Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 < α < 2

References

[1] P. Balasubramaniam, P. Tamilalagan, Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi’s function. Appl. Math. Comput. 256 (2015), 232–246.10.1016/j.amc.2015.01.035Search in Google Scholar

[2] Y.-K. Chang, A. Pereira, R. Ponce, Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators. Fract. Calc. Appl. Anal. 20, No 4 (2017), 963–987; 10.1515/fca-2017-0050; https://www.degruyter.com/view/j/fca.2017.20.issue-4/issue-files/fca.2017.20.issue-4.xml.Search in Google Scholar

[3] H. Deinz, On the behavious of measure of noncompactness with respect to differentiation and integration of vector-valued functions. Nonliear Anal.: TMA7 (1983), 1351–1371.10.1016/0362-546X(83)90006-8Search in Google Scholar

[4] Z. Fan, Existence and regularity of solutions for evolution equations with Riemann-Liouville fractional derivatives. Indagat. Math. 25, No 3 (2014), 516–524.10.1016/j.indag.2014.01.002Search in Google Scholar

[5] T. Guendouzi, L. Bousmaha, Approximate controllability of fractional neutral stochastic functional integro-differential inclusions with infinite delay. Qual. Theory Dyn. Syst. 13, No 1 (2014), 89–119.10.1007/s12346-014-0107-ySearch in Google Scholar

[6] Y.C. Guo, X.-B. Shu, Y.J. Li, F. Xu, The existence and Hyers-Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1 < β < 2. Bound. Value Probl. 2019, No 59 (2019), 1–18.10.1186/s13661-019-1172-6Search in Google Scholar

[7] E. Hernández, D. O’Regan, K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. 73, No 10 (2010), 3462–3471.10.1016/j.na.2010.07.035Search in Google Scholar

[8] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar

[9] X. Liu, Z. Liu, M. Bin, The solvability and optimal controls for some fractional impulsive equations of order 1 < α < 2. Abstr. Appl. Anal. 2014 (2014), 9 pp., ID 142067.Search in Google Scholar

[10] Z. Liu, X. Li, Approximate controllability of fractional evolution systems with Riemann-Liouville fractional derivatives. SIAM J. Control Optim. 53, No 4 (2015), 1920–1933.10.1137/120903853Search in Google Scholar

[11] C. Lizama, A. Pereira, R. Ponce, On the compactness of fractional resolvent operator functions. Semigroup Forum93, No 2 (2016), 363–374.10.1007/s00233-016-9788-7Search in Google Scholar

[12] N.I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim. 42, No 5 (2003), 1604–1622.10.1137/S0363012901391688Search in Google Scholar

[13] N.I. Mahmudov, Finite-approximate controllability of fractional evolution equations: variational approach. Fract. Calc. Appl. Anal. 21, No 4 (2018), 919–936; 10.1515/fca-2018-0050; https://www.degruyter.com/view/j/fca.2018.21.issue-4/issue-files/fca.2018.21.issue-4.xml.Search in Google Scholar

[14] N.I. Mahmudov, S. Zorlu, Approximate controllability of fractional integro-differential equations involving nonlocal initial conditions. Bound. Value Probl. 2013, No 118 (2013), 1–16.10.1186/1687-2770-2013-118Search in Google Scholar

[15] Z.D. Mei, J.G. Peng, Y. Zhang, An operator theoretical approach to Riemann-Liouville fractional Cauchy problem.Math. Nachr. 288, No 7 (2015), 784–797.10.1002/mana.201200191Search in Google Scholar

[16] F. Mokkedem, X. Fu, Approximate controllability for a semilinear stochastic evolution system with infinite delay in L^p space. Appl. Math. Opt. 75, No 2 (2017), 253–283.10.1007/s00245-016-9332-xSearch in Google Scholar

[17] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach space. Nonlinear Anal. 4, No 5 (1980), 985–999.10.1016/0362-546X(80)90010-3Search in Google Scholar

[18] T. Mur, H.R. Henrí quez, Controllability of abstract systems of fractional order. Fract. Calc. Appl. Anal. 18, No 6 (2015), 1379–1398; 10.1515/fca-2015-0080; https://www.degruyter.com/view/j/fca.2015.18.issue-6/issue-files/fca.2015.18.issue-6.xml.Search in Google Scholar

[19] R. Ponce, Existence of mild solutions to nonlocal fractional Cauchy problems via compactness. Abstr. Appl. Anal. 2016 (2016), 15 pp., ID 4567092.10.1155/2016/4567092Search in Google Scholar

[20] D. Prato, G. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992).10.1017/CBO9780511666223Search in Google Scholar

[21] H. Qin, X. Zuo, J. Liu, L. Liu, Approximate controllability and optimal controls of fractional dynamical systems of order 1 < q < 2 in Banach spaces. Adv. Differ. Equ. 2015, No 73 (2015), 1–17; 10.1186/s13662–015–0399–5.Search in Google Scholar

[22] R. Sakthivel, R. Ganesh, Y. Ren, S.M. Anthoni, Approximate controllability of nonlinear fractional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 18, No 12 (2013), 3498–3508.10.1016/j.cnsns.2013.05.015Search in Google Scholar

[23] R. Sakthivel, R. Ganesh, S. Suganya, Approximate controllability of fractional neutral stochastic system with infinite delay. Rep. Math. Phys. 70, No 3 (2012), 291–311.10.1016/S0034-4877(12)60047-0Search in Google Scholar

[24] A. Shukla, N. Sukavanam, D.N. Pandey, Approximate controllability of fractional semilinear stochastic systems of order α ∈ (1, 2] . J. Dyn. Control Syst. 23, No 4 (2017), 679–691.10.1007/s10883-016-9350-7Search in Google Scholar

[25] X.-B. Shu, Y.J. Shi, A study on the mild solution of impulsive fractional evolution equations. Appl. Math. Comput. 273 (2016), 465–476.10.1016/j.amc.2015.10.020Search in Google Scholar

[26] X.-B Shu, Q.Q Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1 < α < 2. Comput. Math. Appl. 64, No 6 (2012), 2100–2110.10.1016/j.camwa.2012.04.006Search in Google Scholar

[27] X.-B Shu, F. Xu, Upper and lower solution method for fractional evolution equations with order 1 < α < 2. J. Korean Math. Soc. 51, No 6 (2014), 1123–1139.10.4134/JKMS.2014.51.6.1123Search in Google Scholar

[28] X.-B Shu, F. Xu, Y.J Shi, S-asymptotically ω-positive periodic solutions for a class of neutral fractional differential equations. Appl. Math. Comput. 270 (2015), 768–776.10.1016/j.amc.2015.08.080Search in Google Scholar

[29] Z.L. Wei, Q.D. Li, J.L. Che, Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. and Appl. 367, No 1 (2010), 260–272.10.1016/j.jmaa.2010.01.023Search in Google Scholar