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December  2015, 4(4): 507-524. doi: 10.3934/eect.2015.4.507

Controllability for fractional evolution inclusions without compactness

Yong Zhou 1, , V. Vijayakumar 2, and  R. Murugesu 3,

1. 

School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
2. 

Department of Mathematics, Info Institute of Engineering, Kovilpalayam, Coimbatore - 641 107Tamil Nadu, India
3. 

Department of Mathematics, SRMV College of Arts and Science, Coimbatore - 641 020, Tamil Nadu, India

Received  August 2015 Revised  October 2015 Published  November 2015

In this paper, we study the existence and controllability for fractional evolution inclusions in Banach spaces. We use a new approach to obtain the existence of mild solutions and controllability results, avoiding hypotheses of compactness on the semigroup generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Finally, two examples are given to illustrate our theoretical results.
Keywords: fractional differential inclusions, semigroup theory, Existence, controllability, multivalued map..
Mathematics Subject Classification: Primary: 34A08, 34G20, 34G25, 47H10, 47H20, 93B0.
Citation: Yong Zhou, V. Vijayakumar, R. Murugesu. Controllability for fractional evolution inclusions without compactness. Evolution Equations & Control Theory, 2015, 4 (4) : 507-524. doi: 10.3934/eect.2015.4.507
References:
[1]

P. R. Agarwal, M. Belmekki and M. Benchohra, Existence results for semilinear functional differential inclusions involving Riemann-Liouville fractional derivative, Dynamics of Continuous, Discrete Impulsive Systems, Series A: Mathematical Analysis, 17 (2010), 347-361.  Google Scholar

[2]

I. Benedetti, V. Obukhovskii and V. Taddei, Controllability for systems governed by semilinear evolution inclusions without compactness, Nonlinear Differential Equations and Applications, 21 (2014), 795-812. doi: 10.1007/s00030-014-0267-0.  Google Scholar

[3]

I. Benedetti, L. Malaguti and V. Taddei, Semilinear evolution equations in abstract spaces and applications, Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 44 (2012), 371-388.  Google Scholar

[4]

S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly valued functions, Annals of Mathematics, 39 (1938), 913-944. doi: 10.2307/1968472.  Google Scholar

[5]

H. Brezis, Analyse Fonctionelle, Théorie et Applications, Masson Editeur, Paris, 1983.  Google Scholar

[6]

K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[7]

N. Dunford and J. T. Schwartz, Linear Operators, John Wiley and Sons, Inc., New York, 1988.  Google Scholar

[8]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[9]

I. Ekeland and R. Teman, Convex Anaysis and Variational Problems, North Holland, Amsterdam, 1976. Google Scholar

[10]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, New York, 2000.  Google Scholar

[11]

M. Fečkan, J. R. Wang and Y. Zhou, Controllability of fractional evolution equations of Sobolev type via characteristic solution, Journal of Optimization Theory and Applications, 156 (2013), 79-95. doi: 10.1007/s10957-012-0174-7.  Google Scholar

[12]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications, 7 Walter de Gruyter, Berlin, 2001. doi: 10.1515/9783110870893.  Google Scholar

[13]

L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.  Google Scholar

[14]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.  Google Scholar

[15]

M. Krasnoschok and N. Vasylyeva, On a non classical fractional boundary-value problem for the Laplace operator, Journal of Differential Equations, 257 (2014), 1814-1839. doi: 10.1016/j.jde.2014.05.022.  Google Scholar

[16]

Z. Liu, J. Lv and R. Sakthivel, Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces, IMA Journal of Mathematical Control and Information, 31 (2014), 363-383. doi: 10.1093/imamci/dnt015.  Google Scholar

[17]

J. A. Machado, C. Ravichandran, M. Rivero and J. J. Trujillo, Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory and Applications, 66 (2013), 1-16. doi: 10.1186/1687-1812-2013-66.  Google Scholar

[18]

D. O'Regan, Fixed point theorems for weakly sequentially closed maps, Archivum Mathematicum, 36 (2000), 61-70.  Google Scholar

[19]

