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New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative

Yıl 2022, Cilt: 5 Sayı: 1, 78 - 86, 31.03.2022
https://doi.org/10.53006/rna.1020895

Öz

In this manuscript, we are concerned with the existence result of nonlinear hybrid differential equations involving $\psi-$Caputo fractional derivatives of an arbitrary order $\alpha\in(0,1)$. By applying Krasnoselskii fixed point theorem and some fractional analysis techniques, we prove our main result. As application, a nontrivial example is given to demonstrate the effectiveness of our theoretical result.

Teşekkür

The authors would like to express their sincere apreciation to the editor and to the referees for their very helpful suggestions and many kind comments.

Kaynakça

  • [1] R.P. Agarwal, Y. Zhou, J. Wang and X. Luo. Fractional functional differential equations with causal operators in Banach spaces. Mathematical and Compututer Modelling. 54 (2011) 1440–1452.
  • [2] R.P. Agarwal, S.K. Ntouyas, B. Ahmad and A.K. Alzahrani. Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments. Advances in Difference Equations. 1 (2016) 1–15.
  • [3] R.P. Agarwal, S. Hristova and D. O’Regan. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19 (2016) 290–318.
  • [4] R. Almeida. Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul.44 (2017) 460–481.
  • [5] R. Almeida, A.B. Malinowska, and M.T.T. Monteiro. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Mathematical Methods in the Applied Sciences. 41(1) (2018) 336–352.
  • [6] D. Baleanu, H. Jafari, H. Khan and S.J Johnston. Results for mild solution of fractional coupled hybrid boundary value problems. Open Math.13 (2015) 601–608.
  • [7] D. Baleanu, H. Khan, H. Jafari and R.A. Alipour. On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions. Advances in Difference Equations . 1(2015) 1–14.
  • [8] A. Belarbi, M. Benchohra, A. Ouahab. Uniqueness results for fractional functional differential equations with infinite delay in Frechet spaces. Appl. Anal. 58 (2006) 1459–1470.
  • [9] M. Caputo. Linear models of dissipation whose Q is almost frequency independent. International Journal of Geographical Information Science. 13(5)(1967) 529–539.
  • [10] B.C. Dhage. On α−condensing mappings in Banach algebras. Math. Student 63(1994)146–152.
  • [11] B.C. Dhage. On a fixed point theorem in Banach algebras with applications. Appl. Math. Lett. 18(2005) 273–280.
  • [12] B.C. Dhage and V. Lakshmikantham. Basic results on hybrid differential equations. Nonlinear Anal. Hybrid 4(2010) 414–424.
  • [13] K. Diethelm. The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin, 2010.
  • [14] T.L. Guo and M. Jiang. Impulsive fractional functional differential equations. Comput. Math. Appl.64 (2012)3414–3424.
  • [15] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. North-Holland Mathematical studies 204. Ed van Mill. Amsterdam. 2006). Elsevier Science B.V. Amsterdam, 2006.
  • [16] R. Hilfer. Applications of Fractional Calculus in Physics .Singapore. 2000.
  • [17] Y. Luchko and J.J. Trujillo. Caputo-type modification of the Erdelyi-Kober fractional derivative. Fract. Calc. Appl. Anal. 10( 2007) 249–267. [18] F. Mainardi. Fractals and Fractional Calculus Continuum Mechanics Springer Verlag.1997.
  • [19] I. Podlubny. Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press. New York. 1999.
  • [20] J. Schauder. Der Fixpunktsatz in Functionalraiumen. Studia Math. 2(1930)171–180.
Yıl 2022, Cilt: 5 Sayı: 1, 78 - 86, 31.03.2022
https://doi.org/10.53006/rna.1020895

