Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations

Priscila Santos Ramos; J. Vanterler da C. Sousa; E. Capelas de Oliveira

doi:10.3934/eect.2020100

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February 2022, 11(1): 1-24. doi: 10.3934/eect.2020100

Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations

Priscila Santos Ramos , J. Vanterler da C. Sousa ^, and E. Capelas de Oliveira

Department of Applied Mathematics, Imecc-State University of Campinas, 13083-859, Campinas, SP, Brazil

^* Corresponding author: vanterler@ime.unicamp.br

Received May 2019 Revised August 2020 Published February 2022 Early access October 2020

We discuss the existence and uniqueness of mild solutions for a class of quasi-linear fractional integro-differential equations with impulsive conditions via Hausdorff measures of noncompactness and fixed point theory in Banach space. Mild solution controllability is discussed for two particular cases.

Keywords: Fractional integro-differential equations, mild solution, existence and uniqueness, Hausdorff measure noncompactness.

Mathematics Subject Classification: Primary: 34A08, 34A37, 34K05, 34K30, 47H10.

Citation: Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations and Control Theory, 2022, 11 (1) : 1-24. doi: 10.3934/eect.2020100

References:

[1]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481. doi: 10.1016/j.cnsns.2016.09.006.
[2]	R. Almeida, A. M. C. Brito da Cruz, N. Martins and M. T. T. Monteiro, An epidemiological MSEIR model described by the Caputo fractional derivative, Inter. J. Dyn. Control, 7 (2019), 776-784. doi: 10.1007/s40435-018-0492-1.
[3]	R. Almeida, A. B. Malinowska and Tatiana Odzijewicz, On systems of fractional differential equations with the $\psi$-Caputo derivative and their applications, Math. Meth. Appl. Sci., (2019), 1-16.
[4]	J. M. Ayerbe Toledano, T. Domínguez Benavides and G. López Acedo, Measure of Nonompactness in Metric Fixed Point Theorem, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8920-9.
[5]	D. Bahuguna, R. Sakthivel and A. Chadha, Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay, Stochc. Anal. Appl., 35 (2017), 63-88. doi: 10.1080/07362994.2016.1249285.
[6]	J. Banaś and K. Goebel, Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Marcle Dekker, New York, 1980.
[7]	J. Banaś and W. G. El-Sayed, Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation, J. Math. Anal. Appl., 167 (1992), 133-151. doi: 10.1016/0022-247X(92)90241-5.
[8]	M. Benchohra, S. Litimein and J. J. Nieto, Semilinear fractional differential equations with infinite delay and non-instantaneous impulses, J. Fixed Point Theory Appl., 21 (2019), Paper No. 21, 16 pp. doi: 10.1007/s11784-019-0660-8.
[9]	A. Boudjerida, D. Seba and G. M. N'Guérékata, Controllability of coupled systems for impulsive $\varphi$-Hilfer fractional integro-differential inclusions., Applicable Anal., (2020), 1-18.
[10]	D. Bothe, Multivalued perturbation of $m$-accretive differential inclusions, Israel J. Math., 108 (1998), 109-138. doi: 10.1007/BF02783044.
[11]	L. Byszewski and V. Lakshmikantham, Theorems about the existence and uniqueness of solutions of a nonlocal Cauchy problem in Banach spaces, Applicable Anal., 40 (1990), 11-19. doi: 10.1080/00036819008839989.
[12]	D. N. Chalishajar, K. Malar and R. Ilavarasi, Existence and controllability results of impulsive fractional neutral integro-differential equation with sectorial operator and infinite delay, AIP Conference Proceedings, AIP Publishing LLC, 2159 (2019), 030006. doi: 10.1063/1.5127471.