B. J. Pettis, On the integration in vector spaces, Transactions of the American Mathematical Society, 44 (1938), 277-304. doi: 10.1090/S0002-9947-1938-1501970-8.  Google Scholar

[20]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar

[21]

R. Ponce, Hölder continuous solutions for fractional differential equations and maximal regularity, Journal of Differential Equations, 255 (2013), 3284-3304. doi: 10.1016/j.jde.2013.07.035.  Google Scholar

[22]

L. Schwartz, Cours d'Analyse I, 2nd ed. Hermann, Paris, 1981.  Google Scholar

[23]

V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, 2010. doi: 10.1007/978-3-642-14003-7.  Google Scholar

[24]

J. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Analysis: Real World Analysis, 12 (2011), 3642-3653. doi: 10.1016/j.nonrwa.2011.06.021.  Google Scholar

[25]

R. N. Wang, D. H. Chen and Ti-Jun Xiao, Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252 (2012), 202-235. doi: 10.1016/j.jde.2011.08.048.  Google Scholar

[26]

R. N. Wang, Q. M. Xiang and P. X. Zhu, Existence and approximate controllability for systems governed by fractional delay evolution inclusions, Optimization, 63 (2014), 1191-1204. doi: 10.1080/02331934.2014.917303.  Google Scholar

[27]

V. Vijayakumar, C. Ravichandran and R. Murugesu, Existence of mild solutions for nonlocal Cauchy problem for fractional neutral evolution equations with infinite delay, Surveys in Mathematics and its Applications 9 (2014), 117-129.  Google Scholar

[28]

V. Vijayakumar, C. Ravichandran and R. Murugesu, Nonlocal controllability of mixed Volterra-Fredholm type fractional semilinear integro-differential inclusions in Banach spaces, Dynamics of Continuous, Discrete Impulsive Systems, Series B: Applications & Algorithms, 20 (2013), 485-502.  Google Scholar

[29]

L. Zhang and Y. Zhou, Fractional Cauchy problems with almost sectorial operators, Applied Mathematics and Computation, 257 (2015), 145-157. doi: 10.1016/j.amc.2014.07.024.  Google Scholar

[30]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.  Google Scholar

[31]

Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier & Academic Press, 2015. Google Scholar

[32]

Y. Zhou, L. Zhang and X. H. Shen, Existence of mild solutions for fractional evolution equations, Journal of Integral Equations and Applications, 25 (2013), 557-586. doi: 10.1216/JIE-2013-25-4-557.  Google Scholar

[33]

Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Analysis, 11 (2010), 4465-4475. doi: 10.1016/j.nonrwa.2010.05.029.  Google Scholar

show all references

References:
[1]

P. R. Agarwal, M. Belmekki and M. Benchohra, Existence results for semilinear functional differential inclusions involving Riemann-Liouville fractional derivative, Dynamics of Continuous, Discrete Impulsive Systems, Series A: Mathematical Analysis, 17 (2010), 347-361.  Google Scholar

[2]

I. Benedetti, V. Obukhovskii and V. Taddei, Controllability for systems governed by semilinear evolution inclusions without compactness, Nonlinear Differential Equations and Applications, 21 (2014), 795-812. doi: 10.1007/s00030-014-0267-0.  Google Scholar

[3]

I. Benedetti, L. Malaguti and V. Taddei, Semilinear evolution equations in abstract spaces and applications, Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 44 (2012), 371-388.  Google Scholar

[4]

S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly valued functions, Annals of Mathematics, 39 (1938), 913-944. doi: 10.2307/1968472.  Google Scholar

[5]

H. Brezis, Analyse Fonctionelle, Théorie et Applications, Masson Editeur, Paris, 1983.  Google Scholar

[6]

K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[7]

N. Dunford and J. T. Schwartz, Linear Operators, John Wiley and Sons, Inc., New York, 1988.  Google Scholar

[8]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255. doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[9]

I. Ekeland and R. Teman, Convex Anaysis and Variational Problems, North Holland, Amsterdam, 1976. Google Scholar

[10]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, New York, 2000.  Google Scholar

