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November  2018, 17(6): 2479-2493. doi: 10.3934/cpaa.2018118

Coupled systems of Hilfer fractional differential inclusions in banach spaces

Saïd Abbas 1, , Mouffak Benchohra 2, and  John R. Graef 3,,

1. 

Laboratory of Mathematics, Geometry, Analysis, Control and Applications, Tahar Moulay University of Saïda, P.O. Box 138, EN-Nasr, 20000 Saïda, Algeria
2. 

Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, 22000, Algeria
3. 

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA

* Corresponding author

Received  October 2017 Revised  January 2018 Published  June 2018

This paper deals with some existence results in Banach spaces for Hilfer and Hilfer-Hadamard fractional differential inclusions. The main tools used in the proofs are Mönch's fixed point theorem and the concept of a measure of noncompactness.

Keywords: Fractional differential inclusion, coupled system, left-sided mixed Riemann-Liouville and Hadamard integrals of fractional order, Hilfer fractional derivative, Hilfer-Hadamard fractional derivative, measure of noncompactness.
Mathematics Subject Classification: Primary: 26A33, 34A60.
Citation: Saïd Abbas, Mouffak Benchohra, John R. Graef. Coupled systems of Hilfer fractional differential inclusions in banach spaces. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2479-2493. doi: 10.3934/cpaa.2018118
References:
[1]

S. Abbas and M. Benchohra, Stability results for fractional differential equations with not instantaneous impulses and state-dependent delay, Math. Slovaca, 67 (2017), 875-894. 

[2]

S. Abbas, M. Benchohra and M. A. Darwish, Upper and lower solutions method for partial discontinuous fractional differential inclusions with not instantaneous impulses, Discus. Math. Diff. Incl., Contr. Optim., 36 (2016), 155-179. 

[3]

S. Abbas, M. Benchohra, J. R. Graef and J. E. Lazreg, Implicit Hadamard fractional differential equations with impulses under weak topologies, to appear.

[4] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012. 
[5] S. Abbas, M. Benchohra and G.M. N'Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015. 
[6] J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin-Heidelberg, New York, 1984. 
[7] J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. 
[8]

J. M. Ayerbee Toledano, T. Dominguez Benavides and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory, Advances and Applications, vol 99, Birkhäuser, Basel, Boston, Berlin, 1997.

[9]

J. Bana and K. Goebel, Measures of Noncompactness in Banach Spaces, Dekker, New York, 1980.

[10]

M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for functional differential equations of fractional order, J. Math. Anal. Appl., 338 (2008), 1340-1350. 

[11]

M. Benchohra, J. Henderson and D. Seba, Measure of noncompactness and fractional differential equations in Banach spaces, Commun. Appl. Anal., 12 (2008), 419-428. 

[12]

M. Benchohra and D. Seba, Integral equations of fractional order with multiple time delays in Banach spaces, Electron. J. Differential Equations, 2012 (2012), 8 pp. 

[13] K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin-New York, 1992. 
[14]

K. M. Furati, M. D. Kassim. Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differential Equations, 235 (2013), 10 pp. 

[15]

K. M. Furati, M. D. Kassim and N. e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616-1626. 

[16]

J. R. Graef, N. Guerraiche and S. Hamani, Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces, Studia Universitatis BabeşBolyai Mathematica, 62 (2017), 427-438. 

[17]

J. R. Graef, N. Guerraiche and S. Hamani, Initial value problems for fractional functional differential inclusions with Hadamard type derivatives in Banach spaces, Surv. Math. Appl., 13 (2018), 27-40. 

[18]

H. P. Heinz, On the behaviour of measure of noncompacteness with respect of differentiation and integration of vector-valued function, Nonlinear. Anal., 7 (1983), 1351-1371. 

[19] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. 
[20]

Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, 1997.

[21]

R. Kamocki and C. Obcz′nnski, On fractional Cauchy-type problems containing Hilfer's derivative, Electron. J. Qual. Theory Differ. Equ., 50 (2016), 1-12. 

[22]

A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204. 

[23] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006. 
[24]

V. Lakshmikantham and J. Vasundhara Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1 (2008), 38-45. 

[25]

V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677-2682. 

[26]

V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett., 21 (2008), 828-834. 

[27]

A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equation, Bull. Accd. Pol. Sci., Ser. Sci. Math. Astronom. Phys., 13 (1965), 781-786. 

[28]

D. O'Regan and R. Precup, Fixed point theorems for set-valued maps and existence principles for integral inclusions, J. Math. Anal. Appl., 245 (2000), 594-612. 

