Existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations

Yıl 2019, Cilt: 2 Sayı: 3, 136 - 142, 01.10.2019

Abdelouaheb Ardjouni , Ahcene Djoudi

Öz

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

Anahtar Kelimeler

Implicit fractional differential equations, Caputo-Hadamard fractional derivatives, Fixed point theorems

Kaynakça

• [1] R. P. Agarwal, Y. Zhou, Y. He, Existence of fractional functional differential equations, Computers and Mathematics with Applications, 59 (2010), 1095-1100.[2] B. Ahmad, S. K. NTouyas, Initial-value problems for fractional differential equations, Electronic Journal of Differential Equations, 2014 (161), 1-8, (2014).[3] B. Ahmad, S. Sivasundaram, Some existence results for fractional integro-differential equations with nonlocal conditions, Communications in Applied Analysis, 12 (2008), 107-112.[4] H. Boulares, A. Ardjouni, Y. Laskri, Positive solutions for nonlinear fractional differential equations, Positivity, 21 (2017), 1201-1212.[5] H. Boulares, A. Ardjouni, Y. Laskri, Stability in delay nonlinear fractional differential equations, Rend. Circ. Mat. Palermo, 65 (2016), 243-253.[6] D. B. Dhaigude, S. P. Bhairat, On Ulam type stability for nonlinear implicit fractional differential equations, arXiv: 1707.07597v1, [math.CA] 24 Jul 2017.[7] K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, Springer-verlag, Berlin, Heidelberg, (2010).[8] J. Dong, Y. Feng and J. Jiang, A note on implicit fractional differential equations, Mathematica Aeterna, 7(3) (2017), 261-267.[9] 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), 09-13[10] D. Henry, Geometric theory of semi linear parabolic equations, Springer -Verlag, Berlin, Heidelberge, New York, (1981).[11] 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).[12] 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.[13] 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.[14] 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.[15] I. Podlubny, Fractional differential equations, Academic Press, San Diego, (1999).[16] 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.[17] 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.