EXISTENCE OF MILD SOLUTION FOR $(k, \Psi)$-HILFER FRACTIONAL CAUCHY VALUE PROBLEM OF SOBOLEV TYPE
Keywords:
$(\alpha, \beta, k)$-resolvent family, $(k, \Psi)$-Hilfer Sobolev type fractional differential equations, Mild solution, existenceDOI:
https://doi.org/10.17654/0974324324024Abstract
In the context of solving $(k, \Psi)$-Hilfer fractional differential equations with Sobolev type, we initially explore a more generalized version of the $(\alpha, \beta, k)$-resolvent family. Subsequently, we present various properties associated with this resolvent family. Specific instances of this resolvent family, such as the $C_0$ semigroup, sine family, cosine family and others, have been previously discussed in other academic papers. By combining this resolvent family with the $(k, \Psi)$-Hilfer fractional derivative, we examine the existence of mild solutions to $(k, \Psi)$-Hilfer Sobolev type fractional evolution equations, without requiring the existence of the inverse of $E$. Ultimately, two existence theorems are derived.
Received: June 2, 2024
Accepted: July 26, 2024
References
Y. K. Chang, A. Pereira and R. Ponce, Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators, Fract. Calc. Appl. Anal. 20 (2017), 963-987.
K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, Scientific Computing in Chemical Engineering II, Springer-Verlag, Heidelberg (1999), 217-224.
L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Process. 5 (1991), 81-88.
W. G. Glockle and T. F. Nonnenmacher, A fractional calculus approach of self- similar protein dynamics, Biophys. J. 68 (1995), 46-53.
I. Haque, J. Ali and M. Mursaleen, Existence of solutions for an infinite system of Hilfer fractional boundary value problems in tempered sequence spaces, Alx. Eng. J. 65 (2023), 575-583.
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000.
F. Jarad and T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete. Contin. Dyn. Syst. 13 (2012), 709-722.
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006.
K. D. Kucche and A. D. Mali, On the nonlinear -Hilfer fractional differential equations, Chaos Soliton Fract. 152 (2021), 111335.
Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah and S. M. Kang, Generalized Riemann-Liouville k-fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities, IEEE Access 6 (2018), 64946-64953.
L. S. Liu, F. Guo, C. X. Wu and Y. H. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005), 638-649.
B. Maayah, A. Moussaoui, S. Bushnaq and O. Abu Arqub, The multistep Laplace optimized decomposition method for solving fractional-order coronavirus disease model (COVID-19) via the Caputo fractional approach, Demonstratio Mathematica 55 (2022), 963-977.
R. Ponce, Existence of mild solutions to nonlocal fractional Cauchy problems via compactness, Abatr. Appl. Anal. (2016), 4567092.
J. Sabatier, O. P. Agrawal and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
J. Sausa and E. De Oliveira, On the -Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 60 (2018), 72-91.
L. W. Weis, A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory, J. Math. Anal. Appl. 129 (1988), 6-23.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Pushpa Publishing House, Prayagraj, India
This work is licensed under a Creative Commons Attribution 4.0 International License.
Attribution: Credit Pushpa Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pushpa Publishing House for more info or permissions.
