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Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations
Department of Applied Mathematics, Imecc-State University of Campinas, 13083-859, Campinas, SP, Brazil |
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.
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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