[13]	J. Dabas and A. Chauhan, Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Math. Computer Model., 57 (2013), 754-763. doi: 10.1016/j.mcm.2012.09.001.
[14]	A. Debbouche, Fractional evolution integro-differential systems with nonlocal conditions, Adv. Dyn. Syst. Appl., 5 (2010), 49-60.
[15]	A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442-1450. doi: 10.1016/j.camwa.2011.03.075.
[16]	K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229-248. doi: 10.1006/jmaa.2000.7194.
[17]	R. Dhayal, M. Malik, S. Abbas, A. Kumar and R. Sakthivel, Approximation theorems for controllability problem governed by fractional differential equation, Evolution Equ. Control Theory, (2019). doi: 10.3934/eect.2020073.
[18]	G. Emmanuele, Measures of weak noncompactness and fixed point theorems, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 25 (1981), 353-358.
[19]	Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 258 (2010), 1709-1727. doi: 10.1016/j.jfa.2009.10.023.
[20]	H. Gou and B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 204-214. doi: 10.1016/j.cnsns.2016.05.021.
[21]	E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649. doi: 10.1090/S0002-9939-2012-11613-2.
[22]	L. Hu, Y. Ren and R. Sakthivel, Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum, 79 (2009), 507-514. doi: 10.1007/s00233-009-9164-y.
[23]	S. Ji and G. Li, Existence results for impulsive differential inclusions with nonlocal conditions, Comput. Math. Appl., 62 (2011), 1908-1915. doi: 10.1016/j.camwa.2011.06.034.
[24]	R. Joice Nirmala, K. Balachandran and J. J. Trujillo, Null controllability of fractional dynamical systems with constrained control, Frac. Calc. Appl. Anal., 20 (2017), 553-565. doi: 10.1515/fca-2017-0029.
[25]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, New York, 2006.
[26]	V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Meth. Appl., 69 (2008), 2677-2682. doi: 10.1016/j.na.2007.08.042.
[27]	Z. Liu, J. Lv and R. Sakthivel, Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces, IMA J. Math. Control Infor., 31 (2014), 363-383. doi: 10.1093/imamci/dnt015.
[28]	M. Mallika Arjunan, V. Kavitha and S. Selvi, Existence results for impulsive differential equations with nonlocal conditions via measures of noncompactness, J. Nonlinear Sci. Appl., 5 (2012), 195-205. doi: 10.22436/jnsa.005.03.04.
[29]	B. P. Moghaddam, J. A. Tenreiro Machado and M. L. Morgado, Numerical approach for a class of distributed order time fractional partial differential equations, Appl. Numer. Math., 136 (2019), 152-162. doi: 10.1016/j.apnum.2018.09.019.
[30]	G. M. Mophou and G. M. N'Guérékata, Existence of the mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum, 79 (2009), 315-322. doi: 10.1007/s00233-008-9117-x.
[31]	S. Nemati, P. M. Lima and D. F. M. Torres, A numerical approach for solving fractional optimal control problems using modified hat functions, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104849, 14pp. doi: 10.1016/j.cnsns.2019.104849.
[32]	S. K. Ntouyas and P. Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210 (1997), 679-687. doi: 10.1006/jmaa.1997.5425.
[33]	M. D. Ortigueira, D. Valério and J. T. Machado, Variable order fractional systems, Commun. Nonlinear Sci. Numer. Simul., 71 (2019), 231-243. doi: 10.1016/j.cnsns.2018.12.003.
[34]	R. Ponce, Bounded mild solutions to fractional integro-differential equations in Banach spaces, Semigroup Forum, 87 (2013), 377-392. doi: 10.1007/s00233-013-9474-y.
[35]	R. Sakthivel, N. I. Mahmudov and J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334-10340. doi: 10.1016/j.amc.2012.03.093.
