On Nonlinear Periodic Problems with Caputo's Exponential Fractional Derivative
Yıl 2023,
Cilt: 7 Sayı: 1, 103 - 120, 31.03.2023
Mohamed Chohri
Soufyane Bouriah
Salim Abdelkrim
,
Mouffak Benchohra
Öz
In this article, we employ Mawhin's theory of degree of coincidence to provide an existence result for a class of problems involving non-linear implicit fractional differential equations with the exponentially fractional derivative of Caputo. Two examples are provided to demonstrate the applicability of our results.
Kaynakça
- [1] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York,
2012.
- [2] S. Abbas, M. Benchohra and G M. N'Guérékata, Advanced Fractional Di?erential and Integral Equations, Nova Science
Publishers, New York, 2014.
- [3] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a
fractional differential equation, Math. Meth. Appl. Sci. (2020), 1-12.
- [4] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential
equations with multi-point and integral boundary conditions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM,
(2021), 115-155.
- [5] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solutions of fractional differential equations via Geraghty type
hybrid contractions, Appl. Comput. Math, 20 (2021), 313-333.
- [6] H. Afshari and E. Karapinar, A solution of the fractional differential equations in the setting of b-metric space. Carpathian
Math. Publ. 13 (2021), 764-774. https://doi.org/10.15330/cmp.13.3.764-774
- [7] H. Afshari and E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via
ψ-Hilfer fractional derivative on b-metric spaces, Adv. Di?erence Equ., 2020 (2020), 616. https://doi.org/10.1186/s13662-
020-03076-z
- [8] G.A. Anastassiou, Advances on Fractional Inequalities, Springer, New York, 2011.
- [9] D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculs Models and Numerical Methods, World Scientific
Publishing, New York, 2012.
- [10] D. Baleanu, Z.B. Güvenç and J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications,
Springer, New York, 2010.
- [11] D. Baleanu, J.A.T. Machado and A.C.-J. Luo, Fractional Dynamics and Control, Springer, 2012.
- [12] M. Benchohra and S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential
equations of fractional order, Moroccan J. Pure. Appl. Anal., 1 (1), (2015), 22-36.
- [13] M. Benchohra, S. Bouriah and J.J. Nieto, Existence of periodic solutions for nonlinear implicit Hadamard fractional
di?erential equations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 112 (1) (2018), 25-35.
- [14] M. Benchohra, S. Bouriah and J.R. Graef, Nonlinear implicit differential equation of fractional order at resonance, Electron.
J. Differential Equations Vol. 2016 (2016), No. 324, pp. 1-10.
- [15] M. Benchohra and J.E. Lazreg, Nonlinear fractional implicit di?erential equations. Commun. Appl. Anal., 17 (2013),
471-482.
- [16] S. Bouriah, D. Foukrach, M. Benchohra and J. Graef, Existence and uniqueness of periodic solutions for some non-
linear fractional pantograph differential equations with ψ-Caputo derivative, Arab. J. Math., 10 (2021), 575-587.
https://doi.org/10.1007/s40065-021-00343-z
- [17] C. Derbazi, H. Hammouche, A. Salim and M. Benchohra, Measure of noncompactness and fractional hybrid differential
equations with Hybrid conditions, Differ. Equ. Appl., 14 (2022), 145-161. http://dx.doi.org/10.7153/dea-2022-14-09
- [18] Y. Feng and Z. Bai, Solvability of some nonlocal fractional boundary value problems at resonance in R n , Fractal Fract., 6
(2022), 16pages. https://doi.org/10.3390/fractalfract6010025
- [19] D. Foukrach, S. Bouriah, M. Benchohra and E. Karapinar, Some new results for ψ-Hilfer fractional pantograph-type
differential equation depending on ψ-Riemann-Liouville integral, J. Anal (2021). https://doi.org/10.1007/s41478-021-00339-
0
- [20] D. Foukrach, S. Bouriah, S. Abbas and M. Benchohra, Periodic solutions of nonlinear fractional pantograph integro-
differential equations with Ψ-Caputo derivative, Ann. Univ. Ferrara (2022). https://doi.org/10.1007/s11565-022-00396-8
- [21] R. Hermann, Fractional Calculus: An Introduction For Physicists, World Scienti?c Publishing Co. Pte. Ltd. 2011.
- [22] A. Heris, A. Salim, M. Benchohra and E. Karapinar, Fractional partial random differential equations with infinite delay,
Results in Physics (2022). https://doi.org/10.1016/j.rinp.2022.105557
- [23] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- [24] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differenatial Equations, North-
Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
- [25] N. Laledj, A. Salim, J.E. Lazreg, S. Abbas, B. Ahmad and M. Benchohra, On implicit fractional q-difference equations:
Analysis and stability, Math. Meth. Appl. Sci., 2 (2022), 1-23. https://doi.org/10.1002/mma.8417
- [26] J.E. Lazreg, M. Benchohra and A. Salim, Existence and Ulam stability of k-Generalized ψ-Hilfer Fractional Problem, J.
