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On Nonlinear Periodic Problems with Caputo's Exponential Fractional Derivative

Yıl 2023, Cilt: 7 Sayı: 1, 103 - 120, 31.03.2023
https://doi.org/10.31197/atnaa.1130743

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
https://doi.org/10.31197/atnaa.1130743

Öz

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
Toplam 38 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Mohamed Chohri Bu kişi benim 0000-0003-1902-4507

Soufyane Bouriah Bu kişi benim 0000-0002-6077-7992

Salim Abdelkrim 0000-0003-2795-6224

Mouffak Benchohra 0000-0003-3063-9449

Yayımlanma Tarihi 31 Mart 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 7 Sayı: 1

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