Research ArticleOpen AccessGoogle Scholar indexed
On the Nonlinear Neutral Conformable Fractional Integral-Differential Equation
School of Mathematical Sciences, Anhui University, Hefei, China
School of Mathematical Sciences, Anhui University, Hefei, China
School of Mathematical Sciences, Anhui University, Hefei, China
School of Mathematical Sciences, Anhui University, Hefei, China
- 1 School of Mathematical Sciences, Anhui University, Hefei, China
- 2 School of Mathematical Sciences, Anhui University, Hefei, China
- 3 School of Mathematical Sciences, Anhui University, Hefei, China
- 4 School of Mathematical Sciences, Anhui University, Hefei, China
Applied Mathematics·Volume 11 (2020)·Pages 1041–1051·Published 30 September 2020·DOI10.4236/am.2020.1110069
Copy link · social · email
Abstract
In this paper, we investigate the nonlinear neutral fractional integral-differential equation involving conformable fractional derivative and integral. First of all, we give the form of the solution by lemma. Furthermore, existence results for the solution and sufficient conditions for uniqueness solution are given by the Leray-Schauder nonlinear alternative and Banach contraction mapping principle. Finally, an example is provided to show the application of results.
KeywordsConformable Fractional DerivativeDelayExistence and UniquenessFunctional Differential Equation
- Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations, Vol. 204. North Holland Mathematics. Elsevier, Amsterdam.
- Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Differential Equations. John Wiley, Hoboken.
- Podlubny, I. (1999) Fractional Differential Equations. Academic Press, Cambridge.
- Khalil, R., Al Horani, M., Yousef, A. and Sababheh, M. (2014) A New Definition of Fractional Derivative. Computational and Applied Mathematics, 264, 65-70. https://doi.org/10.1016/j.cam.2014.01.002
- Abdeljawad, T. (2015) On Conformable Fractional Calculus. Computational and Applied Mathematics, 279, 57-66. https://doi.org/10.1016/j.cam.2014.10.016
- Ziaei, E. and Farahi, M.H. (2019) The Approximate Solution of Non-Linear Time-Delay Fractional Optimal Control Problems by Embedding Process. Mathematical and Information, 36, 713-727. https://doi.org/10.1093/imamci/dnx063
- Alharbi, F.M., Baleanu, D. and Ebaid, A. (2019) Physical Properties of the Projectile Motion Using the Conformable Derivative. Chinese Journal of Physics, 58, 18-28. https://doi.org/10.1016/j.cjph.2018.12.010
- Senol, M., Tasbozan, O. and Kurt, A. (2019) Numerical Solutions of Fractional Burgers’ Type Equations with Conformable Derivative. Chinese Journal of Physics, 58, 75-84. https://doi.org/10.1016/j.cjph.2019.01.001
- Li, X., Liu, S. and Jiang, W. (2011) Positive Solutions for Boundary Value Problem of Nonlinear Fractional Functional Differential Equations. Applied Mathematics and Computation, 217, 9278-9285. https://doi.org/10.1016/j.amc.2011.04.006
- Guo, T. and Jiang, W. (2012) Impulsive Fractional Functional Differential Equations. Computers and Mathematics with Applications, 64, 3414-3424. https://doi.org/10.1016/j.camwa.2011.12.054
- Zhang, S., Jiang, W. and Huang, Y. (2013) Existence of Positive Periodic Solutions for a Class of Higher-Dimension Functional Differential Equations with Impulses. Abstract and Applied Analysis, 2013, Article ID: 396509. https://doi.org/10.1155/2013/396509
- Shen, G., Sakthivel, R. and Ren, Y. (2020) Controllability and Stability of Fractional Stochastic Functional Systems Driven by Rosenblatt Process. Collectanea Mathematica, 71, 63-82. https://doi.org/10.1007/s13348-019-00248-3
- Zhou, Y., Jiao, F. and Li, J. (2009) Existence and Uniqueness for Fractional Neutral Differential Equations with Infinite Delay. Nonlinear Analysis: Theory, Methods and Applications, 71, 3249-3256. https://doi.org/10.1016/j.na.2009.01.202