On Nonlinear Periodic Problems with Caputo's Exponential Fractional Derivative

On Nonlinear Periodic Problems with Caputo's Exponential Fractional Derivative

Authors

  • Mohamed CHOHRİ, Soufyane BOURİAH, Salim ABDELKRİM*, Mouffak BENCHOHRA

Keywords:

Caputo, implicit fractional differential equations, fractional integral, existence, coincidence degree theory

Abstract

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.

References

S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.

S. Abbas, M. Benchohra and G M. N'Guérékata, Advanced Fractional Di?erential and Integral Equations, Nova Science Publishers, New York, 2014.

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.

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.

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.

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

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

G.A. Anastassiou, Advances on Fractional Inequalities, Springer, New York, 2011.

D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculs Models and Numerical Methods, World Scientific Publishing, New York, 2012.

D. Baleanu, Z.B. Güvenç and J.A.T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.

D. Baleanu, J.A.T. Machado and A.C.-J. Luo, Fractional Dynamics and Control, Springer, 2012.

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.

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.

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.

M. Benchohra and J.E. Lazreg, Nonlinear fractional implicit di?erential equations. Commun. Appl. Anal., 17 (2013), 471-482.

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

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

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

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

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

R. Hermann, Fractional Calculus: An Introduction For Physicists, World Scienti?c Publishing Co. Pte. Ltd. 2011.

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

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

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.

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

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.

A.J. Luo and V. Afraimovich, Long-range Interactions, Stochasticity and Fractional Dynamics, Springer, New York, Dor- drecht, Heidelberg, London, 2010.

J. Mawhin, NSFCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1979.

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.

D. O'Regan, Y.J. Chao, Y.Q. Chen, Topological Degree Theory and Application, Taylor and Francis Group, Boca Raton, London, NewYork, 2006.

M.D. Otigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84. Springer, Dordrecht, 2011.

I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer Heidelberg Dordrecht London New York, 2011.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

P. Sahoo, T. Barman and J.P. Davim, Fractal Analysis in Machining, Springer, New York, Dordrecht, Heidelberg, London, 2011.

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

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

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

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

Downloads

Published

2023-08-01

Issue

Section

Articles

How to Cite

On Nonlinear Periodic Problems with Caputo’s Exponential Fractional Derivative . (2023). Advances in the Theory of Nonlinear Analysis and Its Application, 7(1), 103-120. https://atnaea.org/index.php/journal/article/view/22