[36]	F. P. Samuel and K. Balachandran, Existence of solutions for quasi-linear impulsive functional integrodifferential equations in Banach spaces, J. Nonlinear Sci. Appl., 7 (2014), 115-125. doi: 10.22436/jnsa.007.02.05.
[37]	G. Shen, R. Sakthivel, Y. Ren and M. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collectanea Math., 71 (2020), 63-82. doi: 10.1007/s13348-019-00248-3.
[38]	X.-B. Shu and Y. Shi, A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465-476. doi: 10.1016/j.amc.2015.10.020.
[39]	G. Sales Teodoro, J. A Tenreiro Machado and E. Capelas de Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388 (2019), 195-208. doi: 10.1016/j.jcp.2019.03.008.
[40]	M. R. Sidi Ammi and D. F. M. Torres, Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives, Comput. Math. Appl., 78 (2019), 1507-1516. doi: 10.1016/j.camwa.2019.03.043.
[41]	N. H. Tuan, E. Nane, D. O'Regan and N. D. Phuong, Approximation of mild solutions of a semilinear fractional differential equation with random noise, Proc. Amer. Math. Soc., 148 (2020), 3339-3357. doi: 10.1090/proc/15029.
[42]	N. H. Tuan, A. Debbouche and T. B. Ngoc, Existence and regularity of final value problems for time fractional wave equations, Comput. Math. Appl., 78 (2019), 1396-1414. doi: 10.1016/j.camwa.2018.11.036.
[43]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91. doi: 10.1016/j.cnsns.2018.01.005.
[44]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81 (2018), 50-56. doi: 10.1016/j.aml.2018.01.016.
[45]	J. Vanterler da C. Sousa, K. D. Kucche and E. Capelas de Oliveira, Stability of $\psi$-Hilfer impulsive fractional differential equations, Appl. Math. Lett., 88 (2019), 73-80. doi: 10.1016/j.aml.2018.08.013.
[46]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the $\psi$-Hilfer operator, J. Fixed Point Theory Appl., 20 (2018), Paper No. 96, 21 pp. doi: 10.1007/s11784-018-0587-5.
[47]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, Leibniz type rule: $\psi$-Hilfer fractional operator, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 305-311. doi: 10.1016/j.cnsns.2019.05.003.
[48]	J. Vanterler da C. Sousa, M. N. N. dos Santos, L. A. Magna and E. Capelas de Oliveira, Validation of a fractional model for erythrocyte sedimentation rate, Comput. Appl. Math., 37 (2018), 6903-6919. doi: 10.1007/s40314-018-0717-0.
[49]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the $\Psi$-fractional integral and applications, Comput. Appl. Math., 38 (2019), Paper No. 4, 22 pp. doi: 10.1007/s40314-019-0774-z.
[50]	J. Vanterler da C. Sousa, E. Capelas de Oliveira and L. A. Magna, Fractional calculus and the ESR test, AIMS Math., 2 (2017), 692.
[51]	J. Vanterler da C. Sousa, D. S. Oliveira and E. Capelas de Oliveira, A note on the mild solutions of Hilfer impulsive fractional differential equations, arXiv: 1811.09256, (2018).
[52]	J. Vanterler da C. Sousa, D. Ferreira Gomes and E. Capelas de Oliveira, A new class of mild and strong solutions of integro-differential equation of arbitrary order in Banach space, arXiv: 1812.11197, (2018).
[53]	J. Vanterler da C. Sousa, Existence of mild solutions to Hilfer fractional evolution equations in Banach space, arXiv: 1812.02213, (2018).
[54]	J. Wang, W. Wei and Y. Yang, Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control, Opuscula Math., 30 (2010), 217-234. doi: 10.7494/OpMath.2010.30.2.217.
[55]	X.-J. Yang, F. Gao, J. A. Tenreiro Machado and D. Baleanu, Exact travelling wave solutions for local fractional partial differential equations in mathematical physics, Math. Meth. Engineering. Nonlinear Sys. Compl., 24 (2019), 175-191.
[56]	Y. Zhang, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.

