Topological properties of solution sets for Sobolev-type fractional stochastic differential inclusions with Poisson jumps

References

• Abbas S, Benchohra M, N'Guérékata GM. Topics in fractional differential equations. New York: Springer; 2012.
   Google Scholar
• Zhou Y. Fractional evolution equations and inclusions: analysis and control. New York: Elsevier; 2016.
   Google Scholar
• Knoche C. Mild solutions of SPDEs driven by Poisson noise in infinite dimension and their dependence on initial conditions [Thesis]. Bielefeld University; 2005.
   Google Scholar
• Muthukumar P, Thiagu K. Existence of solutions and approximate controllability of fractional nonlocal neutral impulsive stochastic differential equations of order 1<q<2 with infinite delay and Poisson jumps. J Dyn Control Syst. 2017;23:213–235. doi: 10.1007/s10883-015-9309-0
   Web of Science ®Google Scholar
• Ren Y, Zhou Q, Chen L. Existence, uniqueness, and stability of mild solutions for time-dependent stochastic evolution equations with Poisson jumps and infinite delay. J Optim Theory Appl. 2011;149:315–331. doi: 10.1007/s10957-010-9792-0
   Web of Science ®Google Scholar
• Ren Y, Sakthivel R. Existence, uniqueness, and stability of mild solutions for second-order neutral stochastic evolution equations with infinite delay and Poisson jumps. J Math Phys. 2012;53:835–863.
   Web of Science ®Google Scholar
• Taniguchi T, Luo JW. The existence and asymptotic behaviour of mild solutions to stochastic evolution equations with infinite delays driven by Poisson jumps. Stoch Dyn. 2009;9:217–229. doi: 10.1142/S0219493709002646
   Web of Science ®Google Scholar
• Lightbourne J, Rankin S. A partial functional differential equation of Sobolev type. J Math Anal Appl. 1983;93:328–337. doi: 10.1016/0022-247X(83)90178-6
   Web of Science ®Google Scholar
• Revathi P, Sakthivel R, Ren Y. Stochastic functional differential equations of Sobolev-type with infinite delay. Statist Probab Lett. 2016;109:68–77. doi: 10.1016/j.spl.2015.10.019
   Web of Science ®Google Scholar
• Ahmed H. Sobolev-type fractional stochastic integrodifferential equations with nonlocal conditions in Hilbert space. J Theor Probab. 2017;30:771–783. doi: 10.1007/s10959-016-0665-9
   Web of Science ®Google Scholar
• Benchaabane A, Sakthivel R. Sobolev-type fractional stochastic differential equations with non-Lipschitz coefficients. J Comput Appl Math. 2017;312:65–73. doi: 10.1016/j.cam.2015.12.020
   Web of Science ®Google Scholar
• Debbouche A, Nieto JJ. Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls. Appl Math Comput. 2014;245:74–85.
   Web of Science ®Google Scholar
• Debbouche A, Torres DFM. Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions. Fract Calc Appl Anal. 2015;18:95–121. doi: 10.1515/fca-2015-0007
   Web of Science ®Google Scholar
• Fekan M, Wang JR, Zhou Y. Controllability of fractional functional evolution equations of Sobolev type via characteristic solution operators. J Optim Theory Appl. 2013;156:79–95. doi: 10.1007/s10957-012-0174-7
   Web of Science ®Google Scholar
• Andres J, Gabor G, Górniewicz L. Topological structure of solution sets to multi-valued asymptotic problems. Z Anal Anwend. 2000;19:35–60. doi: 10.4171/ZAA/937
   Web of Science ®Google Scholar
• Andres J, Pavlačková M. Topological structure of solution sets to asymptotic boundary value problems. J Differ Equ. 2010;248:127–150. doi: 10.1016/j.jde.2009.08.010
   Web of Science ®Google Scholar
• Chen DH, Wang RN, Zhou Y. Nonlinear evolution inclusions: topological characterizations of solution sets and applications. J Funct Anal. 2013;265:2039–2073. doi: 10.1016/j.jfa.2013.05.033
   Web of Science ®Google Scholar
