Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Non-Instantaneous Impulses in Banach Spaces

Yıl 2020, Cilt: 4 Sayı: 4, 332 - 348, 30.12.2020

Abdelkrim Salim Mouffak Benchohra Jamal Eddine Lazreg Johnny Henderson

https://doi.org/10.31197/atnaa.825294

Cited By: 19

Öz

In the present article, we prove some results concerning the existence of solutions for a class of
initial value problem for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivative in Banach spaces. The results are based on fixed point theorems of Darbo and Monch associated with the technique of measure of noncompactness. An example is included to show the applicability of our results.
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Anahtar Kelimeler

Implicit fractional differential equations, generalized Hilfer fractional derivative, initial value problem, measure of noncompactness, fixed point, non-instantaneous impulses, Ulam stability

Kaynakça

• [1] S. Abbas and M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional di?erential equations with not instantaneous impulses, Appl. Math. Comput. 257 (2015), 190-198.
• [2] S. Abbas, M. Benchohra and M.A. Darwish, New stability results for partial fractional di?erential inclusions with not instantaneous impulses, Frac. Calc. Appl. Anal. 18 (1) (2015), 172-191.
• [3] S. Abbas, M. Benchohra, J.R. Graef and J. Henderson, Implicit Di?erential and Integral Equations: Existence and stability, Walter de Gruyter, London, 2018.
• 4] S. Abbas, M. Benchohra, J. E. Lazreg, A Alsaedi and Y. Zhou, Existence and Ulam stability for fractional di?erential equations of Hilfer-Hadamard type, Adv. Di?erence Equ. (2017) 14p.
• [5] S. Abbas, M. Benchohra, J. E. Lazreg and G. N' Guérékata, Hilfer and Hadamard functional random fractional di?erential inclusions, Cubo 19 (2017), 17-38
• [6] S. Abbas, M. Benchohra, J. E. Lazreg and Y. Zhou, A survey on Hadamard and Hilfer fractional di?erential equations: analysis and stability, Chaos Solitons Fractals 102 (2017), 47-71.
• [7] S. Abbas, M. Benchohra and G M. N'Guérékata, Topics in Fractional Di?erential Equations, Springer-Verlag, New York, 2012.
• [8] S. Abbas, M. Benchohra and G M. N'Guérékata, Advanced Fractional Di?erential and Integral Equations, Nova Science Publishers, New York, 2014.
• [9] R.P. Agarwal, S. Hristova and D. O'Regan, Non-Instantaneous Impulses in Di?erential Equations, Springer, New York, 2017.
• [10] B. Ahmad, A. Alsaedi, S.K. Ntouyas and J. Tariboon, Hadamard-type Fractional Di?erential Equations, Inclusions and Inequalities. Springer, Cham, 2017.
• [11] B. Ahmad and S.K Ntouyas, Fractional di?erential inclusions with fractional separated boundary conditions, Fract. Calc. Appl. Anal. 15 (2012), 362-382.
• [12] R. Almeida, A.B. Malinowska and T. Odzijewicz, Fractional di?erential equations with dependence on the Caputo? Katugampola derivative, J. Comput. Nonlinear Dynam. 11 (6) (2016), 1-11
• [13] D. Baleanu, Z.B. Güvenç and J.A.T. Machado New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.
• [14] J. Banas and K. Goebel, Measures of noncompactness in Banach spaces. Marcel Dekker, New York, 1980.
• [15] M. Benchohra, J. Henderson and S. K. Ntouyas , Impulsive Di?erential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York, 2006.
• [16] M. Benchohra and J. E. Lazreg, Existence and Ulam stability for Nonlinear implicit fractional diferential equations with Hadamard derivative, Stud. Univ. Babes-Bolyai Math. 62 (2017), 27-38.
• [17] M. Benchohra and J. E. Lazreg, On stability for nonlinear implicit fractional difequations, Le Matematiche (Catania) 70 (2015), 49-61.
• [18] P. Egbunonu and M. Guay, Identification of switched linear systems using subspace and integer programming techniques. Nonlinear Anal. Hybrid Syst. 1 (2007), 577-592.
• [19] D.J. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic Publishers, Dordrecht, 1996. [20] K. Goebel, Concise course on Fixed Point Theorems. Yokohama Publishers, Japan, 2002.
• [21] D.H. Hyers. On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), 222-224.
• [22] U. Katugampola, A new approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011), 860-865.
• [23] H. Mönch, BVP for nonlinear ordinary diferential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), 985-999.
• [24] D.S. Oliveira and E. Capelas de Oliveira, Hilfer-Katugampola fractional derivatives, Comput. Appl. Math. 37 (3)(2018), 3672-3690.
• [25] T.M. Rassias. On the stability of the linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
• [26] I.A. Rus, Ulam stability of ordinary diferential equations, Stud. Univ. Babes-Bolyai, Math. LIV(4), (2009), 125-133.
• [27] S.M. Ulam. A collection of mathematical problems. Interscience Publishers, New York, 1968.
• [28] F. Vaadrager and J. Van Schuppen (eds.), Hybrid Systems, Computation and Control. Lecture Notes in Computer Sciences, vol. 1569. Springer, New York, 1999.