show all references

References:

[1]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481. doi: 10.1016/j.cnsns.2016.09.006.
[2]	R. Almeida, A. M. C. Brito da Cruz, N. Martins and M. T. T. Monteiro, An epidemiological MSEIR model described by the Caputo fractional derivative, Inter. J. Dyn. Control, 7 (2019), 776-784. doi: 10.1007/s40435-018-0492-1.
[3]	R. Almeida, A. B. Malinowska and Tatiana Odzijewicz, On systems of fractional differential equations with the $\psi$-Caputo derivative and their applications, Math. Meth. Appl. Sci., (2019), 1-16.
[4]	J. M. Ayerbe Toledano, T. Domínguez Benavides and G. López Acedo, Measure of Nonompactness in Metric Fixed Point Theorem, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8920-9.
[5]	D. Bahuguna, R. Sakthivel and A. Chadha, Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay, Stochc. Anal. Appl., 35 (2017), 63-88. doi: 10.1080/07362994.2016.1249285.
[6]	J. Banaś and K. Goebel, Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Marcle Dekker, New York, 1980.
[7]	J. Banaś and W. G. El-Sayed, Measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation, J. Math. Anal. Appl., 167 (1992), 133-151. doi: 10.1016/0022-247X(92)90241-5.
[8]	M. Benchohra, S. Litimein and J. J. Nieto, Semilinear fractional differential equations with infinite delay and non-instantaneous impulses, J. Fixed Point Theory Appl., 21 (2019), Paper No. 21, 16 pp. doi: 10.1007/s11784-019-0660-8.
[9]	A. Boudjerida, D. Seba and G. M. N'Guérékata, Controllability of coupled systems for impulsive $\varphi$-Hilfer fractional integro-differential inclusions., Applicable Anal., (2020), 1-18.
[10]	D. Bothe, Multivalued perturbation of $m$-accretive differential inclusions, Israel J. Math., 108 (1998), 109-138. doi: 10.1007/BF02783044.
[11]	L. Byszewski and V. Lakshmikantham, Theorems about the existence and uniqueness of solutions of a nonlocal Cauchy problem in Banach spaces, Applicable Anal., 40 (1990), 11-19. doi: 10.1080/00036819008839989.
[12]	D. N. Chalishajar, K. Malar and R. Ilavarasi, Existence and controllability results of impulsive fractional neutral integro-differential equation with sectorial operator and infinite delay, AIP Conference Proceedings, AIP Publishing LLC, 2159 (2019), 030006. doi: 10.1063/1.5127471.
[13]	J. Dabas and A. Chauhan, Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay, Math. Computer Model., 57 (2013), 754-763. doi: 10.1016/j.mcm.2012.09.001.
[14]	A. Debbouche, Fractional evolution integro-differential systems with nonlocal conditions, Adv. Dyn. Syst. Appl., 5 (2010), 49-60.
[15]	A. Debbouche and D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442-1450. doi: 10.1016/j.camwa.2011.03.075.
[16]	K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229-248. doi: 10.1006/jmaa.2000.7194.
[17]	R. Dhayal, M. Malik, S. Abbas, A. Kumar and R. Sakthivel, Approximation theorems for controllability problem governed by fractional differential equation, Evolution Equ. Control Theory, (2019). doi: 10.3934/eect.2020073.
[18]	G. Emmanuele, Measures of weak noncompactness and fixed point theorems, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 25 (1981), 353-358.
[19]	Z. Fan and G. Li, Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal., 258 (2010), 1709-1727. doi: 10.1016/j.jfa.2009.10.023.
[20]	H. Gou and B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 204-214. doi: 10.1016/j.cnsns.2016.05.021.
[21]	E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 141 (2013), 1641-1649. doi: 10.1090/S0002-9939-2012-11613-2.
[22]	L. Hu, Y. Ren and R. Sakthivel, Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum, 79 (2009), 507-514. doi: 10.1007/s00233-009-9164-y.
[23]	S. Ji and G. Li, Existence results for impulsive differential inclusions with nonlocal conditions, Comput. Math. Appl., 62 (2011), 1908-1915. doi: 10.1016/j.camwa.2011.06.034.
[24]	R. Joice Nirmala, K. Balachandran and J. J. Trujillo, Null controllability of fractional dynamical systems with constrained control, Frac. Calc. Appl. Anal., 20 (2017), 553-565. doi: 10.1515/fca-2017-0029.
[25]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, New York, 2006.
[26]	V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Meth. Appl., 69 (2008), 2677-2682. doi: 10.1016/j.na.2007.08.042.
[27]	Z. Liu, J. Lv and R. Sakthivel, Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces, IMA J. Math. Control Infor., 31 (2014), 363-383. doi: 10.1093/imamci/dnt015.
[28]	M. Mallika Arjunan, V. Kavitha and S. Selvi, Existence results for impulsive differential equations with nonlocal conditions via measures of noncompactness, J. Nonlinear Sci. Appl., 5 (2012), 195-205. doi: 10.22436/jnsa.005.03.04.
[29]	B. P. Moghaddam, J. A. Tenreiro Machado and M. L. Morgado, Numerical approach for a class of distributed order time fractional partial differential equations, Appl. Numer. Math., 136 (2019), 152-162. doi: 10.1016/j.apnum.2018.09.019.
