- [31] C. Wang, H. Zhang, S. Wang, Positive solution of a nonlinear fractional differential equation involving Caputo derivative,
Discrete Dynamics in Natural and Society 2012 (2012), Art ID425408.
- [32] S. Zhang, Existence results of positive solutions to boundary value problem for fractional differential equation, Positivity,
13(3) (2009), 583-599.
- [33] S. Zhang, The existence of a positive solution for a fractional di?erential equation, J. Math. Anal. Appl. 252 (2000),
804-812.
Year 2022,
, 161 - 168, 30.06.2022
Abdelouaheb Ardjouni
,
Abderrahim Guerfi
References
- [1] M. Adivar, Y.N. Raffoul, Existence of periodic solutions in totally nonlinear delay dynamic equations, Electronic Journal
of Qualitative Theory of Di?erential Equations 2009(1) (2009), 1-20.
- [2] B. Ahmad, S.K. Ntouyas, Existence and uniqueness of solutions for Caputo-Hadamard sequential fractional order neutral
functional differential equations, Electronic Journal of Differential Equations 2017(36) (2017), 1-11.
- [3] A. Ardjouni, Existence and uniqueness of positive solutions for nonlinear Caputo-Hadamard fractional differential equations,
Proyecciones 40(1) (2021), 139-152.
- [4] A. Ardjouni, Asymptotic stability in Caputo-Hadamard fractional dynamic equations, Results in Nonlinear Analysis 4(2)
(2021), 77-86.
- [5] A. Ardjouni, Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions,
AIMS Mathematics 4(4) (2019), 1101-1113.
- [6] A. Ardjouni, A. Djoudi, Positive solutions for first-order nonlinear Caputo-Hadamard fractional relaxation differential
equations, Kragujevac Journal of Mathematics 45(6) (2021), 897-908.
- [7] A. Ardjouni, A. Djoudi, Initial-value problems for nonlinear hybrid implicit Caputo fractional differential equations, Malaya
Journal of Matematik 7(2) (2019), 314-317.
- [8] A. Ardjouni, A. Djoudi, Approximating solutions of nonlinear hybrid Caputo fractional integro-differential equations via
Dhage iteration principle, Ural Mathematical Journal 5(1) 2019, 3-12.
- [9] A. Ardjouni, A. Djoudi, Existence and uniqueness of positive solutions for first-order nonlinear Liouville-Caputo fractional
differential equations, São Paulo J. Math. Sci. 14 (2020), 381-390.
- [10] A. Ardjouni, A Djoudi, Existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional
di?erential equations, Results in Nonlinear Analysis 2(3) (2019) 136-142.
- [11] A. Ardjouni, A. Lachouri, A. Djoudi, Existence and uniqueness results for nonlinear hybrid implicit Caputo-Hadamard
fractional differential equations, Open Journal of Mathematical Analysis 3(2) (2019), 106-111.
- [12] Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal.
Appl. 311 (2005) 495-505.
- [13] Z.B. Bai, T.T. Qiu, Existence of positive solution for singular fractional differential equation, Appl. Math. Comput. 215
(2009), 2761-2767.
- [14] H. Boulares, A. Ardjouni, Y. Laskri, Positive solutions for nonlinear fractional differential equations, Positivity 21 (2017),
1201?1212.
- [15] B. Bordj, A. Ardjouni, Periodic and asymptotically periodic solutions in nonlinear coupled Volterra integro-dynamic
systems with in nite delay on time scales, Advances in the Theory of Nonlinear Analysis and its Applications 5(2) (2021)
180-192.
- [16] T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.
- [17] D. Delbosco, L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl.
204 (1996), 609-625.
- [18] C. Derbazi, Z. Baitiche, M. Benchohra, A. Cabada, Initial value problem for nonlinear fractional differential equations with
ψ-Caputo derivative via monotone iterative technique, Axioms 9(57) (2020), 55-67.
- [19] C. Derbazi, Z. Baitiche, M. Feckan, Some new uniqueness and Ulam stability results for a class of multiterms fractional
differential equations in the framework of generalized Caputo fractional derivative using the Φ-fractional Bielecki-type
norm, Turk. J. Math. 45 (2021), 2307-2322.
- [20] C. Derbazi, Z. Baitiche, A. Zada, Existence and uniqueness of positive solutions for fractional relaxation equation
in terms of ψ-Caputo fractional derivative, International Journal of Nonlinear Sciences and Numerical Simulation,
https://doi.org/10.1515/ijnsns-2020-0228.
- [21] E. Kaufmann, E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation,
Electron. J. Qual. Theory Differ. Equ. 3 (2008), 1-11.
- [22] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Ams-
terdam 2006.
- [23] C. Kou, H. Zhou, Y. Yan, Existence of solutions of initial value problems for nonlinear fractional differential equations on
the half-axis, Nonlinear Anal. 74 (2011), 5975-5986.
- [24] K.Q. Lan, W. Lin, Positive solutions of systems of Caputo fractional differential equations, Communications in Applied
Analysis 17(1) (2013), 61-86.
- [25] M. Matar, On existence of positive solution for initial value problem of nonlinear fractional differential equations of order
1 < α ≤ 2, Acta Math. Univ. Comenianae, LXXXIV(1) (2015), 51-57.
- [26] K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York,
1993.
- [27] S. Niyom, S.K. Ntouyas, S. Laoprasittichok, J. Tariboon, Boundary value problems with four orders of Riemann-Liouville
fractional derivatives, Adv. Difference Equ., 2016(165) (2016), 1-14.
- [28] S.K. Ntouyas, J. Tariboon, Fractional boundary value problems with multiple orders of fractional derivatives and integrals,
Electronic Journal of Differential Equations, 2017(100) (2017), 1-18.
- [29] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
- [30] C. Wang, R. Wang, S. Wang, C. Yang, Positive Solution of Singular Boundary Value Problem for a Nonlinear Fractional
Differential Equation, Bound. Value Probl. 2011 (2011), Art ID 297026.
- [31] C. Wang, H. Zhang, S. Wang, Positive solution of a nonlinear fractional differential equation involving Caputo derivative,
Discrete Dynamics in Natural and Society 2012 (2012), Art ID425408.
- [32] S. Zhang, Existence results of positive solutions to boundary value problem for fractional differential equation, Positivity,
13(3) (2009), 583-599.
- [33] S. Zhang, The existence of a positive solution for a fractional di?erential equation, J. Math. Anal. Appl. 252 (2000),
804-812.