Katugampola Fractional Differential Equation with Erdelyi-Kober Integral Boundary Conditions

Yıl 2021, Cilt: 5 Sayı: 2, 215 - 228, 30.06.2021

Naas Adjimi Maamar Benbachır

https://doi.org/10.31197/atnaa.711191

Cited By: 1

Öz

In this paper, we study the existence and uniqueness of solutions for nonlinear fractional Katugampola differential equation with Erdely-Kober fractional integral conditions, new existence and uniqueness results are established using Banach's contraction principle, nonlinear contractions, Krasnoselskii's and Leray-Schauder's fixed theorems. Four examples are given in order to clarify theoretical results.

Anahtar Kelimeler

Katugampola Fractional Derivative, Erdelyi-Kober Fractional integral, Boundary value problem, fixed point theorem

Kaynakça

• [1] B. Ahmad, S.K. Ntouyas, J. Tariboon, A. Alsaedi: Caputo type fractional di?erential equations with nonlocal Riemann- Liouville and Erdelyi-Kober integral boundary conditions, Filomat (2017), 4515-4529.
• [2] B. Ahmad, S.K. Ntouyas, A. Alsaedi: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Difference Equ., 2011, Art. ID 107384, 11 pp.
• [3] B. Ahmad, S.K. Ntouyas: A four-point nonlocal integral boundary value problem for fractional differential equations of arbitrary order, Electron. J. Qual. Theory Differ. Equ. 22(2011), pp. 1-15.
• [4] Y. Arioua, B. Basti and N. Benhamidouche: Initial value problem for nonlinear implicit fractional differential equation with Katugampola derivative. Appl. Math. E-Notes, 19(2019), 397-412.
• [5] B. Basti, Y. Arioua, N. Benhamidouche: Existence and Uniqueness of Solutions for Nonlinear Katugampola Fractional Di?erential Equations, Journal of Mathematics and Applications, No 42, pp 35-61 (2019).
• [6] M. Benchohra, S. Hammani and S.K. Ntouyas: Boundary value problems for differential equation with fractional order and nonlocal conditions, Nonlinear Anal. TMA 71 (2009), 2391-2396.
• [7] A. Boutiara, M. Benbachir, K. Guerbati: Boundary value problems for hilfer fractional differential equations with katugam- pola fractional integral and anti-periodic conditions, to appear (http://math.ubbcluj.ro/ mathjour/accepted.html).
• [8] D.W. Boyd and J.S.W. Wong: On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464.
• [9] M. Janaki, E. M. Elsayed, K. Kanagarajan: Katugampola-type fractional differential equations with delay and impulses, Open Access Journal of Mathematical and Theoretical Physics, Volume 1 Issue 3 - 2018.
• [10] M. Janaki, K. kanagarajan, D. Viek: Existence results for Katugampola fractional differential equations via measure of noncompactness, Journal nonlinear Analysis and Applications 2018, No.2(2018)184-191.
• [11] S.L. Kalla, V.S. Kiryakova: An H-function generalized fractional calculus based upon compositions of Erdelyi-Kober operators in LP; Math. Japonica 35 (1990), 1-21.
• [12] K. Logeswari, C. Ravichandran: A new exploration on existence of fractional neutral integro- differential equations in the concept of Atangana-Baleanu derivative, Physica A: Statistical Mechanics and its Applications, Volume 544, 15 April 2020, 123454.
• [13] N.I. Mahmudov, S. Emin: Fractional-order boundary value problems with Katugampola fractional integral conditions. Adv Differ Equ 2018, 81 (2018).
• [14] K. Rajendra Prasad , L. D and M. Khuddush , "Existence and Uniqueness of Positive Solutions for System of (p,q,r)- Laplacian Fractional Order Boundary Value Problems", Advances in the Theory of Nonlinear Analysis and its Application, vol. 5, no. 1, pp. 138-157, Mar. 2021, doi:10.31197/atnaa.703304
• [15] C. Ravichandran, N. Valliammal, Juan J. Nieto: New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces, Journal of the Franklin Institute, 356(3), 2019, 1535-1565.
• [16] A. Saadi and M. Benbachir: Positive solutions for three-point nonlinear fractional boundary value problems, Electron. J. Qual. Theory Differ. Equ. 2011, No. 2, 1-19.
• [17] F. Si Bachir , A. Said , M. Benbachir and M. Benchohra , "Hilfer-Hadamard Fractional Differential Equations; Existence and Attractivity", Advances in the Theory of Nonlinear Analysis and its Application, vol. 5, no. 1, pp. 49-57, Mar. 2021, doi:10.31197/atnaa.848928
• [18] I.N. Sneddon, The use in mathematical analysis of Erdelyi-Kober operators and some of their applications, pp. 37-79 in Fractional calculus and its applications (West Haven, Connecticut, 1974), Lecture Notes in Math.
• [19] N. Thongsalee, S.K. Ntouyas, J. Tariboon: Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions, Frac. Calc. Appl. Anal. Volume 19, Issue 2 (2016).
• [20] N. Valliammal, C. Ravichandran, Ju H. Park: On the controllability of fractional neutral integrodifferential delay equations with nonlocal conditions, Math. Methods Appl. Sci. 40 (14) (2017), 5044-5055.
• [21] S. Zeng, D. Baleanu, Y. Bai, G. Wu: Fractional differential equations of Caputo-Katugampola type and numerical solutions, Applied Mathematics and computation 315(2017) 549-554.