EXISTENCE AND UNIQUENESS RESULTS FOR IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

TitleEXISTENCE AND UNIQUENESS RESULTS FOR IMPLICIT FRACTIONAL DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS
Publication TypeJournal Article
Year of Publication2016
AuthorsBENCHOHRA, MOUFFAK, MAAZOUZ, KADDA
Secondary TitleCommunications in Applied Analysis
Volume20
Issue3
Start Page355
Pagination12
Date Published10/2016
Type of Workscientific: mathematics
ISSN1083-2564
AMS26A33, 34A08, 34B15
Abstract

This paper deals with the existence and uniqueness of solutions for implicit fractional differential equations involving the Caputo fractional derivative in a Banach space by using the Banach contraction principle and Schauder’s fixed point theorem.

URLhttp://www.acadsol.eu/en/articles/20/3/7.pdf
DOI10.12732/caa.v20i3.7
Short TitleImplicit Fractional Differential Equations
Alternate JournalCAA
Refereed DesignationRefereed
Full Text

REFERENCES

[1] S. Abbas, M. Benchohra and G M. N’Gu´er´ekata, Advanced Fractional Differential
and Integral Equations, Nova Science Publishers, New York, 2015.
[2] S. Abbas, M. Benchohra and G M. N’Gu´er´ekata, Topics in Fractional Differential
Equations, Springer-Verlag, New York, 2012.
[3] B. Ahmed and J.J. Nieto, Existence results for nonlinear boundary value problems
of fractional integrodifferential equations with integral boundary coditions,
Boundary value problems (2009), Article ID 708576, 11 pages.
[4] D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculs Models
and Numerical Methods, World Scientific Publishing, New York, 2012.
[5] D. Baleanu, Z.B. G¨uven¸c and J.A.T. Machado, New Trends in Nanotechnology
and Fractional Calculus Applications, Springer, New York, 2010.
[6] M. Benchohra, and F. Berhoun and G.M. N’Gu´er´ekata, Bounded solutions for
fractional order differential equations on the half-line, Bul. Math. Appl. 146
(2008), 9-20.
[7] M. Benchohra, and J.R. Graef and S. Hamani, Existence results for boundary
value problems with nonlinear fractional differential equations, Appl. Anal. 87
(7) (2008), 851-863.
[8] M. Benchohra, and S. Hamani, Boundary value problems for differential equations
with fractional order, Surv. Math. Appl. 3 (2008), 1-12.
[9] M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results
for functional differential equations of fractional order, J. Math. Anal. Appl. 338
(2008), 1340-1350.
[10] M. Benchohra, and J. E. Lazreg, Existence and uniqueness results for nonlinear
implicit fractional differential equations with boundary conditions, Romanian J.
Math. Comput. Sc. 4 (1) (2014), 60-72.
[11] M. Benchohra, and F. Ouaar, Existence results for nonlinear fractional differential
equations with integral boundary conditions, Bul. Math. Anal. Appl.
2(4)(2010), 7-15.
[12] R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations
of fractional order. Fractals and fractional calculus in continuum mechanics
(Udine, 1996), 223–276, CISM Courses and Lectures, 378, Springer, Vienna,
1997.
[13] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore,
2000.
[14] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional
Differenatial Equations. North-Holland Mathematics Studies, 204. Elsevier
Science B.V., Amsterdam, 2006.
[15] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic
Systems, Cambridge Academic Publishers, Cambridge, 2009.
[16] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction
to Mathematical Models. Imperial College Press, London, 2010.
[17] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential
Equations, John Wiley, New York, 1993.
[18] M.D. Otigueira, Fractional Calculus for Scientists and Engineers. Lecture Notes
in Electrical Engineering, 84. Springer, Dordrecht, 2011.
[19] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[20] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives.
Theory and Applications, Gordon and Breach, Yverdon, 1993.
[21] I. Stakgold, Green’s Functions and Boundary Value Problems, Wiley Interscience,
New York, 1979.
[22] V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics
of Particles, Fields and Media, Springer, Heidelberg; Higher Education
Press, Beijing, 2010.
[23] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional
differential equations, Electron. J. Differential Equations, 26 (2006). 1-12.
[24] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific,
Singapore 2014.