Öz

Kaynakça

  • [1] R.P. Agarwal, Y. Zhou, J. Wang and X. Luo. Fractional functional differential equations with causal operators in Banach spaces. Mathematical and Compututer Modelling. 54 (2011) 1440–1452.
  • [2] R.P. Agarwal, S.K. Ntouyas, B. Ahmad and A.K. Alzahrani. Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments. Advances in Difference Equations. 1 (2016) 1–15.
  • [3] R.P. Agarwal, S. Hristova and D. O’Regan. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19 (2016) 290–318.
  • [4] R. Almeida. Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul.44 (2017) 460–481.
  • [5] R. Almeida, A.B. Malinowska, and M.T.T. Monteiro. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Mathematical Methods in the Applied Sciences. 41(1) (2018) 336–352.
  • [6] D. Baleanu, H. Jafari, H. Khan and S.J Johnston. Results for mild solution of fractional coupled hybrid boundary value problems. Open Math.13 (2015) 601–608.
  • [7] D. Baleanu, H. Khan, H. Jafari and R.A. Alipour. On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions. Advances in Difference Equations . 1(2015) 1–14.
  • [8] A. Belarbi, M. Benchohra, A. Ouahab. Uniqueness results for fractional functional differential equations with infinite delay in Frechet spaces. Appl. Anal. 58 (2006) 1459–1470.
  • [9] M. Caputo. Linear models of dissipation whose Q is almost frequency independent. International Journal of Geographical Information Science. 13(5)(1967) 529–539.
  • [10] B.C. Dhage. On α−condensing mappings in Banach algebras. Math. Student 63(1994)146–152.
  • [11] B.C. Dhage. On a fixed point theorem in Banach algebras with applications. Appl. Math. Lett. 18(2005) 273–280.
  • [12] B.C. Dhage and V. Lakshmikantham. Basic results on hybrid differential equations. Nonlinear Anal. Hybrid 4(2010) 414–424.
  • [13] K. Diethelm. The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin, 2010.
  • [14] T.L. Guo and M. Jiang. Impulsive fractional functional differential equations. Comput. Math. Appl.64 (2012)3414–3424.
  • [15] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. North-Holland Mathematical studies 204. Ed van Mill. Amsterdam. 2006). Elsevier Science B.V. Amsterdam, 2006.
  • [16] R. Hilfer. Applications of Fractional Calculus in Physics .Singapore. 2000.
  • [17] Y. Luchko and J.J. Trujillo. Caputo-type modification of the Erdelyi-Kober fractional derivative. Fract. Calc. Appl. Anal. 10( 2007) 249–267. [18] F. Mainardi. Fractals and Fractional Calculus Continuum Mechanics Springer Verlag.1997.
  • [19] I. Podlubny. Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press. New York. 1999.
  • [20] J. Schauder. Der Fixpunktsatz in Functionalraiumen. Studia Math. 2(1930)171–180.
Toplam 19 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Alı El Mfadel

Said Melliani

M'hamed Elomari

Yayımlanma Tarihi 31 Mart 2022
Yayımlandığı Sayı Yıl 2022 Cilt: 5 Sayı: 1

Kaynak Göster

APA El Mfadel, A., Melliani, S., & Elomari, M. (2022). New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative. Results in Nonlinear Analysis, 5(1), 78-86. https://doi.org/10.53006/rna.1020895
AMA El Mfadel A, Melliani S, Elomari M. New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative. RNA. Mart 2022;5(1):78-86. doi:10.53006/rna.1020895
Chicago El Mfadel, Alı, Said Melliani, ve M’hamed Elomari. “New Existence Results for Nonlinear Functional Hybrid Differential Equations Involving the $\psi-$Caputo Fractional Derivative”. Results in Nonlinear Analysis 5, sy. 1 (Mart 2022): 78-86. https://doi.org/10.53006/rna.1020895.
EndNote El Mfadel A, Melliani S, Elomari M (01 Mart 2022) New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative. Results in Nonlinear Analysis 5 1 78–86.
IEEE A. El Mfadel, S. Melliani, ve M. Elomari, “New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative”, RNA, c. 5, sy. 1, ss. 78–86, 2022, doi: 10.53006/rna.1020895.
ISNAD El Mfadel, Alı vd. “New Existence Results for Nonlinear Functional Hybrid Differential Equations Involving the $\psi-$Caputo Fractional Derivative”. Results in Nonlinear Analysis 5/1 (Mart 2022), 78-86. https://doi.org/10.53006/rna.1020895.
JAMA El Mfadel A, Melliani S, Elomari M. New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative. RNA. 2022;5:78–86.
MLA El Mfadel, Alı vd. “New Existence Results for Nonlinear Functional Hybrid Differential Equations Involving the $\psi-$Caputo Fractional Derivative”. Results in Nonlinear Analysis, c. 5, sy. 1, 2022, ss. 78-86, doi:10.53006/rna.1020895.
Vancouver El Mfadel A, Melliani S, Elomari M. New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative. RNA. 2022;5(1):78-86.

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