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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 & 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.   Google Scholar

[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.   Google Scholar

[3]

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

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

[9]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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

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

[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.   Google Scholar

[22]

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

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

[25]

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

[26]

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

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

[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.   Google Scholar

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

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.   Google Scholar

[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.   Google Scholar

[3]

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

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

[9]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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

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

[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.   Google Scholar

[22]

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

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

[25]

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

[26]

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

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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

[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.   Google Scholar

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