Innov. Appl. Math. Comput. Sci., 2 (2022), 01-13.
- [27] A.J. Luo and V. Afraimovich, Long-range Interactions, Stochasticity and Fractional Dynamics, Springer, New York, Dor-
drecht, Heidelberg, London, 2010.
- [28] J. Mawhin, NSFCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1979.
- [29] S.K. Ntouyas, J. Tariboon, C. Sawaddee, Nonlocal initial and boundary value problems via fractional calculus with expo-
nential singular kernel, J. Nonlinear Sci. Appl., 11 (2018), 1015-1030.
- [30] D. O'Regan, Y.J. Chao, Y.Q. Chen, Topological Degree Theory and Application, Taylor and Francis Group, Boca Raton,
London, NewYork, 2006.
- [31] M.D. Otigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84. Springer,
Dordrecht, 2011.
- [32] I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Heidelberg Dordrecht London
New York, 2011.
- [33] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
- [34] P. Sahoo, T. Barman and J.P. Davim, Fractal Analysis in Machining, Springer, New York, Dordrecht, Heidelberg, London,
2011.
- [35] A. Salim, M. Benchohra, J.R. Graef and J.E. Lazreg, Initial value problem for hybrid ψ-Hilfer fractional implicit differential
equations, J. Fixed Point Theory Appl., 24 (2022), 14 pp. https://doi.org/10.1007/s11784-021-00920-x
- [36] A. Salim, M. Benchohra, J.E. Lazreg and J. Henderson, On k-generalized ψ-Hilfer boundary value problems with retardation
and anticipation, Adv. Theor. Nonl. Anal. Appl., 6 (2022), 173-190. https://doi.org/10.31197/atnaa.973992
- [37] A. Salim, M. Benchohra, J.E. Lazreg and E. Karapinar, On k-generalized ψ-Hilfer impulsive boundary value problem with
retarded and advanced arguments, J. Math. Ext., 15 (2021), 1-39. https://doi.org/10.30495/JME.SI.2021.2187
- [38] A. Salim, J.E. Lazreg, B. Ahmad, M. Benchohra and J.J. Nieto, A Study on k-Generalized ψ-Hilfer Derivative Operator,
Vietnam J. Math., (2022). https://doi.org/10.1007/s10013-022-00561-8
Yıl 2023,
Cilt: 7 Sayı: 1, 103 - 120, 31.03.2023
Mohamed Chohri
Soufyane Bouriah
Salim Abdelkrim
,
Mouffak Benchohra
Kaynakça
- [1] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York,
2012.
- [2] S. Abbas, M. Benchohra and G M. N'Guérékata, Advanced Fractional Di?erential and Integral Equations, Nova Science
Publishers, New York, 2014.
- [3] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a
fractional differential equation, Math. Meth. Appl. Sci. (2020), 1-12.
- [4] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, Uniqueness of solution for higher-order nonlinear fractional differential
equations with multi-point and integral boundary conditions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM,
(2021), 115-155.
- [5] R.S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solutions of fractional differential equations via Geraghty type
hybrid contractions, Appl. Comput. Math, 20 (2021), 313-333.
- [6] H. Afshari and E. Karapinar, A solution of the fractional differential equations in the setting of b-metric space. Carpathian
Math. Publ. 13 (2021), 764-774. https://doi.org/10.15330/cmp.13.3.764-774
- [7] H. Afshari and E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via
ψ-Hilfer fractional derivative on b-metric spaces, Adv. Di?erence Equ., 2020 (2020), 616. https://doi.org/10.1186/s13662-
020-03076-z
- [8] G.A. Anastassiou, Advances on Fractional Inequalities, Springer, New York, 2011.
- [9] D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculs Models and Numerical Methods, World Scientific
Publishing, New York, 2012.
- [10] D. Baleanu, Z.B. Güvenç and J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications,
Springer, New York, 2010.
- [11] D. Baleanu, J.A.T. Machado and A.C.-J. Luo, Fractional Dynamics and Control, Springer, 2012.
- [12] M. Benchohra and S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential
equations of fractional order, Moroccan J. Pure. Appl. Anal., 1 (1), (2015), 22-36.
- [13] M. Benchohra, S. Bouriah and J.J. Nieto, Existence of periodic solutions for nonlinear implicit Hadamard fractional
di?erential equations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 112 (1) (2018), 25-35.
- [14] M. Benchohra, S. Bouriah and J.R. Graef, Nonlinear implicit differential equation of fractional order at resonance, Electron.
