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Stability of nonlinear differential systems with delay
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 |
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, 17 (2010), 347.
|
[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.
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.
|
[4] |
S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly valued functions,, Annals of Mathematics, 39 (1938), 913.
doi: 10.2307/1968472. |
[5] |
H. Brezis, Analyse Fonctionelle, Théorie et Applications,, Masson Editeur, (1983).
|
[6] |
K. Diethelm, The Analysis of Fractional Differential Equations,, Lecture Notes in Mathematics, (2010).
doi: 10.1007/978-3-642-14574-2. |
[7] |
N. Dunford and J. T. Schwartz, Linear Operators,, John Wiley and Sons, (1988).
|
[8] |
S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations,, Journal of Differential Equations, 199 (2004), 211.
doi: 10.1016/j.jde.2003.12.002. |
[9] |
I. Ekeland and R. Teman, Convex Anaysis and Variational Problems,, North Holland, (1976). Google Scholar |
[10] |
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (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.
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, (2001).
doi: 10.1515/9783110870893. |
[13] |
L. V. Kantorovich and G. P. Akilov, Functional Analysis,, Pergamon Press, (1982).
|
[14] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Elsevier, (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.
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.
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.
doi: 10.1186/1687-1812-2013-66. |
[18] |
D. O'Regan, Fixed point theorems for weakly sequentially closed maps,, Archivum Mathematicum, 36 (2000), 61.
|
[19] |
B. J. Pettis, On the integration in vector spaces,, Transactions of the American Mathematical Society, 44 (1938), 277.
doi: 10.1090/S0002-9947-1938-1501970-8. |
[20] |
I. Podlubny, Fractional Differential Equations,, Academic Press, (1999).
|
[21] |
R. Ponce, Hölder continuous solutions for fractional differential equations and maximal regularity,, Journal of Differential Equations, 255 (2013), 3284.
doi: 10.1016/j.jde.2013.07.035. |
[22] |
L. Schwartz, Cours d'Analyse I,, 2nd ed. Hermann, (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.
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.
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.
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), 9 (2014), 117.
|
[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, 20 (2013), 485.
|
[29] |
L. Zhang and Y. Zhou, Fractional Cauchy problems with almost sectorial operators,, Applied Mathematics and Computation, 257 (2015), 145.
doi: 10.1016/j.amc.2014.07.024. |
[30] |
Y. Zhou, Basic Theory of Fractional Differential Equations,, World Scientific, (2014).
doi: 10.1142/9069. |
[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.
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.
doi: 10.1016/j.nonrwa.2010.05.029. |
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, 17 (2010), 347.
|
[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.
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.
|
[4] |
S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly valued functions,, Annals of Mathematics, 39 (1938), 913.
doi: 10.2307/1968472. |
[5] |
H. Brezis, Analyse Fonctionelle, Théorie et Applications,, Masson Editeur, (1983).
|
[6] |
K. Diethelm, The Analysis of Fractional Differential Equations,, Lecture Notes in Mathematics, (2010).
doi: 10.1007/978-3-642-14574-2. |
[7] |
N. Dunford and J. T. Schwartz, Linear Operators,, John Wiley and Sons, (1988).
|
[8] |
S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations,, Journal of Differential Equations, 199 (2004), 211.
doi: 10.1016/j.jde.2003.12.002. |
[9] |
I. Ekeland and R. Teman, Convex Anaysis and Variational Problems,, North Holland, (1976). Google Scholar |
[10] |
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Graduate Texts in Mathematics, (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.
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, (2001).
doi: 10.1515/9783110870893. |
[13] |
L. V. Kantorovich and G. P. Akilov, Functional Analysis,, Pergamon Press, (1982).
|
[14] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Elsevier, (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.
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.
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.
doi: 10.1186/1687-1812-2013-66. |
[18] |
D. O'Regan, Fixed point theorems for weakly sequentially closed maps,, Archivum Mathematicum, 36 (2000), 61.
|
[19] |
B. J. Pettis, On the integration in vector spaces,, Transactions of the American Mathematical Society, 44 (1938), 277.
doi: 10.1090/S0002-9947-1938-1501970-8. |
[20] |
I. Podlubny, Fractional Differential Equations,, Academic Press, (1999).
|
[21] |
R. Ponce, Hölder continuous solutions for fractional differential equations and maximal regularity,, Journal of Differential Equations, 255 (2013), 3284.
doi: 10.1016/j.jde.2013.07.035. |
[22] |
L. Schwartz, Cours d'Analyse I,, 2nd ed. Hermann, (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.
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.
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.
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), 9 (2014), 117.
|
[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, 20 (2013), 485.
|
[29] |
L. Zhang and Y. Zhou, Fractional Cauchy problems with almost sectorial operators,, Applied Mathematics and Computation, 257 (2015), 145.
doi: 10.1016/j.amc.2014.07.024. |
[30] |
Y. Zhou, Basic Theory of Fractional Differential Equations,, World Scientific, (2014).
doi: 10.1142/9069. |
[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.
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.
doi: 10.1016/j.nonrwa.2010.05.029. |
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