L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative
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Benoumran Telli
Abstract
In this paper, we study the existence of integrable solutions for initial value problems for fractional order implicit differential equations with Hadamard fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.
Acknowledgements
The authors are very grateful to the referees for their careful reading of the manuscript and for valuable comments which improved the quality of the paper. The authors are also very grateful for the help from the editor.
References
[1] S. Abbas, M. Benchohra and G. M. N’Guérékata, Topics in Fractional Differential Equations, Dev. Math. 27, Springer, New York, 2012. 10.1007/978-1-4614-4036-9Suche in Google Scholar
[2] S. Abbas, M. Benchohra and G. M. N’Guerekata, Advanced Fractional Differential and Integral Equations, Math. Res. Dev., Nova Science, New York, 2015. Suche in Google Scholar
[3] R. P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann–Liouville fractional derivative, Adv. Difference Equ. 2009 (2009), Article ID 981728. 10.1155/2009/981728Suche in Google Scholar
[4] R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010), no. 3, 973–1033. 10.1007/s10440-008-9356-6Suche in Google Scholar
[5] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, Ser. Complex. Nonlinearity Chaos 3, World Scientific, Hackensack, 2012. 10.1142/8180Suche in Google Scholar
[6] A. Belarbi, M. Benchohra and A. Ouahab, Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces, Appl. Anal. 85 (2006), no. 12, 1459–1470. 10.1080/00036810601066350Suche in Google Scholar
[7] M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2008), no. 2, 1340–1350. 10.1016/j.jmaa.2007.06.021Suche in Google Scholar
[8] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 10.1007/978-0-387-70914-7Suche in Google Scholar
[9] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. 10.1007/978-3-662-00547-7Suche in Google Scholar
[10]
A. M. A. El-Sayed and S. A. Abd El-Salam,
[11] A. M. A. El-Sayed and H. H. G. Hashem, Integrable and continuous solutions of a nonlinear quadratic integral equation, Electron. J. Qual. Theory Differ. Equ. 2008 (2008), Paper No. 25. 10.14232/ejqtde.2008.1.25Suche in Google Scholar
[12] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, 2000. 10.1142/3779Suche in Google Scholar
[13] A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191–1204. Suche in Google Scholar
[14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Suche in Google Scholar
[15] M. Kirane and B. T. Torebek, Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations, Fract. Calc. Appl. Anal. 22 (2019), no. 2, 358–378. 10.1515/fca-2019-0022Suche in Google Scholar
[16] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, 2009. Suche in Google Scholar
[17] L. Ma, Comparison theorems for Caputo–Hadamard fractional differential equations, Fractals 27 (2019), no. 3, Article ID 1950036. 10.1142/S0218348X19500361Suche in Google Scholar
[18] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College, London, 2010. 10.1142/p614Suche in Google Scholar
[19] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Lect. Notes Electr. Eng. 84, Springer, Dordrecht, 2011. 10.1007/978-94-007-0747-4Suche in Google Scholar
[20] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Suche in Google Scholar
[21] V. E. Tarasov, Fractional dynamics of relativistic particle, Internat. J. Theoret. Phys. 49 (2010), no. 2, 293–303. 10.1007/s10773-009-0202-zSuche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative
- On tangential approximations of the solution set of set-valued inclusions
- Stabilization of the wave equation with a nonlinear delay term in the boundary conditions
- Fixed point to fixed circle and activation function in partial metric space
- A note on the validity of the Schrödinger approximation for the Helmholtz equation
- Certain classes of analytic functions defined by Hurwitz–Lerch zeta function
- A new factor theorem on absolute matrix summability method
- On a solution to a functional equation
- Hydromagnetic effects on non-Newtonian Hiemenz flow
- Stability of a class of entropies based on fractional calculus
- Asymptotic behavior of solution of Whitham–Broer–Kaup type equations with negative dispersion
- Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts
- Weak solutions to the time-fractional g-Navier–Stokes equations and optimal control
- On the asymptotic formulas for perturbations in the eigenvalues of the Stokes equations due to the presence of small deformable inclusions
- Extended homogeneous balance conditions in the sub-equation method
Artikel in diesem Heft
- Frontmatter
- L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative
- On tangential approximations of the solution set of set-valued inclusions
- Stabilization of the wave equation with a nonlinear delay term in the boundary conditions
- Fixed point to fixed circle and activation function in partial metric space
- A note on the validity of the Schrödinger approximation for the Helmholtz equation
- Certain classes of analytic functions defined by Hurwitz–Lerch zeta function
- A new factor theorem on absolute matrix summability method
- On a solution to a functional equation
- Hydromagnetic effects on non-Newtonian Hiemenz flow
- Stability of a class of entropies based on fractional calculus
- Asymptotic behavior of solution of Whitham–Broer–Kaup type equations with negative dispersion
- Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts
- Weak solutions to the time-fractional g-Navier–Stokes equations and optimal control
- On the asymptotic formulas for perturbations in the eigenvalues of the Stokes equations due to the presence of small deformable inclusions
- Extended homogeneous balance conditions in the sub-equation method