[29]

M. D. Qassim, K. M. Furati and N. -e. Tatar, On a differential equation involving HilferHadamard fractional derivative, Abstr. Appl. Anal., Vol. 2012, Article ID 391062, 17 pages, 2012.

[30]

M. D. Qassim and N. -e. Tatar, Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal., Vol. 2013, Article ID 605029, 12 pages, 2013.

[31]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1987, Engl. Trans. from the Russian.

[32] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg, Higher Education Press, Beijing, 2010. 
[33]

Ž. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms Spec. Funct., 21 (2010), 797-814. 

[34]

J.-R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850-859. 

[35] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. 

show all references

References:
[1]

S. Abbas and M. Benchohra, Stability results for fractional differential equations with not instantaneous impulses and state-dependent delay, Math. Slovaca, 67 (2017), 875-894. 

[2]

S. Abbas, M. Benchohra and M. A. Darwish, Upper and lower solutions method for partial discontinuous fractional differential inclusions with not instantaneous impulses, Discus. Math. Diff. Incl., Contr. Optim., 36 (2016), 155-179. 

[3]

S. Abbas, M. Benchohra, J. R. Graef and J. E. Lazreg, Implicit Hadamard fractional differential equations with impulses under weak topologies, to appear.

[4] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012. 
[5] S. Abbas, M. Benchohra and G.M. N'Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015. 
[6] J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin-Heidelberg, New York, 1984. 
[7] J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. 
[8]

J. M. Ayerbee Toledano, T. Dominguez Benavides and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory, Advances and Applications, vol 99, Birkhäuser, Basel, Boston, Berlin, 1997.

[9]

J. Bana and K. Goebel, Measures of Noncompactness in Banach Spaces, Dekker, New York, 1980.

[10]

M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for functional differential equations of fractional order, J. Math. Anal. Appl., 338 (2008), 1340-1350. 

[11]

M. Benchohra, J. Henderson and D. Seba, Measure of noncompactness and fractional differential equations in Banach spaces, Commun. Appl. Anal., 12 (2008), 419-428. 

[12]

M. Benchohra and D. Seba, Integral equations of fractional order with multiple time delays in Banach spaces, Electron. J. Differential Equations, 2012 (2012), 8 pp. 

[13] K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin-New York, 1992. 
[14]

K. M. Furati, M. D. Kassim. Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differential Equations, 235 (2013), 10 pp. 

[15]

K. M. Furati, M. D. Kassim and N. e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616-1626. 

[16]

J. R. Graef, N. Guerraiche and S. Hamani, Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces, Studia Universitatis BabeşBolyai Mathematica, 62 (2017), 427-438. 

[17]

J. R. Graef, N. Guerraiche and S. Hamani, Initial value problems for fractional functional differential inclusions with Hadamard type derivatives in Banach spaces, Surv. Math. Appl., 13 (2018), 27-40. 

[18]

H. P. Heinz, On the behaviour of measure of noncompacteness with respect of differentiation and integration of vector-valued function, Nonlinear. Anal., 7 (1983), 1351-1371. 

[19] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. 
[20]

Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, 1997.

[21]

R. Kamocki and C. Obcz′nnski, On fractional Cauchy-type problems containing Hilfer's derivative, Electron. J. Qual. Theory Differ. Equ., 50 (2016), 1-12. 

[22]

A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204. 

[23] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006. 
[24]

V. Lakshmikantham and J. Vasundhara Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math., 1 (2008), 38-45. 

[25]

V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677-2682. 

[26]

V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett., 21 (2008), 828-834. 

[27]

A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equation, Bull. Accd. Pol. Sci., Ser. Sci. Math. Astronom. Phys., 13 (1965), 781-786. 

[28]

D. O'Regan and R. Precup, Fixed point theorems for set-valued maps and existence principles for integral inclusions, J. Math. Anal. Appl., 245 (2000), 594-612. 

[29]

M. D. Qassim, K. M. Furati and N. -e. Tatar, On a differential equation involving HilferHadamard fractional derivative, Abstr. Appl. Anal., Vol. 2012, Article ID 391062, 17 pages, 2012.

[30]

M. D. Qassim and N. -e. Tatar, Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal., Vol. 2013, Article ID 605029, 12 pages, 2013.

[31]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1987, Engl. Trans. from the Russian.

[32] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg, Higher Education Press, Beijing, 2010. 
[33]

Ž. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transforms Spec. Funct., 21 (2010), 797-814. 

[34]

J.-R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850-859. 

[35] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. 
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