• Gabor G, Grudzka A. Structure of the solution set to impulsive functional differential inclusions on the half-line. Nonlinear Differ Equ Appl. 2012;19:609–627. doi: 10.1007/s00030-011-0144-z
   Web of Science ®Google Scholar
• Hu SC, Papageorgiou NS. On the topological regularity of the solution set of differential inclusions with constraints. J Differ Equ. 1994;107:280–289. doi: 10.1006/jdeq.1994.1013
   Web of Science ®Google Scholar
• Wang JR, Zhou Y. Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal. 2011;12:3642–3653. doi: 10.1016/j.nonrwa.2011.06.021
   Google Scholar
• Zhou Y, Peng L, Ahmad B. Topological properties of solution sets for stochastic evolution inclusions. Stoch Anal Appl. 2018;36:114–137. doi: 10.1080/07362994.2017.1374191
   Web of Science ®Google Scholar
• Wang RN, Zhu PX, Ma QH. Multi-valued nonlinear perturbations of time fractional evolution equations in Banach spaces. Nonlinear Dyn. 2015;80:1745–1759. doi: 10.1007/s11071-014-1453-7
   Web of Science ®Google Scholar
• Zhou Y, Peng L, Ahmad B, et al. Topological properties of solution sets of fractional stochastic evolution inclusions. Adv Differ Equ. 2017;2017:90. DOI:10.1186/s13662-017-1142-1
   Google Scholar
• Chang YK, Pereira A, Ponce R. Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators. Fract Calc Appl Anal. 2017;20:963–987. doi: 10.1515/fca-2017-0050
   Web of Science ®Google Scholar
• Dunford N, Schwartz JT. Linear operators. New York: Wiley; 1988.
   Google Scholar
• Kamenskii M, Obukhovskii V, Zecca P. Condensing multi-valued maps and semilinear differential inclusions in banach spaces. New York: Walter de Gruyter; 2001.
   Google Scholar
• Kantorovich LV, Akilov GP. Functional analysis. Oxford: Pergamon Press; 1982.
   Google Scholar
• Chang YK, Pei Y. Degenerate type fractional evolution hemivariational inequalities and optimal controls via fractional resolvent operators. Int J Control. 2018. DOI:10.1080/00207179.2018.1479540.
   Web of Science ®Google Scholar
• Lizama C, Pereira A, Ponce R. On the compactness of fractional resolvent operator functions. Semigroup Forum. 2016;93:363–374. doi: 10.1007/s00233-016-9788-7
   Web of Science ®Google Scholar
• O'Regan D. Fixed point theorems for weakly sequentially closed maps. Arch Math. 2000;36:61–70.
   Google Scholar
• Ponce R. Existence of mild solutions to nonlocal fractional Cauchy problems via compactness. Abstr Appl Anal. 2016. Article ID 4567092, 15 pages.
   Google Scholar
• Lizama C. Regularized solutions for abstract Volterra equations. J Math Anal Appl. 2000;243:278–292. doi: 10.1006/jmaa.1999.6668
   Web of Science ®Google Scholar
• Favini A, Yagi A. Degenerate differential equations in Banach spaces. New York: Dekker; 1999.
   Google Scholar
• Arendt W, Favini A. Integrated solutions to implicit differential equations. Rend Sem Mat Univ Pol Torino. 1993;51:315–329.
   Google Scholar
• Kisielewicz M. Stochastic differential inclusions and applications. New York: Springer; 2013.
   Google Scholar
• Bothe D. Multi-valued perturbations of m-accretive differential inclusions. Isr J Math. 1998;108:109–138. doi: 10.1007/BF02783044
   Web of Science ®Google Scholar
• Górniewicz L, Lassonde M. Approximation and fixed points for compositions of Rδ-maps. Topol Appl. 1994;55:239–250. doi: 10.1016/0166-8641(94)90039-6
   Web of Science ®Google Scholar
• Brezis H. Analyse fonctionelle, théorie et applications. Paris: Masson Editeur; 1983.
   Google Scholar
• Cao GL, He K, Zhang X. Successive approximation of infinite dimensinal SDES with jump. Stoch Dynam. 2005;5:609–619.
   Google Scholar
• Cuesta E. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Discrete and Continuous Dynamical Systems 2007. Dynamical systems and differential equations. Proceedings of the 6th AIMS international conference; suppl. p. 277–285.
   Google Scholar