Existence of mild solutions for semilinear $\psi-$Caputo-type fractional evolution equations with nonlocal conditions in Banach spaces

Year 2022, Volume: 5 Issue: 4, 459 - 472, 30.12.2022

Ali El Mfadel , Fatima Ezzahra Bourhim , M'hamed Elomari

https://doi.org/10.53006/rna.1121916

Cited By: 3

Abstract

The main crux of this manuscript is to establish the existence of mild solutions for a class of semilinear $\psi-$Caputo-type fractional evolution equations in Banach spaces with non-local conditions. The proofs are based on some fixed point theorems, compact semigroup and some basic concepts of $\psi-$fractional analysis. As application, a nontrivial example is given to illustrate our theoretical results.

Keywords

$\psi-$fractional integral, $\psi-$Caputo fractional derivative, fixed point, compact semigroup

References

• [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 Di?erence Equations, 1 (2006)1-15.
• [3] R.P. Agarwal, S. Hristova and D. O'Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional di?erential 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] A. Belarbi, M. Benchohra and A. Ouahab, Uniqueness results for fractional functional di?erential equations with infinite delay in Frechet spaces, Appl. Anal, 58 (2008)1459-1470.
• [7] T.A. Burton, A fixed point theorem of Krasnoselskii. Appl. Math. Lettn, 11(1) (1998)85-88.
• [8] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, International Journal of Geographical Information Science, 13(5) (1967) 529-539.
• [9] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, (2010).
• [10] J. W. Green and F. A. Valentine, On the arzela-ascoli theorem, Mathematics Magazine, 34 (1961) 199-202 .
• [11] A. El Mfadel, S. Melliani and M. Elomari, On the initial value problem for fuzzy nonlinear fractional differential equations, 48(4)(2024) 547-554.
• [12] A. El Mfadel, S. Melliani and M. Elomari, Existence of solutions for nonlinear ψ−Caputo-type fractional hybrid di?erential equations with periodic boundary conditions, Asia Pac. J. Math. (2022).
• [13] A. El Mfadel, S. Melliani, and M. Elomari, Existence and uniqueness results for Caputo fractional boundary value problems involving the p-Laplacian operator. U.P.B. Sci. Bull. Series A, 84(1) (2022) 37-46.
• [14] A. El Mfadel, S. Melliani, and M. Elomari, New existence results for nonlinear functional hybrid differential equations involving the ψ− Caputo fractional derivative. Results in Nonlinear Analysis, 5 (1) (2022) 78-86.
• [15] A. El Mfadel, S. Melliani and M. Elomari, Existence results for nonlocal Cauchy problem of nonlinear ψ−Caputo type fractional differential equations via topological degree methods. Advances in the Theory of Nonlinear Analysis and its Application, 6(2) (2022) 270-279.
• [16] A. El Mfadel, S. Melliani and M. Elomari, On the Existence and Uniqueness Results for Fuzzy Linear and Semilinear Fractional Evolution Equations Involving Caputo Fractional Derivative. Journal of Function Spaces. (2021).
• [17] A. El Mfadel, S. Melliani and M. Elomari, A Note on the Stability Analysis of Fuzzy Nonlinear Fractional Differential Equations Involving the Caputo Fractional Derivative. International Journal of Mathematics and Mathematical Sciences, (2021).
• [18] G.M.A. Guérékata, Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Analysis: Theory, Methods and Applications, 70 (5) (2009)1873-1876.
• [19] T.L. Guo and M. Jiang, Impulsive fractional functional differential equations, Comput. Math. Appl,64(2012) 3414-3424.
• [20] R. Hilfer, Applications of Fractional Calculus in Physics , Singapore, (2000).
• [21] F. Jarad and T. Abdeljawad, Generalized fractional derivatives and Laplace transform. Discrete Contin. Dyn. Syst. Ser, 13(3)(2019) 709-722.
• [22] 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. Elsevier Science B.V. Amsterdam (2006).
• [23] A. Lachouri, A. Ardjouni, F. Jarad and M.S. Abdo, Semilinear Fractional Evolution Inclusion Problem in the Frame of a Generalized Caputo Operator. Journal of Function Spaces, (2021).
• [24] Y. Luchko and J.J. Trujillo, Caputo-type modi?cation of the Erdelyi-Kober fractional derivative, Fract. Calc. Appl. Anal, 10(2007) 249-267.
• [25] F. Mainardi, Fractals and Fractional Calculus Continuum Mechanics, Springer Verlag,(1997).
• [26] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation. Waves and Stability in Continuous Media, World Scienti?c, Singapore. (1994) 246-251.
• [27] S.K. Ntouyas, boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions, Opuscula Math., 331(2013)117-138.
• [28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, (1983).
• [29] I. Podlubny, Fractional Di?erential Equations, Mathematics in Science and Engineering, Academic Press, New York, (1999).
• [30] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives. Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam , (1998).
• [31] A. Suechoei and P. Sa Ngiamsunthorn, Optimal feedback control for fractional evolution equations with nonlinear pertur- bation of the time-fractional derivative term. Boundary Value Problems, 2022(1) (2022)1-26.
• [32] A. Suechoei and P. Sa Ngiamsunthorn, Existence uniqueness and stability of mild solutions for semilinear ψ-Caputo fractional evolution equations, Advances in Di?erence Equations, 2020(1)(2020)1-28.
• [33] Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear analysis: real world applica- tions. 11(5)(2010) 4465-4475 .
• [34] J. Zhao, P. Wang, and W. Ge, Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces, Nonlinear Sci. Numer. Simul, 16(2011) 402-410.
• [35] S. Zhang, Existence of solutions for a boundary value problem of fractional order, Acta Math. Scientia, 6( 2006) 220-228.
• [36] W. Zhon and W. Lin, Nonlocal and multiple-point boundary value problem for fractional differential equations, and Mathematics with Applications,593 (2010) 1345-1351.