[11]

M. Fečkan, J. R. Wang and Y. Zhou, Controllability of fractional evolution equations of Sobolev type via characteristic solution, Journal of Optimization Theory and Applications, 156 (2013), 79-95. doi: 10.1007/s10957-012-0174-7.  Google Scholar

[12]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications, 7 Walter de Gruyter, Berlin, 2001. doi: 10.1515/9783110870893.  Google Scholar

[13]

L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.  Google Scholar

[14]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.  Google Scholar

[15]

M. Krasnoschok and N. Vasylyeva, On a non classical fractional boundary-value problem for the Laplace operator, Journal of Differential Equations, 257 (2014), 1814-1839. doi: 10.1016/j.jde.2014.05.022.  Google Scholar

[16]

Z. Liu, J. Lv and R. Sakthivel, Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces, IMA Journal of Mathematical Control and Information, 31 (2014), 363-383. doi: 10.1093/imamci/dnt015.  Google Scholar

[17]

J. A. Machado, C. Ravichandran, M. Rivero and J. J. Trujillo, Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions, Fixed Point Theory and Applications, 66 (2013), 1-16. doi: 10.1186/1687-1812-2013-66.  Google Scholar

[18]

D. O'Regan, Fixed point theorems for weakly sequentially closed maps, Archivum Mathematicum, 36 (2000), 61-70.  Google Scholar

[19]

B. J. Pettis, On the integration in vector spaces, Transactions of the American Mathematical Society, 44 (1938), 277-304. doi: 10.1090/S0002-9947-1938-1501970-8.  Google Scholar

[20]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  Google Scholar

[21]

R. Ponce, Hölder continuous solutions for fractional differential equations and maximal regularity, Journal of Differential Equations, 255 (2013), 3284-3304. doi: 10.1016/j.jde.2013.07.035.  Google Scholar

[22]

L. Schwartz, Cours d'Analyse I, 2nd ed. Hermann, Paris, 1981.  Google Scholar

[23]

V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, 2010. doi: 10.1007/978-3-642-14003-7.  Google Scholar

[24]

J. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Analysis: Real World Analysis, 12 (2011), 3642-3653. doi: 10.1016/j.nonrwa.2011.06.021.  Google Scholar

[25]

R. N. Wang, D. H. Chen and Ti-Jun Xiao, Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252 (2012), 202-235. doi: 10.1016/j.jde.2011.08.048.  Google Scholar

[26]

R. N. Wang, Q. M. Xiang and P. X. Zhu, Existence and approximate controllability for systems governed by fractional delay evolution inclusions, Optimization, 63 (2014), 1191-1204. doi: 10.1080/02331934.2014.917303.  Google Scholar

[27]

V. Vijayakumar, C. Ravichandran and R. Murugesu, Existence of mild solutions for nonlocal Cauchy problem for fractional neutral evolution equations with infinite delay, Surveys in Mathematics and its Applications 9 (2014), 117-129.  Google Scholar

[28]

V. Vijayakumar, C. Ravichandran and R. Murugesu, Nonlocal controllability of mixed Volterra-Fredholm type fractional semilinear integro-differential inclusions in Banach spaces, Dynamics of Continuous, Discrete Impulsive Systems, Series B: Applications & Algorithms, 20 (2013), 485-502.  Google Scholar

[29]

L. Zhang and Y. Zhou, Fractional Cauchy problems with almost sectorial operators, Applied Mathematics and Computation, 257 (2015), 145-157. doi: 10.1016/j.amc.2014.07.024.  Google Scholar

[30]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069.  Google Scholar

[31]

Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier & Academic Press, 2015. Google Scholar

[32]

Y. Zhou, L. Zhang and X. H. Shen, Existence of mild solutions for fractional evolution equations, Journal of Integral Equations and Applications, 25 (2013), 557-586. doi: 10.1216/JIE-2013-25-4-557.  Google Scholar

[33]

Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Analysis, 11 (2010), 4465-4475. doi: 10.1016/j.nonrwa.2010.05.029.  Google Scholar

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