Existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions

Year 2019, , 46 - 52, 31.03.2019

Abdelouaheb Ardjouni , Ahcene Djoudi

https://doi.org/10.31197/atnaa.501118

Cited By: 10

Abstract

In this paper, we use the contraction mapping principle to obtain the existence, interval of existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions. We also use the generalization of Gronwall's inequality to show the estimate of the solutions.

Keywords

Implicit fractional differential equations, Nonlocal conditions, Caputo-Hadamard fractional derivatives, Fixed point theorems, Existence, Uniqueness

References

• 1) R. P. Agarwal, Y. Zhou, Y. He, Existence of fractional functional differential equations, Computers and Mathematics with Applications, 59 (2010), 1095--1100. 2) A. Bashir, S. Sivasundaram, Some existence results for fractional integro-differential equations with nonlocal conditions, Communications in Applied Analysis, 12 (2008), 107--112.3) H. Boulares, A. Ardjouni, Y. Laskri, Positive solutions for nonlinear fractional differential equations, Positivity, 21 (2017), 1201--1212. 4) H. Boulares, A. Ardjouni, Y. Laskri, Stability in delay nonlinear fractional differential equations, Rend. Circ. Mat. Palermo, 65 (2016), 243--253. 5) D. B. Dhaigude, S. P. Bhairat, On Ulam type stability for nonlinear implicit fractional differential equations, arXiv: 1707.07597v1, [math.CA] 24 Jul 2017. 6) K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, Springer-verlag, Berlin, Heidelberg, (2010). 7) J. Dong, Y. Feng and J. Jiang, A note on implicit fractional differential equations, Mathematica Aeterna, 7(3) (2017), 261--267. 8) M. Haoues, A. Ardjouni and A. Djoudi, Existence, interval of existence and uniqueness of solutions for nonlinear implicit Caputo fractional differential equations, TJMM, 10(1) (2018), 09--139) D. Henry, Geometric theory of semi linear parabolic equations, Springer -Verlag, Berlin, Heidelberge, New York, (1981). 10) A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Editor: Jan Van Mill, Elsevier, Amsterdam, The Netherlands, (2006). 11) K. D. Kucche, J. J. Nieto and V. Venktesh, Theory of nonlinear implicit fractional differential equations, Differ. Equ. Dyn. Syst., DOI 10.1007/s12591-016-0297-7. 12) K. D. Kucche, S. T. Sutar, On existence and stability results for nonlinear fractional delay differential equations, Bol. Soc. Paran. Mat. (3s.) v., 36 (4) (2018), 55--75. 13) K. D. Kucche, S. S. Sutar, Stability via successive approximation for nonlinear implicit fractional differential equations, Moroccan J. Pure Appl. Anal., 3(1) (2017), 36--55. 14) I. Podlubny, Fractional differential equations, Academic Press, San Diego, (1999). 15) S. T. Sutar, K. D. Kucche, Global existence and uniqueness for implicit differential equations of arbitrary order, Fractional Differential Calculus, 5(2) (2015), 199-208. 16) J. Wang, L. Lv, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun Nonlinear Sci Numer Simulat, 17 (2012), 2530-2538.