Nonlocal Fractional Differential Equation On The Half Line in Banach Space

Yıl 2022, Cilt: 6 Sayı: 1, 118 - 134, 31.03.2022

Kheireddine Benia El Hadi Ait Dads Moustafa Beddani Benaouda Hedia

https://doi.org/10.31197/atnaa.942349

Öz

Our aim in this paper is to study the existence of solution sets and its topological structure for non-local
fractional differential equations on the half-line in a Banach space using Riemann-Liouville definition. The
main result is based on Meir-Keeler fixed point theorem for condensing operators combined with measure of
non-compactness. An example is given to illustrate the feasibility of our main result.

Anahtar Kelimeler

Nonlocal boundary value problem, measure of non-compactness, unbounded domain, Banach space, fixed point theorem, Riemann-Liouville fractional derivative, Meir-Keeler condensing operator, fractional differential equations

Destekleyen Kurum

University of Tiaret, Laboratory of mathematics and informatics

Proje Numarası

COOL03UN140130180002

Teşekkür

The authors would like to express their thanks to the editor and anonymous referees for his/her suggestions and comments that improved the quality of the paper.

Kaynakça

• [1] S. Abbas, Y. Xia, Existence and attractivity of k-almost automorphic solutions of model of cellular neutral network with delay, Acta. Math. Sci., 1 (2013), 290-302.
• [2] H. Afshari, Solution of fractional differential equations in quasi-b- metric and b-metric-like spaces, Adv. Differ. Equ. 2018, 285 (2018). https://doi.org/10.1186/s13662-019-2227-9.
• [3] H. Afshari, M. Atapour, E. Karapinar, A discussion on a generalized Geraghty multi- valued mappings and applications, Adv. Differ. Equ. 2020, 356 (2020). https://doi. org/10.1186/s13662-020-02819-2.
• [4] H. Afshari, H. Hosseinpour, H.R. Marasi, Application of some new contractions for existence and uniqueness of differential equations involving Caputo Fabrizio derivative, Adv. Differ. Equ. 2021, 321 (2021).
• [5] H. Afshari, E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via ψ- Hilfer fractional derivative on b-metric spaces, Adv. Di?er. Equ. 2020, 616 (2020). https://doi. org/10.1186/s13662-020-03076-z.
• [6] H. Afshari, S. Kalantari, D. Baleanu, Solution of fractional differential equations via α − φ− Geraphty type mappings, Adv. Differ. Equ. 2018, 347 (2018). https://doi. org/10.1186/s13662-018-1807-4.
• [7] A. Aghajani, M. Mursaleen and A. Shole Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measure of non-compactness, Acta Math. Sci. Ser. 35 (2015), 552-556.
• [8] R. P. Agarwal, B. Hedia and M. Beddani, Structure of solutions sets for impulsive fractional differential equation, J. Fractional Cal. Appl Vol. 9(1) Jan. (2018), pp. 15-34.
• [9] Md. Asaduzzamana, Md. Zul?kar Alib, Existence of Solution to Fractional Order Impulsive Partial Hyperbolic Di?erential Equations with In?nite Delay, Adv. Theory Nonlinear Anal. Appl., 4 (2020) No. 2, 77-91. [10] J. Bana? s, K. Goebel, Measures of non-compactness in Banach spaces, Lecture Note in Pure App. Math, 60, Dekker, New York, 1980. [11] F. Z. Berrabah, B. Hedia and J. Henderson, Fully Hadamard and Erdélyi-Kober-type integral boundary value problem of a coupled system of implicit differential equations, Turk. J. Math. 43 (2019), 1308-1329.
• [12] M. Beddani and B. Hedia, Solution sets for fractional differential inclusions, J. Fractional Calc. Appl. 10 (2) July 2019, 273-289.
• [13] M. Benchohra, M. Slimane, Fractional Differential Inclusions with Non Instantaneous Impulses in Banach Spaces, Results in Nonlinear Anal., 2 (2019) No. 1, 36-47.
• [14] L. Byszewski, Existence and uniqueness of mild and classical solutions of semilinear functional differential evolution non- local Cauchy problem, Selected problems of mathematics,50th Anniv. Cracow Univ. Technol. Anniv. Issue 6, Cracow Univ. Technol. Krakow, (1995), 25-33.
• [15] L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a non-local abstract Cauchy problem in a Banach space, Appl. Anal. 40 (1991), 11-19.
• [16] G. Christopher, Existence and uniqueness of solutions to a fractional difference equation with non-local conditions, Comput. Math. with Appl. 61 (2011), 191-202.
• [17] C. Derbazi, Z. Baitiche, M. Benchohra, Cauchy problem with ψ-Caputo fractional derivative in Banach spaces, Adv. Theory Nonlinear Anal. Appl.4 (2020), 349-361.
• [18] D. J. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, 1996.
• [19] B. Hedia, Non-local Conditions for Semi-linear Fractional Differential Equations with Hilfer Derivative, Springer proceeding in mathematics and statistics 303, ICFDA 2018, Amman, Jordan, July 16-18, 69-83.
• [20] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B. V. Amsterdam, 2006.
• [21] S. Muthaiaha, M. Murugesana, N.G. Thangaraja, Existence of Solutions for Nonlocal Boundary Value Problem of Hadamard Fractional Differential Equations, Adv. Theory Nonlinear Anal. Appl. 3 (2019) No. 3, 162-173.
• [22] I. Podlubny, Fractional Differential Equations, in: Mathematics in Science and Engineering, vol. 198, Academic Press, New York, London, Toronto, 1999.
• [23] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. Di?er- ential Equations, 36 (2006), 1-12.
• [24] Y. Zhou, F. Jiao, J. Pecaric, On the Cauchy problem for fractional functional differential equations in Banach spaces. Topol. Methods Nonlinear Anal. 42 (2013), 119-136.
• [25] Z. Baitiche, C. Derbazi, M.Benchohra, ψ-Caputo Fractional Differential Equations with Multi-point Boundary Conditions by Topological Degree Theory, Results in Nonlinear Anal.3 (2020) No. 4, 167-178.