Existence results for a ψ-Hilfer-type fractional Langevin differential inclusion.

Hamid Lmou; Khalid Hilal; Ahmed Kajouni

doi:10.5269/bspm.68348

Hamid Lmou Sultan Moulay Slimane University https://orcid.org/0000-0002-8786-2230
Khalid Hilal Sultan Moulay Slimane University https://orcid.org/0000-0002-0806-2623
Ahmed Kajouni Sultan Moulay Slimane University https://orcid.org/0000-0001-8484-6107

Abstract

In this paper we deals with the existence of solution for a new kind of Langevin inclusion involving ψ-Hilfer fractional derivative. The suggested study is based on some basic definitions of fractional calculus and multivalued analysis. The existence result is obtained by making use of the nonlinear alternative of Leray-Schauder type. In the end, we are giving an example to illustrate our results.

Downloads

Download data is not yet available.

Author Biographies

Hamid Lmou, Sultan Moulay Slimane University

Laboratory of Applied Mathematics and Scientific Competing

Khalid Hilal, Sultan Moulay Slimane University

Laboratory of Applied Mathematics and Scientific Computing

Ahmed Kajouni, Sultan Moulay Slimane University

Laboratory of Applied Mathematics and Scientific Computing

References

R. P, Agarwal. M, Benchohra. S, Hamani. A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010), 973–1033.

A, Alsaedi. S. K, Ntouyas. B, Ahmad. (2013, January). Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions. In Abstract and Applied Analysis (Vol. 2013). Hindawi.

T.M, Atanackovic. S, Pilipovic. B, Stankovic. D, Zorica.(2014). Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles. Wiley, New York.

W. T, Coffey. Y. P, Kalmykov. J. T,Waldron. The Langevin Equation, vol. 14 of World Scientific Series in Contemporary Chemical Physics, World Scientific Publishing, River Edge, NJ, USA, 2nd edition, 2004.

S. I. Denisov, H. Kantz, and P. Hanggi, Langevin equation with super-heavy-tailed noise, Journal of Physics A, vol. 43, no. 28, Article ID 285004, 2010

A, Granas. J, Dugundji. Fixed Point Theory, Springer-Verlag, New York, 2005.

K, Hilal. A, Kajouni. H, Lmou. (2022). Boundary Value Problem for the Langevin Equation and Inclusion with the Hilfer Fractional Derivative. International Journal of Differential Equations, 2022.

R, Hilfer. Applications of Fractional Calculs in Physics, World Scientific, Singapore, 2000.

S, Hu, N. S, Papageorgiou. Handbook of Multivalued Analysis, vol. I of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.

A. A, Kilbas. H. M, Srivastava. J. J, Trujillo. Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.

A. Lasota, Z. Opial. An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull Acad Polon Sci Ser Sci Math Astronom Phys. 13 (1965), pp. 781-786.

R.L, Magin.(2006). Fractional Calculus in Bioengineering. Begell House Inc. Publisher.

I, Podlubny. (1993).Fractional Differential Equations, Academic Press, New York, NY, USA.

R, Rizwan. (2019). Existence theory and stability analysis of fractional Langevin equation. International Journal of Nonlinear Sciences and Numerical Simulation, 20(7-8), 833-848.

V. E, Tarasov. Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, Springer, HEP, Moscow, 2011.

C, Thaiprayoon. W, Sudsutad. S. K, Ntouyas. Mixed nonlocal boundary value problem for implicit fractional integrodifferential equations via ψ-Hilfer fractional derivative, Adv. Differ. Equ., 2021 (2021), 1–24.

M, Uranagase. T, Munakata. Generalized Langevin equation revisited: mechanical random force and self-consistent structure, Journal of Physics A, vol. 43, no. 45, Article ID 455003, 2010.

J, da Vanterler. C, Sousa. E, Capelas de Oliveira. On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72-91 (2018).

Y, Zhou.(2014). Basic Theory of Fractional Differential Equations, Xiangtan University, China.

H. Lmou, K. Hilal, A. Kajouni. On a class of fractional Langevin inclusion with multi-point boundary conditions. Boletim da Sociedade Paranaense de Matem´atica. (2023), 41

H. Lmou, K. Hilal, A. Kajouni, A New Result for ψ-Hilfer Fractional Pantograph-Type Langevin Equation and Inclusions. Journal of Mathematics, 2022.

K. Hilal, A. Kajouni, H. Lmou. Existence and stability results for a coupled system of Hilfer fractional Langevin equation with non local integral boundary value conditions. filomat. 37, 1241-1259 (2023)

K. Hilal, A. Kajouni, H. Lmou. (2023). Existence and Uniqueness Results for Hilfer Langevin Fractional Pantograph Differential Equations and Inclusions. International Journal of Difference Equations (IJDE), 18(1), 145-162.