[30]	G. M. Mophou and G. M. N'Guérékata, Existence of the mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum, 79 (2009), 315-322. doi: 10.1007/s00233-008-9117-x.
[31]	S. Nemati, P. M. Lima and D. F. M. Torres, A numerical approach for solving fractional optimal control problems using modified hat functions, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104849, 14pp. doi: 10.1016/j.cnsns.2019.104849.
[32]	S. K. Ntouyas and P. Ch. Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J. Math. Anal. Appl., 210 (1997), 679-687. doi: 10.1006/jmaa.1997.5425.
[33]	M. D. Ortigueira, D. Valério and J. T. Machado, Variable order fractional systems, Commun. Nonlinear Sci. Numer. Simul., 71 (2019), 231-243. doi: 10.1016/j.cnsns.2018.12.003.
[34]	R. Ponce, Bounded mild solutions to fractional integro-differential equations in Banach spaces, Semigroup Forum, 87 (2013), 377-392. doi: 10.1007/s00233-013-9474-y.
[35]	R. Sakthivel, N. I. Mahmudov and J. J. Nieto, Controllability for a class of fractional-order neutral evolution control systems, Appl. Math. Comput., 218 (2012), 10334-10340. doi: 10.1016/j.amc.2012.03.093.
[36]	F. P. Samuel and K. Balachandran, Existence of solutions for quasi-linear impulsive functional integrodifferential equations in Banach spaces, J. Nonlinear Sci. Appl., 7 (2014), 115-125. doi: 10.22436/jnsa.007.02.05.
[37]	G. Shen, R. Sakthivel, Y. Ren and M. Li, Controllability and stability of fractional stochastic functional systems driven by Rosenblatt process, Collectanea Math., 71 (2020), 63-82. doi: 10.1007/s13348-019-00248-3.
[38]	X.-B. Shu and Y. Shi, A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465-476. doi: 10.1016/j.amc.2015.10.020.
[39]	G. Sales Teodoro, J. A Tenreiro Machado and E. Capelas de Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388 (2019), 195-208. doi: 10.1016/j.jcp.2019.03.008.
[40]	M. R. Sidi Ammi and D. F. M. Torres, Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives, Comput. Math. Appl., 78 (2019), 1507-1516. doi: 10.1016/j.camwa.2019.03.043.
[41]	N. H. Tuan, E. Nane, D. O'Regan and N. D. Phuong, Approximation of mild solutions of a semilinear fractional differential equation with random noise, Proc. Amer. Math. Soc., 148 (2020), 3339-3357. doi: 10.1090/proc/15029.
[42]	N. H. Tuan, A. Debbouche and T. B. Ngoc, Existence and regularity of final value problems for time fractional wave equations, Comput. Math. Appl., 78 (2019), 1396-1414. doi: 10.1016/j.camwa.2018.11.036.
[43]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91. doi: 10.1016/j.cnsns.2018.01.005.
[44]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81 (2018), 50-56. doi: 10.1016/j.aml.2018.01.016.
[45]	J. Vanterler da C. Sousa, K. D. Kucche and E. Capelas de Oliveira, Stability of $\psi$-Hilfer impulsive fractional differential equations, Appl. Math. Lett., 88 (2019), 73-80. doi: 10.1016/j.aml.2018.08.013.
[46]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the $\psi$-Hilfer operator, J. Fixed Point Theory Appl., 20 (2018), Paper No. 96, 21 pp. doi: 10.1007/s11784-018-0587-5.
[47]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, Leibniz type rule: $\psi$-Hilfer fractional operator, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 305-311. doi: 10.1016/j.cnsns.2019.05.003.
[48]	J. Vanterler da C. Sousa, M. N. N. dos Santos, L. A. Magna and E. Capelas de Oliveira, Validation of a fractional model for erythrocyte sedimentation rate, Comput. Appl. Math., 37 (2018), 6903-6919. doi: 10.1007/s40314-018-0717-0.
[49]	J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the $\Psi$-fractional integral and applications, Comput. Appl. Math., 38 (2019), Paper No. 4, 22 pp. doi: 10.1007/s40314-019-0774-z.
[50]	J. Vanterler da C. Sousa, E. Capelas de Oliveira and L. A. Magna, Fractional calculus and the ESR test, AIMS Math., 2 (2017), 692.
[51]	J. Vanterler da C. Sousa, D. S. Oliveira and E. Capelas de Oliveira, A note on the mild solutions of Hilfer impulsive fractional differential equations, arXiv: 1811.09256, (2018).
[52]	J. Vanterler da C. Sousa, D. Ferreira Gomes and E. Capelas de Oliveira, A new class of mild and strong solutions of integro-differential equation of arbitrary order in Banach space, arXiv: 1812.11197, (2018).
[53]	J. Vanterler da C. Sousa, Existence of mild solutions to Hilfer fractional evolution equations in Banach space, arXiv: 1812.02213, (2018).
[54]	J. Wang, W. Wei and Y. Yang, Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control, Opuscula Math., 30 (2010), 217-234. doi: 10.7494/OpMath.2010.30.2.217.
[55]	X.-J. Yang, F. Gao, J. A. Tenreiro Machado and D. Baleanu, Exact travelling wave solutions for local fractional partial differential equations in mathematical physics, Math. Meth. Engineering. Nonlinear Sys. Compl., 24 (2019), 175-191.
[56]	Y. Zhang, J. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.

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