J. Differential Equations Vol. 2016 (2016), No. 324, pp. 1-10.
- [15] M. Benchohra and J.E. Lazreg, Nonlinear fractional implicit di?erential equations. Commun. Appl. Anal., 17 (2013),
471-482.
- [16] S. Bouriah, D. Foukrach, M. Benchohra and J. Graef, Existence and uniqueness of periodic solutions for some non-
linear fractional pantograph differential equations with ψ-Caputo derivative, Arab. J. Math., 10 (2021), 575-587.
https://doi.org/10.1007/s40065-021-00343-z
- [17] C. Derbazi, H. Hammouche, A. Salim and M. Benchohra, Measure of noncompactness and fractional hybrid differential
equations with Hybrid conditions, Differ. Equ. Appl., 14 (2022), 145-161. http://dx.doi.org/10.7153/dea-2022-14-09
- [18] Y. Feng and Z. Bai, Solvability of some nonlocal fractional boundary value problems at resonance in R n , Fractal Fract., 6
(2022), 16pages. https://doi.org/10.3390/fractalfract6010025
- [19] D. Foukrach, S. Bouriah, M. Benchohra and E. Karapinar, Some new results for ψ-Hilfer fractional pantograph-type
differential equation depending on ψ-Riemann-Liouville integral, J. Anal (2021). https://doi.org/10.1007/s41478-021-00339-
0
- [20] D. Foukrach, S. Bouriah, S. Abbas and M. Benchohra, Periodic solutions of nonlinear fractional pantograph integro-
differential equations with Ψ-Caputo derivative, Ann. Univ. Ferrara (2022). https://doi.org/10.1007/s11565-022-00396-8
- [21] R. Hermann, Fractional Calculus: An Introduction For Physicists, World Scienti?c Publishing Co. Pte. Ltd. 2011.
- [22] A. Heris, A. Salim, M. Benchohra and E. Karapinar, Fractional partial random differential equations with infinite delay,
Results in Physics (2022). https://doi.org/10.1016/j.rinp.2022.105557
- [23] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- [24] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differenatial Equations, North-
Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
- [25] N. Laledj, A. Salim, J.E. Lazreg, S. Abbas, B. Ahmad and M. Benchohra, On implicit fractional q-difference equations:
Analysis and stability, Math. Meth. Appl. Sci., 2 (2022), 1-23. https://doi.org/10.1002/mma.8417
- [26] J.E. Lazreg, M. Benchohra and A. Salim, Existence and Ulam stability of k-Generalized ψ-Hilfer Fractional Problem, J.
Innov. Appl. Math. Comput. Sci., 2 (2022), 01-13.
- [27] A.J. Luo and V. Afraimovich, Long-range Interactions, Stochasticity and Fractional Dynamics, Springer, New York, Dor-
drecht, Heidelberg, London, 2010.
- [28] J. Mawhin, NSFCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1979.
- [29] S.K. Ntouyas, J. Tariboon, C. Sawaddee, Nonlocal initial and boundary value problems via fractional calculus with expo-
nential singular kernel, J. Nonlinear Sci. Appl., 11 (2018), 1015-1030.
- [30] D. O'Regan, Y.J. Chao, Y.Q. Chen, Topological Degree Theory and Application, Taylor and Francis Group, Boca Raton,
London, NewYork, 2006.
- [31] M.D. Otigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84. Springer,
Dordrecht, 2011.
- [32] I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Heidelberg Dordrecht London
New York, 2011.
- [33] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
- [34] P. Sahoo, T. Barman and J.P. Davim, Fractal Analysis in Machining, Springer, New York, Dordrecht, Heidelberg, London,
2011.
- [35] A. Salim, M. Benchohra, J.R. Graef and J.E. Lazreg, Initial value problem for hybrid ψ-Hilfer fractional implicit differential
equations, J. Fixed Point Theory Appl., 24 (2022), 14 pp. https://doi.org/10.1007/s11784-021-00920-x
- [36] A. Salim, M. Benchohra, J.E. Lazreg and J. Henderson, On k-generalized ψ-Hilfer boundary value problems with retardation
and anticipation, Adv. Theor. Nonl. Anal. Appl., 6 (2022), 173-190. https://doi.org/10.31197/atnaa.973992
- [37] A. Salim, M. Benchohra, J.E. Lazreg and E. Karapinar, On k-generalized ψ-Hilfer impulsive boundary value problem with
retarded and advanced arguments, J. Math. Ext., 15 (2021), 1-39. https://doi.org/10.30495/JME.SI.2021.2187
- [38] A. Salim, J.E. Lazreg, B. Ahmad, M. Benchohra and J.J. Nieto, A Study on k-Generalized ψ-Hilfer Derivative Operator,
Vietnam J. Math., (2022). https://doi.org/10.1007/s10013-022-00561-8