# American Institute of Mathematical Sciences

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

## Controllability for fractional evolution inclusions without compactness

 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.
Citation: Yong Zhou, V. Vijayakumar, R. Murugesu. Controllability for fractional evolution inclusions without compactness. Evolution Equations and 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. [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. [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. [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. [5] H. Brezis, Analyse Fonctionelle, Théorie et Applications, Masson Editeur, Paris, 1983. [6] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, 2010. doi: 10.1007/978-3-642-14574-2. [7] N. Dunford and J. T. Schwartz, Linear Operators, John Wiley and Sons, Inc., New York, 1988. [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. [9] I. Ekeland and R. Teman, Convex Anaysis and Variational Problems, North Holland, Amsterdam, 1976. [10] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, New York, 2000. [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. [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. [13] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. [14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [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. [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. [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. [18] D. O'Regan, Fixed point theorems for weakly sequentially closed maps, Archivum Mathematicum, 36 (2000), 61-70. [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. [20] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [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. [22] L. Schwartz, Cours d'Analyse I, 2nd ed. Hermann, Paris, 1981. [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. [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. [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. [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. [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. [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. [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. [30] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069. [31] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier & Academic Press, 2015. [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. [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.

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##### 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. [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. [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. [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. [5] H. Brezis, Analyse Fonctionelle, Théorie et Applications, Masson Editeur, Paris, 1983. [6] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, 2010. doi: 10.1007/978-3-642-14574-2. [7] N. Dunford and J. T. Schwartz, Linear Operators, John Wiley and Sons, Inc., New York, 1988. [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. [9] I. Ekeland and R. Teman, Convex Anaysis and Variational Problems, North Holland, Amsterdam, 1976. [10] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, New York, 2000. [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. [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. [13] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. [14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [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. [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. [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. [18] D. O'Regan, Fixed point theorems for weakly sequentially closed maps, Archivum Mathematicum, 36 (2000), 61-70. [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. [20] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [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. [22] L. Schwartz, Cours d'Analyse I, 2nd ed. Hermann, Paris, 1981. [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. [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. [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. [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. [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. [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. [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. [30] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. doi: 10.1142/9069. [31] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier & Academic Press, 2015. [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. [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.
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