Existence of solutions to a class of nonlinear boundary value problems with right and left fractional derivarives

Hamza Moffek; Assia Guezane-Lakoud; Hamza Moffek; Assia Guezane-Lakoud

doi:10.3934/math.2020305

AIMS Mathematics

2020, Volume 5, Issue 5: 4770-4780. doi: 10.3934/math.2020305

Previous Article Next Article

Research article

Existence of solutions to a class of nonlinear boundary value problems with right and left fractional derivarives

Hamza Moffek ¹,
Assia Guezane-Lakoud ^{2
,
,}

1.
Department of Exact Sciences, ENS Ouargla, Algeria
2.
Laboratory of Advanced Materials, Department of Mathematics, Badji Mokhtar-Annaba University, P. O. Box 12, 23000 Annaba, Algeria

Received: 24 December 2019 Accepted: 11 May 2020 Published: 01 June 2020
MSC : 34A08, 34B15

We discuss the existence of solutions for a boundary value problem involving both left Riemann-Liouville and right Caputo types derivatives. For this, we convert the posed problem to a sum of two integral operators then we apply Krasnoselskii's fixed point theorem to conclude the existence of nontrivial solutions.
- boundary value problem,
- fractional derivative,
- fixed point theorem,
- existence of solution,
- integral equation
Citation: Hamza Moffek, Assia Guezane-Lakoud. Existence of solutions to a class of nonlinear boundary value problems with right and left fractional derivarives[J]. AIMS Mathematics, 2020, 5(5): 4770-4780. doi: 10.3934/math.2020305

Related Papers:

Abstract

We discuss the existence of solutions for a boundary value problem involving both left Riemann-Liouville and right Caputo types derivatives. For this, we convert the posed problem to a sum of two integral operators then we apply Krasnoselskii's fixed point theorem to conclude the existence of nontrivial solutions.

References

[1]	O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl., 272 (2002), 368-379.
[2]	R. P. Agarwal, M. Bohner, W. T. Li, Nonoscillation and oscillation: theory for functional differential equations, Monographs and Textbooks in Pure and Applied Mathematics, 267, Dekker, New York, 2004.
[3]	B. Ahmad, S. K. Ntouyas, A. Alsaedi, Fractional order differential systems involving right Caputo and left Riemann-Liouville fractional derivatives with nonlocal coupled conditions, Boundary Value Problems, 2019, Article number: 109 (2019).
[4]	B. Ahmad, S. K. Ntouyas, A. Alsaedi, Existence theory for nonlocal boundary value problems involving mixed fractional derivatives. Nonlinear Anal. Model. Control., 24 (2019), 937-957.
[5]	B. Ahmad, A. Broom, A. Alsaedi, et al. Nonlinear integro-differential equations involving mixed right and left fractional derivatives and integrals with nonlocal boundary data, Mathematics, 8 (2020), 336.
[6]	T. M. Atanackovic, B. Stankovic, On a differential equation with left and right fractional derivatives. Fract. Calc. Appl. Anal., 10 (2007), 139-150.
[7]	R. Almeida, D. F. M. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1490-1500. doi: 10.1016/j.cnsns.2010.07.016
[8]	D. Baleanu, K. Diethelm, E. Scalas, et al. Fractional calculus models and numerical methods, World Scientific, Singapore, 2012.
[9]	T. Blaszczyk, M. Ciesielski, Numerical solution of Euler-Lagrange equation with Caputo derivatives, Adv. Appl. Math. Mech, 9 (2017), 173-185.
[10]	G. Bonanno, R. Rodríguez-López, S. Tersian, Existence of solutions to boundary value problem for impulsive fractional differential equations. Fract. Calc. Appl. Anal., 17 (2014), 717-744.
[11]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer New York Dordrecht Heidelberg London, 2010.
[12]	A. Guezane-Lakoud, R. Rodríguez-López, On a fractional boundary value problem in a weighted space, SeMA 75 (2018), 435-443.
[13]	A. Guezane-Lakoud, R. Khaldi, D. F. M. Torres, On a fractional oscillator equation with natural boundary conditions, Prog. Frac. Diff. Appl, 3 (2017), 191-197.
[14]	A. G. Lakoud, R. Khaldi, A. Kılıçman, Existence of solutions for a mixed fractional boundary value problem, Advances in Difference Equations, 2017, 164.
[15]	R. Khaldi, A. Guezane-Lakoud, Higher order fractional boundary value problems for mixed type derivatives, J. Nonlinear Funct. Anal. 2017 (2017), Article ID 30.
[16]	R. Khaldi, A. Guezane-Lakoud, On generalized nonlinear Euler-Bernoulli Beam type equations, Acta Univ. Sapientiae, Mathematica, 10 (2018), 90-100.
[17]	N. Nyamoradi, R. Rodríguez-López, On boundary value problems for impulsive fractional differential equations, Appl. Math. Comput., 271 (2015), 874-892.
[18]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
[19]	I. Podlubny, Fractional Differential Equation, Academic Press, Sain Diego, 1999.
[20]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
[21]	M. A. Krasnoselskii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl., 10 (1958), 345-409.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)