New existence result under weak topology for fractional differential equations

Abstract

This paper deals with the existence of weak solutions for an initial value problem involving Riemann-Liouville-type fractional derivatives. To this end, we transform the posed problem to a sum of two integral operators, then we apply a variant of Krasnoselskii’s fixed point theorem under weak topology to conclude the existence of weak solutions.

Keywords

• Fractional differential equations
• Weak topology
• Existence result
• Fixed pointtheorems

Kaynakça

1. 1] R. (Ed.). Hilfer, Applications of fractional calculus in physics, World Scientific, 2000.
2. [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 204(2006).
3. [3] I. Podlubny, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Math. Sci. Eng. 198(1999).
4. [4] S. Das, I. Pan, Fractional order signal processing: introductory concepts and applications, Springer Science and Business Media, 2011.
5. [5] I. Petrás, Fractional-order nonlinear systems: modeling, analysis and simulation, Springer Science and Business Media, 2011.
6. [6] M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 338(2) 1340-1350.
7. [7] R. P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Computers and Mathematics with Applications, 59(3) (2010) 1095-1100.
8. [8] K. Nouri, M. Nazari, B. Keramati, Existence results of solutions for some fractional neutral functional integro-differential equations with infinite delay, doi:10.20944/preprints201706.0090.v1, 2017.

9. [9] B. Ahmad, S. K. Ntouyas, Existence and uniqueness of solutions for Caputo-Hadamard sequential fractional order neutral functional differential equations, Electronic Journal of Differential Equations, 36(2017) 1-11.
10. [10] A. Hallaci, A. Jeribi, B. Krichen, B. Mefteh, Existence Results for Fractional Differential Equations Under Weak Topology Features, Pan-American Journal of Mathematics, 1 (2022) 14.
11. [11] K. Latrach, M. A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L1 spaces, Nonlinear Analysis, 66(2007) 2325-2333.
12. [12] M. A. Taoudi, N. Salhi, B. Ghribi, Integrable solutions of a mixed type operator equation, Applied Mathematics and Computation, 216(4) (2010) 1150-1157.
13. [13] A. Jeribi, B. Krichen, Nonlinear Functional Analysis in Banach Spaces and Banach Algebras: FixedPoint Theory Under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications, (Vol.12), CRC Press, 2015.
14. [14] F. S. De Blasi, On a property of the unit sphere in a Banach space. Bulletin mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, (1977) 259-262.
15. [15] J. Appell, Su alcuni parametri connessi con la misura di non compattezza di Hausdor? in spazi di funzioni misurabili, 1984.
16. [16] C. S. Barroso, E. V. Teixeira, A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Analysis: Theory, Methods and Applications, 60(4) (2005) 625-650.
17. [17] N. Dunford, J. T. Schwartz, W. G. Bade, R. G. Bartle, Linear Operators: Spectral Theory, (Vol. 7), Interscience Publishers, 1958.