On Implicit Caputo Tempered  Fractional Boundary Value Problems with Delay

Authors

  • Salim Krim Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes,P.O. Box 89, Sidi Bel-Abbes 22000, Algeria
  • Abdelkrim Salim Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes,P.O. Box 89, Sidi Bel-Abbes 22000, Algeria; Faculty of Technology, Hassiba Benbouali University of Chlef, P.O. Box 151 Chlef 02000, Algeria
  • Mouffak Benchohra Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbes,P.O. Box 89, Sidi Bel-Abbes 22000, Algeria Corresponding Author

Keywords:

Fixed point, implicit differential equations, existence, uniqueness, tempered fractional derivative, boundary condition

Abstract

This paper deals with some existence and uniqueness  results for a class of problems for nonlinear Caputo tempered implicit fractional differential equations subject to boundary conditions and delay. The results are based on the fixed point theorems of Banach, Schauder and Schaefer. Furthermore, several illustrations are presented to demonstrate the plausibility of our results.

References

S. Abbas, M. Benchohra, J.R. Graef and J. Henderson, Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter, Berlin, 2018.

S. Abbas, M. Benchohra and G. M. N'Guerekata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.

A. Almalahi and K. Panchal, On the Theory of $psi$-Hilfer Nonlocal Cauchy Problem, J. Sib. Fed. Univ. Math. Phys. 14 (2) (2021), 161-177.

R. Almeida, M. L. Morgado, Analysis and numerical approximation of tempered fractional calculus of variations problems, J. Comput. Appl. Math. 361 (2019), 1-12.

A. Anguraja and M. Latha Maheswari, Existence of solutions for fractional impulsive neutral functional infinite delay integrodifferential equations with nonlocal conditions, J. Nonlinear Sci. Appl. 5 (2012), 271-280.

M. Benchohra, F. Bouazzaoui, E. Karapinar and A. Salim, Controllability of second order functional random differential equations with delay. Mathematics 10 (2022), 16pp. https://doi.org/10.3390/math10071120

N. Benkhettou, K. Aissani, A. Salim,M. Benchohra and C. Tunc, Controllability of fractional integro-differential equations with infinite delay and non-instantaneous impulses, Appl. Anal. Optim. 6 (2022), 79-94.

R. G. Buschman, Decomposition of an integral operator by use of Mikusinski calculus, SIAM J. Math. Anal. 3 (1972), 83-85.

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

C. Li, W. Deng and L. Zhao, Well-posedness and numerical algorithm for the tempered fractional differential equations, Discr. Contin. Dyn. Syst. Ser. B. 24 (2019), 1989-2015.

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.

J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkc. Ekvac. 21 (1978), 11-41.

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

A.A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Amsterdam, 2006.

S. Krim, A. Salim, S. Abbas and M. Benchohra, On implicit impulsive conformable fractional differential equations with infinite delay in $b$-metric spaces. Rend. Circ. Mat. Palermo (2). (2022), 1-14. https://doi.org/10.1007/s12215-022-00818-8

N. Laledj, A. Salim, J. E. Lazreg, S. Abbas, B. Ahmad and M. Benchohra, On implicit fractional $q$-difference equations: Analysis and stability. Math. Methods Appl. Sci. 45 (17) (2022), 10775-10797. https://doi.org/10.1002/mma.8417

M. Medved and E. Brestovanska, Differential Equations with Tempered $psi$-Caputo Fractional Derivative, Math. Model. Anal. 26 (2021), 631-650.

N. A. Obeidat, D. E. Bentil, New theories and applications of tempered fractional differential equations, Nonlinear Dyn. 105 (2021), 1689-1702.

M. D. Ortigueira, G. Bengochea and J. T. Machado, Substantial, tempered, and shifted fractional derivatives: Three faces of a tetrahedron, Math. Methods Appl. Sci. 44 (2021), 9191-9209.

A. Salim, S. Abbas, M. Benchohra and E. Karapinar, Global stability results for Volterra-Hadamard random partial fractional integral equations. Rend. Circ. Mat. Palermo (2). (2022), 1-13. https://doi.org/10.1007/s12215-022-00770-7

A. Salim, M. Benchohra, J. R. Graef and J. E. Lazreg, Initial value problem for hybrid $psi$-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, J. E. Lazreg, B. Ahmad, M. Benchohra and J. J. Nieto, A Study on $k$-Generalized $psi$-Hilfer Derivative Operator, Vietnam J. Math. (2022). https://doi.org/10.1007/s10013-022-00561-8

F. Sabzikar, M. M. Meerschaert and J. Chen, Tempered fractional calculus, J. Comput. Phys. 293 (2015), 14-28.

B. Shiri, G. Wu and D. Baleanu, Collocation methods for terminal value problems of tempered fractional differential equations, Appl. Numer. Math. 156 (2020), 385-395.

V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010.

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.

Downloads

Published

2022-12-27

Issue

1 (2023) No. 1

Section

Articles

License

Copyright (c) 2022 Letters in Nonlinear Analysis and its Applications

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.