### Monotone Iterative Technique for Nonlinear Impulsive Conformable Fractional Differential Equations With Delay

#### Abstract

In this paper, we investigate the existence of solutions for boundary value problems of nonlinear impulsive conformable fractional differential equations with delay. By establishing the associate Green’s function and a comparison result for the linear impulsive problem, we obtain that the lower and upper solutions converge to the extremal solutions via the monotone iterative technique. An example is also presented in the last section.

#### Keywords

#### Full Text:

PDF#### References

T. Abdeljawad, M. Al Horani and R. Khalil, Conformable fractional semigroups of operators, Journal of Semigroup Theory and Applications 2015 (2015), Article ID 7, 9 pages, URL: http://scik.org/index.php/jsta/article/view/2410.

T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015), 57 – 66, DOI: 10.1016/j.cam.2014.10.016.

M. Abu Hammad and R. Khalil, Abel’s formula and Wronskian for conformable fractional differential equations, International Journal of Differential Equations and Applications 13 (2014), 177 – 183, DOI: 10.12732/ijdea.v13i3.1753.

I. Abu Hammad and R. Khalil, Fractional Fourier series with applications, American Journal of Computational and Applied Mathematics 4 (2014), 187 – 191, DOI: 10.5923/j.ajcam.20140406.01.

B. Ahmad and R. P. Agarwal, Some new versions of fractional boundary value problems with slitstrips conditions, Boundary Value Problems 2014 (2014), Article number: 175, DOI: 10.1186/s13661-014-0175-6.

B. Ahmad, S. K. Ntouyas and A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Advances in Difference Equations 2011 (2011) Article ID 107384, 11 pages, DOI: 10.1155/2011/107384.

B. Ahmad and S. K. Ntouyas, Existence results for Caputo type sequential fractional differential inclusions with nonlocal integral boundary conditions, Journal of Applied Mathematics and Computing 50 (2016), 157 – 174, DOI: 10.1007/s12190-014-0864-4.

B. Ahmad, S. K. Ntouyas and J. Tariboon, Fractional differential equations with nonlocal integral and integer-fractional-order Neumann type boundary conditions, Mediterranean Journal of Mathematics 13 (2016), 2365 – 2381, DOI: 10.1007/s00009-015-0629-9.

B. Ahmad and S. K. Ntouyas, Some fractional-order one-dimensional semi-linear problems under nonlocal integral boundary conditions, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 110 (2016), 159 – 172, DOI: 10.1007/s13398-015-0228-4.

A. Alsaedi, S. K. Ntouyas and B. Ahmad, New existence results for fractional integro-differential equations with nonlocal integral boundary conditions, Abstract and Applied Analysis 2015 (2015), Article ID 205452, 10 pages, URL: https://projecteuclid.org/download/pdfview_1/euclid.aaa/1429104834.

A. Alsaedi, S. K. Ntouyas, R. P. Agarwal and B. Ahmad, On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Advances in Difference Equations 2015 (2015), Article number: 33, DOI: 10.1186/s13662-015-0379-9.

D. Anderson and D. Ulness, Newly defined conformable derivatives, Advances in Dynamical Systems and Applications 10 (2015), 109 – 137, URL: https://www.researchgate.net/profile/Douglas_Anderson6/publication/287216142_Newly_Defined_Conformable_Derivatives/links/569e53d908ae3bbb87bce643/Newly-Defined-Conformable-Derivatives.pdf.

S. Asawasamrit, S. N. Ntouyas, P. Thiramanus and J. Tariboon, Periodic boundary value problems for impulsive conformable fractional integro-differential equations, Boundary Value Problems 2016 (2016) Article number: 122, DOI: 10.1186/s13661-016-0629-0.

Z. B. Bai and W. Sun, Existence and multiplicity of positive solutions for singular fractional boundary value problems, Computers & Mathematics with Applications 63 (2012), 1369 – 1381, DOI: 10.1016/j.camwa.2011.12.078.

H. Batarfi, J. Losada, J. J. Nieto and W. Shammakh, Three-point boundary value problems for conformable fractional differential equations, Journal of Function Spaces 2015 (2015), Article ID 706383, 6 pages, DOI: 10.1155/2015/706383.

J. Cao and H. Chen, Impulsive fractional differential equations with nonlinear boundary conditions, Mathematical and Computer Modelling 55 (2012), 303 – 311, DOI: 10.1016/j.mcm.2011.07.037.

J. R. Graef, L. Kong and M. Wang, Existence and uniqueness of solutions for a fractional boundary value problem on a graph, Fractional Calculus and Applied Analysis 17 (2014), 499 – 510, DOI: 10.2478/s13540-014-0182-4.

R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics 264 (2014), 65 – 70, DOI: 10.1016/j.cam.2014.01.002.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier Science B.V., Amsterdam (2006), URL: https://www.elsevier.com/books/theory-and-applications-of-fractionaldifferential-equations/kilbas/978-0-444-51832-3.

G. S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston (1985).

V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Applied Mathematics Letters 21 (2008), 828 – 834, DOI: 10.1016/j.aml.2007.09.006.

J. Mu, Y. Li, Monotone iterative technique for impulsive fractional evolution equations, Journal of Inequalities and Applications 2011 (2011), Article number: 125, DOI: 10.1186/1029-242X-2011-125.

S. K. Ntouyas, J. Tariboon and C. Thaiprayoon, Nonlocal boundary value problems for Riemann-Liouville fractional differential inclusions with Hadamard fractional integral boundary conditions, Taiwanese Journal of Mathematics 20 (2016), 91 – 107, URL: https://projecteuclid.org/download/pdf_1/euclid.twjm/1498874423.

S. K. Ntouyas, S. Etemad and J. Tariboon, Existence of solutions for fractional differential inclusions with integral boundary conditions, Boundary Value Problems 2015 (2015), Article number: 92, DOI: 10.1186/s13661-015-0356-y.

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 1st edition, Vol. 198, Academic Press, San Diego (1998), URL: https://www.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9.

J. Sabatier, O. P. Agrawal and J. A. T. Machado (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht (2007), DOI: 10.1007/978-1-4020-6042-7.

Y. Su and Z. Feng, Existence theory for an arbitrary order fractional differential equation with deviating argument, Acta Applicandae Mathematicae 118 (2012), 81 – 105, DOI: 10.1007/s10440-012-9679-1.

J. Tariboon, S. K. Ntouyas and P. Thiramanus, Riemann-Liouville fractional differential equations with Hadamard fractional integral conditions, International Journal of Applied Mathematics and Statistics 54 (2016), 119 – 134, URL: http://www.ceser.in/ceserp/index.php/ijamas/article/view/3949.

J. Tariboon, S. K. Ntouyas and W. Sudsutad, Fractional integral problems for fractional differential equations via Caputo derivative, Advances in Difference Equations 2014 (2014), Article number: 181, DOI: 10.1186/1687-1847-2014-181.

L. Zhang and Y. Liang, Monotone iterative technique for impulsive fractional evolution equations with noncompact semigroup, Advances in Difference Equations 2015 (2015), Article number: 324, DOI: 10.1186/s13662-015-0665-6.

DOI: http://dx.doi.org/10.26713%2Fcma.v12i1.587

### Refbacks

- There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905